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Unit 1 · Real Numbers
Searches the concepts, worked examples, Exercises 1.1 to 1.3, the Review Exercise and their solutions. The Unit Test and generated papers are not searched.
Concepts
Learn the ideas before attempting the exercise.
Real numbers include every number that can be placed on the number line. This section separates them into rational and irrational numbers, then develops the algebraic and order properties used throughout the chapter.
Rational numbers
A real number is rational when it can be written in the form , where and .
Integers are rational because every integer can be written as .
The decimal test
A terminating decimal has finitely many digits after the decimal point. A recurring decimal continues forever but repeats a fixed block.
A real number is rational exactly when its decimal expansion terminates or eventually repeats. A nonterminating, nonrecurring decimal is irrational.
Irrational and real numbers
An irrational number cannot be expressed as a quotient of two integers. Its decimal expansion is nonterminating and nonrecurring.
The real-number set is the union of the rational and irrational numbers.
Perfect-square test
If is a positive integer and is not a perfect square, then is irrational.
Quick check. The nearest perfect squares around are and . Therefore is irrational and lies between and .
Properties of real numbers
Commutative
Associative
Identity
Inverse
Distributive
Cancellation
Order and inequality
Adding the same real number to both sides does not change the direction of an inequality.
Multiplication or division by a positive number keeps the direction. Multiplication or division by a negative number reverses it.
The reciprocal rule requires nonzero numbers on the same side of zero.
Numbers between numbers
Between any two distinct rational numbers there are infinitely many rational numbers. The average gives one immediately.
Repeating the averaging process gives more and more rational numbers.
Historical note
By about , Greek mathematicians knew that the side and diagonal of a square are incommensurable. In modern language, the diagonal of a unit square has length , and is irrational. The exact individual responsible is uncertain.
Fun fact: a recurring decimal can equal an integer
There is no real number between and ; they are two notations for the same real number.
Additional notes
Written to support the questions below. Not part of the printed textbook.
01The number sets: natural, whole, integer, rational, real
Definition
are the natural numbers. Adding gives the whole numbers. Adding the negatives gives the integers. A rational number is any number of the form with and ; these form . An irrational number is a real number that is not rational. Together they form the real numbers.
What it means
Each set sits inside the next: . Every point of the number line is exactly one real number, and every real number is exactly one point.
The rule, with its conditions
and have no member in common: no number is both rational and irrational.
Worked example
To which sets does belong?
Is it natural or whole?
No, both contain only non-negative numbers.
Is it an integer?
Yes, .
Is it rational?
Yes, because , a quotient of integers with non-zero denominator.
Answer
, and , but and .
Common misconception
“Irrational means it has a square root sign.”
is rational, and is irrational with no radical sign at all. Classification depends on the value, not on how it is written.
Quick check
Is rational or irrational?
Show the answer
Rational. , so the number is .
Did you know?
Notation worth getting right
Write for the set of irrational numbers. The older symbol is ambiguous unless the universal set has been stated, so this book avoids it.
02Classifying a number from its decimal expansion
Definition
Every real number has a decimal expansion, and the expansion tells you immediately whether the number is rational.
The rule, with its conditions
A bar marks the recurring block: and
Worked example
Classify .
Read the bar.
The block repeats for ever, so the expansion is non-terminating.
Is it recurring?
Yes, a fixed block repeats.
Apply the rule.
Non-terminating and recurring means rational.
Answer
Rational.
Common misconception
“A long decimal must be irrational.”
has a recurring block of digits. It looks random on a calculator screen but is perfectly rational.
Quick check
Which of and is irrational?
Show the answer
The first. Its pattern grows (one zero, then two, then three) so no fixed block ever repeats. The second has the repeating block and is rational.
Did you know?
When does a fraction terminate?
in lowest terms has a terminating decimal exactly when the only prime factors of are and . So terminates () but does not ().
03Converting a recurring decimal to form
Definition
Any recurring decimal can be written as a quotient of integers by multiplying, subtracting and solving.
The rule, with its conditions
If some digits come before the recurring block, multiply twice, once to reach the start of the block and once to pass over it.
Worked example
Express as a fraction.
Name the number.
One digit () sits before the block, so multiply by to reach the block.
The block has digit, so multiply by once more.
Subtract; the tails are identical.
Collect like terms.
Divide, then reduce by .
Answer
Common misconception
Using the wrong multiplier
The multiplier is decided by the length of the recurring block, not by the total number of decimal places. For the block has digits, so multiply by .
Quick check
Express as a fraction.
Show the answer
,, so and .
Did you know?
A famous consequence
Apply the method to :, so . That is, exactly, not approximately.
04Why is irrational when is not a perfect square
Definition
For a natural number , the square root is rational exactly when is a perfect square. Otherwise it is irrational.
The rule, with its conditions
Worked example
Prove that is irrational.
Assume the opposite.
Square both sides.
Multiply both sides by .
So is even, which forces to be even. Write .
Divide both sides by .
So is even, hence is even too.
But then and share the factor , contradicting .
Answer
The assumption fails, so is irrational.
Common misconception
“Irrational irrational is always irrational.”
False. and are rational. Similarly is rational. Each case must be checked on its own. **Rational irrational**, however, is always irrational.
Quick check
Is rational or irrational?
Show the answer
Irrational. lies between and , so it is not a perfect square. (In simplified form, .)
Did you know?
A note on history
The Pythagorean school in ancient Greece discovered that the diagonal of a unit square is incommensurable with its side, in modern terms, that is irrational. The story that Hippasus was drowned for revealing it is a much later legend and should not be presented as established fact.
05Properties of the real numbers, and the conditions they carry
Definition
For all the operations and obey a fixed list of properties. Two of them carry conditions that are easy to forget.
The rule, with its conditions
Worked example
Name the property: .
Compare the order of the three terms.
,, appear in the same order on both sides.
Compare the bracketing.
Only the grouping has moved.
Answer
Associative property of addition.
Common misconception
Cancelling without checking
From you may not conclude . The cancellation property holds only when . The same condition is why has no multiplicative inverse.
Quick check
Which property is used in ?
Show the answer
The distributive property of multiplication over addition, followed by evaluating .
Did you know?
Closure
is closed under and (excluding division by ). The irrationals are not closed under any of these: is rational.
06Order properties and inequalities
Definition
The real numbers are ordered: given any two of them, one is smaller, or they are equal.
The rule, with its conditions
Adding never changes the direction. Multiplying or dividing by a negative number always reverses it.
Worked example
Solve .
Subtract from both sides; the direction does not change.
Divide both sides by , a negative number, so reverse the sign.
Answer
Common misconception
Forgetting to reverse
Writing above is the single most common inequality error in this unit. Any time a negative factor is divided out, flip the symbol.
Quick check
If , is always true?
Show the answer
No. It holds only when and have the same sign. For and :.
Did you know?
Density
Between any two distinct real numbers there is another real number, in fact infinitely many rationals and infinitely many irrationals. The number line has no gaps and no smallest step.
07Constructing irrational numbers on the number line
Definition
Every real number, rational or irrational, corresponds to exactly one point on the number line. Irrational lengths such as can be constructed exactly with a straight edge and compasses.
The spiral continues: built from
The rule, with its conditions
Repeating the step on each new hypotenuse produces in turn.
Worked example
Construct on the number line.
Choose legs whose squares add to .
Mark at and at , then draw with .
This makes a right triangle with legs and .
Apply Pythagoras.
Transfer the length onto the line.
With centre and radius , draw an arc cutting the number line at . Then .
Answer
lies between and , since .
Common misconception
Measuring instead of constructing
Marking with a ruler gives an approximation, not . The compass-and-straight-edge construction gives the exact point.
Quick check
Which two integers does the constructed point for lie between?
Show the answer
Between and , because .
Did you know?
A number that cannot be constructed
can be drawn exactly, but cannot: it is transcendental, which is why squaring the circle with straight edge and compasses is impossible. Lindemann settled this in .
Examples
The textbook's own examples, solved as printed.
Example 1
Identify the following decimal numbers as rational or irrational: , , , , and .
Solution
has finitely many digits after the decimal point. It is a terminating decimal and therefore a rational number.
is nonterminating and recurring because the digit repeats. Therefore it is rational.
is nonterminating and recurring. Therefore it is rational.
is nonterminating and nonrecurring. Therefore it is irrational.
is nonterminating and nonrecurring. Therefore it is irrational.
Example 2
Represent on a number line.
Solution
can be located by a geometric construction.
Draw a horizontal number line and mark the origin .
Mark two units to the right of , so .
At , draw a perpendicular and mark one unit above , so .
Join to .
Use the Pythagorean theorem:
With as center and as radius, draw an arc to meet the number line at .
The point represents .
Geometric construction of .Constructing on the number line
Example 3
Express and as rational numbers in the form .
Solution
(i) Let .
Therefore, .
(ii) Let .
Therefore, .
Example 4
Insert two rational numbers between and .
Solution
There are infinitely many rational numbers between and . Take the average to obtain one:
Now take the average of and :
Thus and are two rational numbers between and .
Example 5
If , , and , verify the distributive properties over addition.
Solution
The left distributive property is .
Since the two sides are equal, the left distributive property is verified.
The right distributive property is .
Therefore the right distributive property is also verified.
Example 6
Identify the property that justifies each statement:
and
and
Solution
Adding to both sides of gives . This is the additive property of order.
Adding the two inequalities gives , hence . This is addition of inequalities.
All four numbers are positive, so multiplying corresponding sides gives , hence .
Multiplying by the negative number reverses the inequality: , so .
Exercise 1.1
Try each part first, then open its solution.
Question 1
Identify each of the following as a rational or an irrational number.
(i)
Solution
Count the digits after the decimal point.
There are exactly digits after the point, and the expansion stops. So is a terminating decimal.
Write the terminating decimal as a quotient of integers.
Check the definition of a rational number.
Numerator and denominator are integers and the denominator is not , so the number has the form with ,.
Answer
Rational.
Study notesRule used · Common mistake
Rule used: Terminating decimals are always rational
A decimal that stops after digits equals , which is a quotient of two integers.
Common mistake: Do not confuse this with
stops. never stops. Both are rational, but for different reasons: the first is terminating, the second is non-terminating recurring.
(ii)
Solution
Read the recurring bar.
The bar over means , so the expansion is non-terminating and the single digit repeats for ever.
Name the number so we can work with it.
One digit recurs, so multiply by to shift the block one place.
Subtract the first equation from the second; the identical tails cancel.
Collect like terms on each side.
Divide both sides by .
Cancel the common factor .
Answer
Rational.
Study notesRule used · Check
Rule used: Why multiply by here
Multiply by where is the length of the recurring block. Here the block is the single digit , so and the multiplier is .
Check by division
Divide by :, which returns the original decimal.
(iii)
Solution
Read what the dots tell us.
The dots show the expansion never ends, and no group of digits is marked as repeating.
Apply the decimal test.
A decimal that is non-terminating and non-recurring cannot be written as with integers and .
A calculator shows only about digits. Seeing no pattern on screen is not a proof of irrationality; the classification comes from knowing the number is and that is not a perfect square.
(iv)
Solution
Test whether the radicand is a perfect square.
and , so lies strictly between two consecutive perfect squares and is therefore not a perfect square.
Apply the theorem on square roots.
If a natural number is not a perfect square, then is irrational.
Answer
Irrational.
Study notesRule used · Why it applies
Rule used
For , is rational exactly when is a perfect square. Otherwise is irrational.
Why it applies: Where the rule comes from
Suppose in lowest terms. Squaring gives , so and hence . Writing gives , so and too. Then and share the factor , contradicting lowest terms.
(v)
Solution
Recall what is.
is the base of the natural logarithm.
State the known result.
has a non-terminating, non-recurring decimal expansion; it was proved irrational by Euler in .
Answer
Irrational.
Study notesCommon mistake · Alternative method
Common mistake: The early digits look like a pattern
appears to repeat , but the next digits are The pattern breaks. Recurrence must continue for ever, not for a few digits.
Alternative method: Did you know?
is not merely irrational, it is transcendental: it is not a root of any polynomial equation with integer coefficients. Hermite proved this in .
(vi)
Solution
Recall the decimal expansion.
, non-terminating and non-recurring.
State the known result.
Lambert proved in that cannot be written as a quotient of two integers.
Answer
Irrational.
Study notesCommon mistake
Common mistake: is not
is a rational approximation of , correct only to decimal places. itself is irrational.
(vii)
Solution
Classify the two parts separately.
is rational. is not a perfect square, so is irrational.
Argue by contradiction: suppose the sum were rational.
Subtract from both sides to isolate the radical.
Closure of under subtraction.
and are both rational, so is rational. That would make rational, which is false.
Answer
Irrational.
Study notesRule used · Common mistake
Rule used: Rational irrational is always irrational
If and , then . The proof above works for any such and : isolate and use closure of under subtraction.
Common mistake: This does not extend to two irrationals
Irrational irrational may be rational. For example , which is rational. See part .
(viii)
Solution
Both terms are irrational, so no shortcut rule applies.
Neither nor is a perfect square, so and are both irrational. The sum of two irrational numbers is not automatically irrational, so this case needs a proof of its own.
Argue by contradiction.
Square both sides to remove the outer structure.
Expand the left side using .
Use and .
Add the two integers.
Isolate the radical term.
Divide both sides by .
Closure of under multiplication, subtraction and division.
If were rational then would be rational, forcing to be rational. But is not a perfect square, so is irrational, a contradiction.
Answer
Irrational.
Study notesCommon mistake · Rule used · Check
Common mistake: The textbook claim needs correction
It is not true that the sum of two irrational numbers is always irrational. and are both rational. Each such sum must be settled on its own.
Rule used: Why squaring is the right move
Squaring turns into , which contains only one radical. A single isolated radical can then be forced to be rational, which is the contradiction we want.
Check: Numerical sanity check
and , so the sum is about , and . Consistent.
(ix)
Solution
Check the definition directly.
, and , so is already in the form .
Answer
Rational.
(x)
Solution
Recognise the form before multiplying out.
The two brackets are conjugates: they match the pattern with and .
Apply the difference-of-squares identity .
Evaluate each square; .
Subtract.
Classify the result.
is a quotient of integers with non-zero denominator.
Answer
Rational.
Study notesWhy it applies · Common mistake
Why it applies: Why the radical disappeared
Multiplying by the conjugate squares the surd, and is rational. This is exactly the mechanism behind rationalizing a binomial denominator in Exercise 1.2.
Common mistake: Do not judge by appearance
An expression containing is not automatically irrational. Always simplify first, then classify.
Question 2
Represent the following numbers on the number line.
(i)
Solution
Start from a unit length on the line.
Draw the number line and mark at and at , so .
Create a right angle so Pythagoras can be applied.
At , draw with .
Use Pythagoras' theorem in right triangle .
Put in the two known lengths.
Evaluate the squares and add.
Take the positive square root; a length cannot be negative.
Transfer the length back onto the line.Constructing on the number line
With centre and radius , draw an arc cutting the number line at . Then , so represents .
Answer
lies between and , since .
Study notesRule used · Check
Rule used
Pythagoras' theorem: in a right triangle, .
Check the position
, so . The arc must cut the line between and .
(ii)
Solution
Build on the previous construction.
First construct exactly as in part .
Raise a new unit perpendicular on the hypotenuse just built.
At , draw with .
Apply Pythagoras in right triangle .
Substitute and .
Use and .
Take the positive square root.
Transfer the length to the line.Constructing from
With centre and radius , draw an arc cutting the line at . Then represents .
Answer
lies between and , since .
Study notesAlternative method
Alternative method: The spiral of Theodorus
Repeating this step, a unit perpendicular on each new hypotenuse, produces in turn. The figure is called the spiral of Theodorus.
(iii)
Solution
Change the mixed number to an improper fraction to see its size.
Identify the two consecutive integers it lies between.
, so the point lies between and , close to .
Divide the correct unit interval, not the whole line. on the number line
Divide the segment from to into equal parts and take the first division point.
Answer
The first of the three division points between and .
Study notesRule used
Rule used
For a mixed number with , the point lies between and . Divide that unit interval into equal parts and count of them.
(iv)
Solution
Write it as a single improper fraction.
Decide which side of zero and between which integers.
The number is negative, so it lies to the left of . Since , the point lies between and .
Divide the correct unit interval. on the number line
Divide the segment from to into equal parts and take the first division point moving left from .
Answer
The first division point to the left of .
Study notesCommon mistake
Common mistake: The commonest error with negatives
means ,not. It is therefore further from zero than , so it lies between and and not between and .
(v)
Solution
Compare numerator and denominator.
, so and the point lies between and .
Split the unit interval into as many parts as the denominator. on the number line
Divide the segment from to into equal parts and count of them from .
Answer
The fifth of the eight division points between and .
(vi)
Solution
Write as an improper fraction.
Identify the correct unit interval.
, so the point lies between and ,not between and .
Divide that unit interval into equal parts. on the number line
Divide the segment from to into equal parts and take the third division point.
Answer
The third of the four division points between and .
Study notesCheck
Check with the improper fraction
and , confirming the interval.
Question 3
Express the following recurring decimals as the rational number , where and are integers and .
(i)
Solution
Name the number.
The recurring block has digit, so multiply by .
Subtract to destroy the infinite tail.
Collect like terms.
Divide both sides by .
Answer
is already in lowest terms since .
Study notesRule used
Rule used
If a recurring block of length starts immediately after the decimal point, multiply by , subtract, and solve. Here .
(ii)
Solution
Name the number.
The recurring block has digits, so multiply by .
Subtract; the two tails are identical and cancel.
Collect like terms.
Divide both sides by .
Answer
In lowest terms, since is prime and .
Study notesWhy it applies
Why it applies: Why and not
The multiplier must shift the decimal point by exactly one whole block. The block occupies places, so the multiplier is . Using would leave , whose tail no longer matches , and the subtraction would fail.
(iii)
Solution
Name the number.
Block length , so multiply by .
Subtract to remove the tail.
Collect like terms.
Divide both sides by .
, so cancel the common factor .
Answer
Study notesCommon mistake · Check
Common mistake: is not the final answer
should be given in lowest terms unless the question says otherwise. Always test the numerator and denominator for a common factor at the end.
Check by division
, which is the original decimal.
Question 4
Name the property used in each of the following.
(i)
Solution
The order of the terms is unchanged; only the bracketing moves.
,, stay in the same left-to-right order, so this is a regrouping, not a reordering.
Answer
Associative property of addition.
(ii)
Solution
The two terms have swapped places.
There are no brackets to move; only the order of the addends has changed.
Answer
Commutative property of addition.
(iii)
Solution
Write subtraction as adding the opposite.
A number added to its opposite gives the additive identity.
Answer
Additive inverse property.
(iv)
Solution
Multiplication is spread across a sum.
The factor outside the bracket multiplies each term inside.
Answer
Distributive property of multiplication over addition.
Study notesRule used
Rule used
For all :. This is the single most used step in the whole of Exercise 1.2.
(v)
Solution
Adding left the number unchanged.
is the additive identity of .
Answer
Additive identity property.
(vi)
Solution
Multiplying by left the number unchanged.
is the multiplicative identity of .
Answer
Multiplicative identity property.
(vii)
Solution
Only the bracketing changed.
,, remain in the same order; the grouping moved.
Answer
Associative property of multiplication.
(viii)
Solution
The two factors swapped places.
Order of multiplication changed, no grouping involved.
Answer
Commutative property of multiplication.
Question 5
Name the property used in each of the following.
(i)
Solution
Find what was done to both sides.
The left side went from to and the right side from to . Both increased by .
Adding the same number to both sides preserves the inequality.
Answer
Additive property of inequality (adding the same real number to both sides preserves the order).
(ii)
If , then
Solution
Taking reciprocals reversed the inequality sign.
This is the reciprocal (multiplicative inverse) property of inequalities.
State the condition the textbook omits.
The statement holds only when and have the same sign and neither is .
Show what goes wrong without the condition.
Answer
Reciprocal property of inequality, valid when and are both positive, or both negative.
Valid for
Study notesCommon mistake
Common mistake: The condition is essential
As printed, the statement is false for ,. Always check that and lie on the same side of zero before reversing the sign.
(iii)
If , then
Solution
The same quantity was added to both sides and the sign did not change.
Addition never reverses an inequality, whatever the sign of .
Answer
Additive property of inequality.
(iv)
If and , then
Solution
A common positive factor was cancelled from both sides.
Dividing by the positive number keeps the direction of the inequality.
Answer
Multiplicative cancellation property of inequality for a positive multiplier.
Study notesWhy it applies
Why it applies: Why matters
Dividing by a positive number preserves order; dividing by a negative number reverses it. That is exactly the difference between this part and part .
(v)
If and , then
Solution
A common negative factor was cancelled and the sign flipped.
Dividing both sides by a negative number reverses the inequality.
See it on a concrete case.
Answer
Multiplicative cancellation property of inequality for a negative multiplier (the order reverses).
Study notesCommon mistake
Common mistake: The single most common inequality error
Forgetting to reverse the sign when multiplying or dividing by a negative number. Whenever a negative factor is removed, flip to and to .
(vi)
Either or or
Solution
Exactly one of three mutually exclusive relations must hold.
For any two real numbers, precisely one of ,, is true.
Answer
Trichotomy property of real numbers.
Question 6
Insert two rational numbers between the given pairs.
(i)
Insert two rational numbers between and .
Solution
Decide which number is smaller before averaging.
With equal numerators, the larger denominator gives the smaller fraction, so .
The mean of two numbers always lies strictly between them.
LCM of and is .
Multiply by .
Average again, this time between and , to get a second number.
LCM of and is .
Multiply by .
Answer
Two such numbers are and .
Study notesRule used · Check · Alternative method
Rule used: density of
Between any two distinct rational numbers there is another rational number, namely their average. Repeating the step gives as many as you like.
Check numerically
,,,, correctly ordered.
Alternative method
Rewrite both with a common denominator large enough to leave room: and , so and work immediately.
(ii)
Insert two rational numbers between and .
Solution
Take the mean of the two given numbers.
Average with to obtain a second number.
Write as .
Multiply by .
Answer
Two such numbers are and .
(iii)
Insert two rational numbers between and .
Solution
The denominators already match, so add the numerators.
The denominators already match, so add the numerators inside the bracket.
Multiply the two fractions: numerator times numerator, denominator times denominator.
Average with .
LCM of and is .
Multiply by .
Answer
Two such numbers are and .
Study notesAlternative method
Alternative method
Scale both fractions: and , so and also work.
Concepts
Learn the ideas before attempting the exercise.
Radicals and indices are two notations for the same operations. This section develops their laws, explains surds and conjugates, and shows why rationalizing a denominator works.
Radicals and the th root
A real number is an th root of when . The principal root is written .
In , is the radicand and is the index. If no index is written, the index is .
For even indices in the real-number system, the radicand must be nonnegative and the principal root is nonnegative.
Laws of radicals
For real values for which both sides are defined:
Example..
Laws of indices
Product
Quotient
Power of a power
Power of a product
Zero exponent
Negative exponent
Surds and like terms
A surd is an irrational radical with a rational radicand. Examples include and .
Surds with one term are monomial; expressions such as are binomial surds. Only like surds can be combined by adding coefficients.
Conjugates
The conjugates of and are obtained by changing the sign between the terms.
The middle terms cancel, so the product contains no surd.
Rationalizing the denominator
Multiplying numerator and denominator by the same nonzero expression multiplies the fraction by and does not change its value.
For a binomial denominator, multiply by its conjugate.
Common mistakes
in general.
; the middle term is required.
For even roots, , not automatically .
When rationalizing, multiply both numerator and denominator by the same factor.
Historical note
Ancient Babylonian mathematics contained a remarkably accurate approximation to . Much later, Greek mathematics proved that the diagonal-to-side ratio of a square cannot be a ratio of integers.
Written to support the questions below. Not part of the printed textbook.
01Radicals, surds and their laws
Definition
For ,, the symbol denotes the th root of . A surd is a root that cannot be evaluated exactly as a rational number, such as or .
The rule, with its conditions
For even , the laws require and so that the roots are real.
Worked example
Simplify .
Find the largest perfect-square factor of .
Apply the product law.
Evaluate the exact root.
Answer
Common misconception
is not always true
For :, not . The correct statement for even roots is . This matters whenever a variable could be negative.
Quick check
Simplify .
Show the answer
and , so the difference is .
Did you know?
Why
There is a product law and a quotient law, but no sum law. Test it: , while .
02Rationalizing a denominator
Definition
To rationalize a denominator is to rewrite a fraction so that no radical remains below the line, without changing its value.
The rule, with its conditions
Monomial denominator multiply by the surd itself. Binomial denominator multiply by the conjugate.
Worked example
Rationalize .
The denominator is binomial, so use the conjugate .
Numerator: distribute .
Denominator: difference of squares.
Divide by .
Answer
Common misconception
Using a conjugate on a monomial denominator
For there is nothing to conjugate, multiply by to get . Reach for a conjugate only when the denominator has two terms.
Quick check
Rationalize .
Show the answer
Multiply by . Denominator: . Result: .
Did you know?
Why bother?
Before calculators, dividing by by hand was far harder than dividing by . Today the reason is different: a rationalized form is unique, so two students' answers can be compared directly.
03The laws of exponents
Definition
For and real , powers obey a fixed set of laws. Every simplification in Exercise 1.2 is one of these laws applied one at a time.
The rule, with its conditions
Worked example
Simplify .
Factor out the lowest power, , from the numerator.
Evaluate the bracket.
Write the denominator with the same factor.
Cancel , which is never zero.
Answer
Common misconception
has no law
The laws cover multiplication, division and powers of powers, never addition. is , not . When terms are added, factor instead.
Quick check
Simplify .
Show the answer
Reciprocal first: .
Did you know?
Why
Follow the quotient law: . But any non-zero number divided by itself is . So for every , the definition is forced, not arbitrary.
Examples
The textbook's own examples, solved as printed.
Example 7
Simplify , , and .
Solution
(i)
The textbook assumes the variables are nonnegative. Otherwise .
(ii)
(iii)
Example 8
Rationalize the denominators of and .
Solution
(i)
(ii)
Exercise 1.2
Try each part first, then open its solution.
Question 1
Rationalize the denominators of the following.
(i)
Solution
The denominator is the binomial , so its conjugate is .
Multiply numerator and denominator by the conjugate. This does not change the value, because .
Distribute over the bracket in the numerator.
The denominator matches with ,.
Apply the difference-of-squares identity to the bracket pair.
Evaluate each square; .
Subtract.
Put numerator over denominator.
Cancel the common factor .
Write what is left.
Answer
Study notesRule used · Why it applies · Check
Rule used: conjugate of a binomial surd
The conjugate of is . Their product is rational, because the difference-of-squares identity squares away the radical.
Why it applies: Why the conjugate and not
Multiplying by alone would give , which still contains a radical. Only the conjugate removes it in one step.
Check: Numerical check
, and . They agree.
(ii)
Solution
The denominator is the single surd , so multiply by , not by a conjugate.
Multiply numerator and denominator by .
Distribute over the bracket in the numerator.
Apply the product law for radicals, , to each term.
Multiply under each radical.
The denominator becomes , which is rational.
A factor multiplied by itself is that factor squared.
Squaring a square root undoes it, because .
Combine numerator and denominator.
Answer
Study notesWhy it applies · Common mistake
Why it applies: Why multiply by and not by a conjugate
The denominator here contains the single surd , with nothing added to it. Multiplying by turns it into , which is rational. A conjugate is needed only when the denominator is a binomial such as .
Common mistake: Do not forget to distribute
is , not . Every term inside the bracket must be multiplied.
(iii)
Solution
Single surd in the denominator.
Multiply above and below by .
Distribute over both terms of the numerator.
Apply to the first term.
Multiply under the radical.
The denominator becomes rational.
Combine.
Answer
Study notesWhy it applies · Common mistake
Why it applies: Why multiply by
The denominator is the single surd . Multiplying by turns it into , which is rational. Because the numerator is a two-term bracket, the same must then be distributed over both terms.
Common mistake: cannot be simplified further
You may not cancel the against . Cancellation needs a common factor of the whole numerator, and has no factor of .
(iv)
Solution
The denominator is the binomial ; its conjugate is .
Multiply numerator and denominator by the conjugate.
The numerator becomes the same bracket multiplied by itself.
Apply with ,.
Evaluate each of the three parts separately. Note .
Add the two rational terms and .
The denominator matches .
Evaluate both squares.
Combine numerator and denominator.
Take out the common factor from the numerator so it can be cancelled.
Cancel the common factor .
Write what remains.
Answer
Study notesWhy it applies · Common mistake · Check
Why it applies: Why the numerator became a square
The numerator and the conjugate happen to be the same bracket, . So the multiplication is a squaring, and applies rather than .
Common mistake:
The whole coefficient is squared as well: .
Check: Numerical check
, and .
(v)
Solution
Conjugate of is .
Multiply numerator and denominator by the conjugate.
The numerator becomes a square.
Apply with ,.
Use and .
Add the rational terms.
Denominator by difference of squares.
Evaluate both squares and subtract.
Divide by .
Answer
Study notesRule used · Check
Rule used
For denominators of the form , the conjugate is and the product is , always rational.
Check: Numerical check
, and .
(vi)
Solution
Conjugate of is .
Multiply numerator and denominator by the conjugate.
Distribute over the bracket.
Apply to each term.
Multiply under each radical.
Denominator by difference of squares.
Apply the difference-of-squares identity to the bracket pair.
Combine.
Take out the common factor so the fraction can be reduced.
Divide by ; write and cancel one .
Write what remains.
Answer
Study notesWhy it applies
Why it applies: Why factor before cancelling
cannot be reduced term by term until the common factor is taken outside a bracket. Cancellation is only valid against a factor of the entire numerator.
Question 2
Simplify the following.
(i)
Solution
A negative exponent means the reciprocal: .
Write both numbers as powers of a prime so the fractional exponent can act.
Apply .
Apply ; multiply the exponents.
in both cases.
Evaluate the cubes.
Answer
Study notesRule used · Why it applies
Rule used
, so a fraction raised to a negative power is the reciprocal raised to the positive power.
Why it applies: Why factor into and
The exponent asks for a fourth root followed by a cube. Writing and makes the fourth root exact and removes all guesswork.
(ii)
Solution
Turn the negative exponent into a reciprocal.
Square numerator and denominator separately.
Cube the second fraction.
Substitute both results back into the expression.
Dividing by a fraction is multiplying by its reciprocal.
Cancel against .
Cancel against the .
Cancel against , and against .
Multiply.
Answer
Study notesRule used · Common mistake
Rule used: Order of operations
and have equal precedence and are worked from left to right. The division must be converted to multiplication by the reciprocal before the final is applied.
Common mistake: Do not cancel across a sign
Cancellation is only valid once every operation is a multiplication. Convert into first.
(iii)
Solution
Write the decimal as a fraction so the root can be taken exactly.
Negative exponent means reciprocal.
Write both as perfect cubes.
Apply .
Multiply the exponents: .
Write the result.
Answer
Study notesWhy it applies
Why it applies: Why convert the decimal first
hides the fact that it is . Writing shows both parts as perfect cubes, so the cube root is exact instead of approximate.
(iv)
Solution
Inside the radical, subtract exponents of like bases: .
Do each subtraction.
Write the seventh root as the exponent .
Apply .
Multiply each exponent by .
Simplify each fraction.
Answer
Valid for
Study notesRule used · Check
Rule used: Why and
The original expression has in a denominator, so it is undefined when or . The restriction belongs in the answer even though the simplified form looks harmless.
Check with a quick substitution
Put : the fraction is , and . The answer gives .
(v)
Solution
Every term must sit on one base. Write .
Multiply the exponents.
Rewrite the numerator with this substitution.
Add exponents in the first term: .
Take out the common factor ; note .
Evaluate inside the bracket.
Now the denominator, using the same substitutions.
Add exponents in the first term.
Take out .
Evaluate inside the bracket.
Form the quotient.
Cancel the common factor , which is never .
Cancel the common factor .
Answer
The answer does not depend on .
Study notesRule used · Why it applies
Rule used: one base first
When an expression mixes , and , rewrite everything as a power of the smallest base. Only then can exponents be added, subtracted or factored.
Why it applies: Why may be cancelled
for every real , so it is never zero and the cancellation is always valid, no restriction is needed here.
(vi)
Solution
Choose base :,,.
Convert the second numerator term.
Rewrite the numerator on base .
Take out ; note .
Evaluate the bracket.
Convert the denominator to base .
Multiply powers of the same base by adding exponents.
Collect the exponent.
Separate the constant part so it can cancel: .
Form the quotient.
Cancel , which is positive for all real .
Cancel the common factor .
Answer
Study notesCheck
Check with
Numerator: . Denominator: . So the value is , matching.
(vii)
Solution
Write as a perfect cube.
Apply the exponent: .
Convert to a reciprocal.
Write as a perfect square.
Apply the exponent: .
Convert to a reciprocal.
Dividing by a fraction is multiplying by its reciprocal.
Multiply.
Answer
(viii)
Solution
Write as everywhere.
Multiply powers of the same base by adding exponents (numerator).
Product of powers with the same base: keep the base, add the exponents.
Collect the like terms and in the exponent.
Same for the denominator.
Product of powers with the same base: keep the base, add the exponents.
Remove the brackets and collect like terms in the exponent.
Divide powers of the same base by subtracting exponents.
Expand the bracket carefully: .
Evaluate the power.
Answer
The value is independent of .
Study notesCommon mistake
Common mistake: Watch the double negative
is , not . The minus sign applies to both terms in the bracket.
(ix)
Solution
In the numerator, the lowest power of present is . Take it out.
Evaluate the bracket.
Write so the numerator carries the same factor as the denominator.
In the denominator, take out the common factor .
Form the quotient.
Cancel , which is positive for every real .
Answer
Valid for
Study notesRule used · Common mistake · Check
Rule used
Factor out the lowest power of the base that appears. Here that is above and below.
Common mistake: The denominator does not simplify to a number
contains two different bases, so cannot be reduced further. The answer therefore still contains , unlike parts , and .
Check with
Numerator: . Denominator: . So the value is , and the formula gives . They agree.
Question 3
If , find the value of each of the following.
(i)
If , find .
Solution
Given:
Simplify the surd first: and .
Write down the reciprocal that has to be simplified.
The denominator is a binomial surd, so use its conjugate .
Denominator by with ,.
.
Divide by .
Now add and .
The terms are opposites and cancel.
Add the rational parts.
Answer
Study notesRule used · Why it applies
Rule used
When and , the reciprocal is simply the conjugate . That is exactly what happened here, since .
Why it applies: Why simplify first
and are the same number, but the second form makes the conjugate obvious and the arithmetic easy to see.
(ii)
Find .
Solution
Recall, part (i)
Both values were found in part : the surd was simplified to , and the reciprocal was rationalized to .
Substitute the two values into .
Remove the second bracket; the minus sign changes both signs inside it.
The rational parts cancel and the surd terms add.
Simplify.
Answer
Study notesCommon mistake
Common mistake: The sign trap
is , not . Writing the distribution as a separate line, as above, is the safest way to avoid this.
(iii)
Find .
Solution
Recall, part (i)
From part we already know .
Square the known sum, using with and .
In the middle term, cancels with .
So the middle term is .
Make the subject by subtracting .
Substitute the value from part .
Evaluate.
Answer
Valid for
Study notesRule used · Why it applies
Rule used
. The comes from the middle term .
Why it applies: Why not substitute directly
You could square and rationalize , but that is far longer. Using the result of part turns the whole problem into one subtraction.
(iv)
Find .
Solution
Recall, part (i)
From part .
Recall, part (ii)
From part .
Recognise as a difference of two squares: with ,.
Substitute the two values recalled above.
Multiply the rational coefficients.
Answer
Valid for
Study notesCommon mistake · Check · Rule used
Common mistake: This answer is not
is the value of from part . The difference is a different quantity and is irrational.
Check: Direct check
and . Subtracting gives , confirming the identity route.
Rule used
. Here both factors were already known, so no further expansion was needed.
(v)
Find .
Solution
Recall, part (iii)
From part we found , by squaring and subtracting . That value is the starting point here, it is not a new number.
Square the known value, using with and .
In the middle term, cancels with .
So the middle term equals .
Make the subject.
Substitute , the value inherited from part .
Evaluate , then subtract.
Answer
Valid for
Study notesRecall · Rule used
Recall: Where came from
It is the answer to part :. If you meet this question on its own, derive first from .
Rule used: The doubling pattern
and . Each step squares the previous result and subtracts .
(vi)
Find .
Solution
Recall, part (ii)
From part .
Substitute the value from part into the square.
Square the coefficient and the radical separately: .
and .
Multiply.
Answer
Valid for
Study notesAlternative method
Alternative method
, using part . Both routes agree, which is a useful self-check.
Question 4
4
Find rational numbers and such that .
Solution
Rationalize the left side first; the conjugate of is .
Expand the numerator term by term.
Work out each of the four products; note .
Collect the rational terms and the terms separately.
Expand the denominator by difference of squares.
.
Form the quotient.
Divide each term of the numerator by separately.
Carry out each division, watching the signs.
Compare with the required form . Since are rational and is irrational, the rational parts and the parts must match separately.
Answer
Study notesRule used · Common mistake · Check
Rule used: comparing coefficients
If with , then and . This works because is irrational: otherwise would be rational.
Common mistake: Watch the negative denominator
Dividing by flips the sign of both terms. and .
Check: Numerical check
, and .
Question 5
Simplify the following.
(i)
Solution
Write each base as a power of a prime.
Write the base as a power of a prime.
Power of a power: multiply the two exponents.
Work out the product of the exponents.
Evaluate the power.
Write the base as a power of a prime.
Power of a power: multiply the two exponents.
Work out the product of the exponents.
Evaluate the power.
Write the base as a power of a prime.
Power of a power: multiply the two exponents.
Work out the product of the exponents.
Evaluate the power.
Write the base as a power of a prime.
Power of a power: multiply the two exponents.
Work out the product of the exponents.
Evaluate the power.
Substitute all four values.
Multiply numerator and denominator.
Answer
In lowest terms: and share no factor.
Study notesWhy it applies
Why it applies: Why prime factorization first
A fractional exponent such as asks for a fourth root and then a fifth power. Writing makes the fourth root exact in one step instead of guessing.
(ii)
Solution
Write the cube root as the exponent .
Write as .
Multiply all three exponents: .
The numerator is therefore a single term.
Now the denominator. Write .
Separate the constant part: .
For the second term, .
Divide by .
Add the two denominator terms; both carry the factor .
Both terms share the factor , so take it outside a bracket.
Add the two numbers inside the bracket.
Form the quotient.
Cancel , which is positive for every real .
; cancel it.
Answer
The value does not depend on .
Study notesRule used · Check
Rule used
, then . Converting radicals to exponents turns every root into ordinary index arithmetic.
Check with
Numerator: . Denominator: . So .
(iii)
Solution
Write as a perfect cube.
Write the base as a power of a prime.
Power of a power: multiply the two exponents.
Work out the product of the exponents.
Evaluate the power.
Evaluate the second numerator factor.
Write the decimal as a fraction.
Negative exponent means reciprocal.
Write and multiply exponents.
Power of a power: multiply the exponents, .
Evaluate .
Substitute all three values inside the radical.
Multiply in the numerator.
; cancel it.
Apply .
Take both square roots.
Answer
Study notesRule used · Common mistake
Rule used: quotient law for radicals
for and . It is what lets split into , so both roots can be taken exactly.
Common mistake: is in the denominator
Because it already sits below the line, its negative exponent does not move it up. Evaluate it as and keep it where it is.
(iv)
Solution
Look for the sum-of-cubes pattern by naming the two pieces.
Check that the second bracket matches :,,.
Apply the sum-of-cubes identity .
Put and back.
Multiply the exponents in each term.
Simplify the exponents.
Answer
Study notesRule used · Why it applies · Check
Rule used: sum of cubes
. Recognising it saves expanding six separate products.
Why it applies: How to spot it
The second bracket has three terms, the outer two are perfect squares of the terms in the first bracket, and the middle term is their product with a minus sign. That signature means sum of cubes.
Check with numbers
Put ,:,. Then , and .
Concepts
Learn the ideas before attempting the exercise.
Word problems become manageable when the quantities are named, translated into equations, and checked against the original situation. This section combines linear equations with temperature, percentage, tax, profit, loss, and compound markup.
From words to equations
Name the unknown quantity.
Translate each condition into an equation.
Solve one step at a time.
Substitute the answer back into the original conditions.
Example. If two numbers have sum and difference , write and .
Temperature conversion
The school text rounds to in its exercises. Scientific work normally uses .
The denominator is the cost price because profit or loss is measured relative to what was originally paid.
Simple profit, tax, and ratios
In , use the rate as a decimal and time in years. For ratio sharing, add the ratio parts first, find one part, then multiply.
Compound markup
Here is principal, is the annual rate in decimal form, is the number of compounding periods per year, and is time in years.
Check before accepting an answer
A selling price with profit should exceed the cost price.
A loss percentage should be positive and below in ordinary cases.
A Fahrenheit temperature corresponding to a hot Celsius temperature should be larger numerically.
Ages, lengths, money, and counts must use appropriate units.
Examples
The textbook's own examples, solved as printed.
Example 9
The sum of two real numbers is and their difference is . Find the numbers.
Solution
Let the numbers be and .
Add the equations:
Substitute into :
The required numbers are and .
Study notesCheck
Check
+= and =, so both original conditions are satisfied.
Example 10
Normal human body temperature is . Convert it to Celsius and Kelvin.
Solution
Using the more precise scientific offset would give .
Example 11
Hamail purchased a bicycle for and sold it for . Find the profit percentage.
Solution
Therefore, the profit percentage is approximately .
Example 12
Umair bought a book for and sold it for . Find the loss percentage.
Solution
Therefore, the loss percentage is approximately .
Example 13
Saleem, Nadeem, and Tanveer earned a profit of . Their investments were in the ratio . Find each person’s share of the profit.
Solution
The three shares total .
Example 14
The simple profit on for years is . Find the annual rate of profit.
Solution
For simple profit, . Solving for gives
Therefore, the annual rate of profit is .
Exercise 1.3
Try each part first, then open its solution.
Question 1
1
The sum of three consecutive integers is . Find the integers.
Solution
Choose the middle integer as the variable; this keeps the algebra symmetric.
Turn the words "sum ... is " into an equation.
Collect like terms; and cancel.
Divide both sides by .
Put back into the three expressions.
Answer
Check: . ✓
Study notesAlternative method · Common mistake
Alternative method: Alternative labelling
Taking the integers as ,, gives , so and the same answer follows. Choosing the middle term simply removes the constant.
Common mistake: Consecutive odd integers are different
Three consecutive odd integers are ,,, whose sum is , always odd, so it can never equal .
Question 2
2
In right-angled , cm and area . Find in the form cm.
Right-angled triangle
Solution
and are the two legs, so they are the base and height.
Put in the given area and the given side.
Multiply both sides by to clear the fraction.
Divide both sides by to isolate .
The denominator is a binomial surd; its conjugate is .
Denominator by difference of squares.
Squaring each square root removes it, since both radicands are positive.
Subtract.
The in the numerator cancels the in the denominator.
Expand the product term by term.
Apply to the last two terms.
Simplify each surd: and .
Group the terms and the terms.
Combine each group.
Answer
So and .
Study notesCheck · Why it applies
Check: Numerical check
cm and cm, so the area is cm, which matches .
Why it applies: Why the answer stays positive
and , so as a length must be, even though is negative.
Question 3
3
A rectangle has sides m and m. Express the area as .
Solution
Simplify the first side: and .
The second side has in a denominator, so rationalize it.
Numerator ; denominator .
So the second side simplifies.
Area of a rectangle is length times width.
Expand term by term.
Work out each product; note .
Group rational terms and terms.
Simplify each group.
Answer
So and . Numerically this is about m.
Study notesRule used · Common mistake
Rule used: area of a rectangle, and rationalizing a monomial denominator
Area . Before multiplying, each side is written in the form : the surd is simplified to , and is rationalized by multiplying above and below by .
Common mistake: Simplify the surds before multiplying
Expanding directly produces terms and far more work. Reduce each side to the form first.
Question 4
4
Find two numbers whose sum is and whose difference is .
Solution
Name the unknowns.
Write the two conditions as equations.
The second condition.
Add equations () and (); the terms cancel.
Collect like terms.
Divide both sides by .
Substitute into equation ().
Subtract from both sides.
Answer
Check: and . ✓
Study notesRule used
Rule used: elimination
When one unknown appears with opposite signs in the two equations, adding them removes it in a single step.
Question 5
5
The temperature in Lahore was . Convert it to Fahrenheit.
Solution
State the conversion formula.
Substitute .
Multiply the fraction by ; keep it exact.
Divide.
Add .
Answer
Study notesRule used · Common mistake · Check
Rule used: Celsius to Fahrenheit
. The factor rescales the degree size ( Celsius degrees span Fahrenheit degrees) and the shifts the freezing point from to .
Common mistake: Keep as a fraction
Replacing by is exact here, but replacing by in the reverse formula is not. Work with the fraction and you never lose accuracy.
Check: Reverse check
. ✓
Question 6
6
The father’s and son’s current ages have sum . Six years ago, the father was twice the son’s age. Find the son’s age six years ago.
Solution
Name the present ages.
Use the sum condition to express the father's present age.
Write both ages six years ago.
Turn "the father was twice as old as his son" into an equation.
Expand the right side.
Add to both sides.
Add to both sides.
Divide both sides by .
The question asks for the son's age six years ago.
Answer
Check: six years ago the father was , and . ✓
Study notesCommon mistake
Common mistake: Read what is actually asked
The equation gives the son's present age, . The question asks for his age six years ago, so the final subtraction is part of the answer, not an optional extra.
Question 7
7
Mirha buys a toy for . Find the selling price that gives a profit of .
Solution
Identify which amount is the cost price.
Convert the percentage to a fraction of the cost price.
Profit is that fraction of the cost price.
Cancel the common factor between and .
Cancel into .
Multiply.
Selling price is cost price plus profit.
Add.
Answer
Study notesAlternative method · Rule used
Alternative method: One-step method
. Useful once the two-step method is understood.
Rule used: Profit percentage is always on cost price
Unless the question says otherwise, a stated profit or loss percentage is measured against the cost price, not the selling price.
Question 8
8
Tayyab’s annual income is , and is exempt from tax. Find the tax at .
Solution
Tax is charged only on the part of the income that is not exempt.
Substitute the two given amounts.
Subtract.
Per cent means “out of one hundred”.
Clear the decimal by multiplying top and bottom by .
Reduce the fraction to lowest terms.
Tax is that fraction of the taxable income.
Cancel the common factor :.
Multiply.
Answer
Study notesCommon mistake · Why it applies
Common mistake: Do not tax the whole income
Applying to gives , which is wrong. The exempt amount must be subtracted first.
Why it applies: Why
. Multiplying top and bottom by clears the decimal: , which reduces by to .
Question 9
9
Find the compound markup on for year at compounded annually.
Solution
State the compound amount formula. Here is the amount **after years**, the principal and the annual rate.
Per cent means “out of one hundred”.
Clear the decimal by multiplying top and bottom by .
Reduce the fraction to lowest terms.
Substitute ,,.
Add inside the bracket.
Multiply.
Compound markup is the amount minus the principal.
Subtract.
Answer
Study notesRule used · Why it applies · Check
Rule used: Use , not , for the number of years
In , the exponent counts the compounding periods. Some printings write for the number of periods; either symbol is fine as long as it is defined.
Why it applies: Why simple and compound agree here
For there is only one compounding period, so compound markup equals simple markup: . A difference appears only from the second year onwards.
Check: What year would give
, so the two-year compound markup would be , more than twice the first year's figure.
Review Exercise 1
The review exercise printed at the end of the unit.
Question 1
Four options are given. Tick the correct one.
(i)
is a/an:
(A)integer
(B)rational number
(C)irrational number
(D)natural number
Answer
Correct option: (C) irrational number
is an irrational number because it is the square root of a "non-perfect-square" number.
(ii)
and are:
(A)natural numbers
(B)integers
(C)rational numbers
(D)irrational numbers
Answer
Correct option: (D) irrational numbers
and are well-known irrational numbers. Neither can be expressed as a fraction of two integers.
(iii)
If is not a perfect square, then is:
(A)rational
(B)natural number
(C)integer
(D)irrational
Answer
Correct option: (D) irrational
If is not a perfect square, then is an irrational number, e.g., .
(iv)
is:
(A)whole number
(B)integer
(C)rational
(D)irrational
Answer
Correct option: (D) irrational
To justify this specific sum, suppose is rational. Then , which would make rational, a contradiction. Therefore is irrational.
(v)
For all , . This is:
(A)reflexive
(B)transitive
(C)symmetric
(D)trichotomy
Answer
Correct option: (A) reflexive
This is the reflexive property: every real number equals itself, i.e., for all .
(vi)
and . This is:
(A)trichotomy
(B)transitive
(C)additive
(D)multiplicative
Answer
Correct option: (B) transitive
This is the transitive property of order: if and , then .
(vii)
, then :
(A)
(B)
(C)
(D)
Answer
Correct option: (A)
Therefore, .
(viii)
and is the ___ property.
(A)reflexive
(B)symmetric
(C)transitive
(D)additive
Answer
Correct option: (B) symmetric
This is the symmetric property of equality: if , then .
(ix)
equals:
(A)
(B)
(C)
(D)
Answer
Correct option: (D)
(x)
Product of is:
(A)prime
(B)odd
(C)irrational
(D)rational
Answer
Correct option: (D) rational
Using the difference of squares:
Since is rational, the answer is (d) rational.
Question 2
If , verify associative property for addition and multiplication.
Solution
Textbook solution, reproduced as printed.
Calculate :
Calculate :
Since , the associative property of addition is verified. Associative Property of Multiplication: The associative property of real numbers with respect to multiplication states that . Calculate :
Calculate :
Since , the associative property of multiplication is verified.
Question 3
Is a rational number? Explain.
Solution
Textbook solution, reproduced as printed.
Yes, is a rational number because it can be expressed as the quotient of two integers, such as or .
Question 4
State the trichotomy property.
Solution
Textbook solution, reproduced as printed.
The trichotomy property of real numbers states that for any two real numbers and , exactly one of the following is true: (i) , (ii) , or (iii) . This property ensures that any two real numbers can be compared, and one of the three possible relationships will hold.
Question 5
Find two rational numbers between and .
Solution
Textbook solution, reproduced as printed.
There are infinitely many rational numbers between and . One such number is the average of the two numbers, i.e.,
Another rational number can be found by taking the average of and , i.e.,
Therefore, two rational numbers between and are and .
Question 6i
Solution
Textbook solution, reproduced as printed.
Question 6ii
Solution
Textbook solution, reproduced as printed.
Question 6iii
Solution
Textbook solution, reproduced as printed.
Question 7
Sum of three consecutive odd integers is . Find them.
Solution
Textbook solution, reproduced as printed.
Let the three consecutive odd integers be , , and . The sum of these integers is
Given that the sum is , we can write
Solving for gives
Therefore, the three consecutive odd integers are
Question 8
balls in two buckets, one has more. How many in each?
Solution
Textbook solution, reproduced as printed.
Let the number of balls in the first bucket be . Then the number of balls in the second bucket is . The total number of balls is
Solving for gives
Therefore, the number of balls in the first bucket is , and the number of balls in the second bucket is
Question 9
Salma invested at for years, then for years. Find total amount.
Solution
Textbook solution, reproduced as printed.
First convert the mixed percent into decimal form.
Now find the first profit for years.
For the next years at ,
Total amount = principal + both profits.
Therefore, the total amount is .
Unit Test 1
Real Numbers. The printed unit test, with the errors found in the printed version corrected.
Unit Test 1
Class 9 · Unit 1 Real NumbersTotal marks: Time: MCQ 20 minutes · written 2 hours 10 minutes
Section A: Multiple choice marks
Q.1Attempt all parts. Each part carries mark. Suggested time: minutes.
(i)
is a/an:
(A)integer
(B)rational number
(C)irrational number
(D)natural number
(ii)
For , if and , then:
(A)
(B)
(C)
(D)
(iii)
If , then :
(A)
(B)
(C)
(D)
(iv)
is a/an:
(A)whole number
(B)integer
(C)rational number
(D)irrational number
(v)
The decimal expansion of an irrational number is always:
(A)terminating
(B)recurring but nonterminating
(C)nonrecurring and nonterminating
(D)either recurring or terminating
(vi)
Which expression represents a rational number?
(A)
(B)
(C)
(D)
(vii)
Let and . Find .
(A)
(B)
(C)
(D)
(viii)
If is irrational, which expression is always irrational?
(A)
(B)
(C)
(D)
(ix)
Which expression disproves the statement “the sum of two irrational numbers is always irrational”?
(A)
(B)
(C)
(D)
(x)
The product is a/an:
(A)prime number
(B)odd number
(C)irrational number
(D)rational number
(xi)
equals:
(A)
(B)
(C)
(D)
(xii)
The result of is:
(A)
(B)
(C)
(D)
(xiii)
The multiplicative inverse of is:
(A)
(B)
(C)
(D)
(xiv)
If , then . Which property is used?
(A)transitive
(B)trichotomy
(C)additive
(D)multiplicative
(xv)
A rational number between and is:
(A)
(B)
(C)
(D)
Section B: Short questions (Group I) marks
Q.2Attempt any six of the following nine parts. Each part carries marks.
(i)
Is ? Why or why not?
(ii)
Insert two rational numbers between and .
(iii)
State the identity properties of real numbers for addition and multiplication.
(iv)
State the right distributive property of multiplication over addition.
(v)
Represent on the number line.
(vi)
Express as a fraction.
(vii)
If ,, and , verify .
(viii)
Name the property: if and , then .
(ix)
Define the set of rational numbers.
Section B: Short questions (Group II) marks
Q.3Attempt any six of the following nine parts. Each part carries marks.
(i)
Define the th root of a real number .
(ii)
Simplify .
(iii)
Simplify .
(iv)
If , find .
(v)
Why is not a surd?
(vi)
Give an example showing that the product of two conjugate surds is rational.
(vii)
Rationalize .
(viii)
For ,, and , verify the associative property.
(ix)
Find .
Section B: Short questions (Group III) marks
Q.4Attempt any six of the following nine parts. Each part carries marks.
(i)
State the formula to convert Fahrenheit into Kelvin.
(ii)
Normal human body temperature is . Convert it into Celsius and Kelvin.
(iii)
Simplify .
(iv)
The sum of three consecutive odd integers is . Find them.
(v)
The sum of two real numbers is and their difference is . Find them.
(vi)
Write the formula for percentage loss.
(vii)
Show that .
(viii)
Why can a real number not be divided by zero?
(ix)
What is the intersection of rational and irrational numbers?
Section C: Long questions marks
Q.5Attempt any three questions. Question is compulsory.
(i)
Express as a recurring decimal. Then write the decimal forms of .
(ii)
Simplify .
(iii)
Find rational numbers and such that .
(iv)
Simplify .
(v)
State the laws of radicals and the laws of indices.
(vi)
Find the compound markup on for one year at compounded annually.
(vii)
Simplify .
(viii)
Tayyab’s annual income is and is exempted. Find tax at .
(ix)
Write the six order properties of real numbers.
(x)
Represent on the number line.
Practice
Build a paper from the Unit 1 item bank. Coverage, difficulty and paper format are chosen independently, and the generator refuses any request the bank cannot fill.