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11, S, F, O, O, R, , P, E, AG, , P, D, E, T, C, , RE, R, O, , C, N, U, , Surds and indices, , This chapter deals with defining the system of real numbers, distinguishing between, rational and irrational numbers, performing operations with surds, using integers, and fractions for index notation, and converting between surd and index forms., After completing this chapter you should be able to:, ▶ define real numbers and distinguish, between rational and irrational, numbers, ▶ simplify expressions involving surds, ▶ expand expressions involving surds, ▶ rationalise the denominators of, simple surds, ▶ use the index laws to define, fractional indices, , ▶ translate expressions in surd form, and index form, ▶ evaluate numerical expressions, involving fractional indices, ▶ use the calculator to evaluate, fractional powers of numbers, ▶ evaluate a fraction raised to the, power of –1, ▶ prove general properties of real numbers., , NUMBER & ALGEBRA – ACMNA 264

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A Square root of a number, The square root of a number, x, is the number, that when multiplied by itself is equal to x., For example, the square root of 9 is 3 or −3,, since 32 = 9 and (−3) 2 = 9., , 11004 An aerial photo of a square, , __, , house block (say 900 square metres)., , √x, , is the positive, square root of, x., __, __, __, For example, √9 = 3 and − √9 = −3. −√x is, then the negative square root of x., , Needs to be square., , EXAMPLE 1, Find the following., a the square root of 81, , b √8 1, , a the square root of 81 = 9 or −9, , b √81 = 9, , S, F, O, O, R c −√81 = −9, , Exercise 11A, , __, , ___, , c −√81, , P, E, AG, , ___, , P, D, E, T, C, , RE, R, O, , 1 Find the following., a i the square root of 4, b i the square root of 25, c i the square root of 49, d i the square root of 64, , C, N, U, , ii, ii, ii, ii, , __, , 4, √___, , iii, iii, iii, iii, , 25, √___, 49, √___, √64, , ___, , __, , −√___, 4, −√___, 25, −√___, 49, −√64, ___, , Since there is___, no number that when multiplied by itself is equal to −9, it is not possible to find √−9 ., We say that √−9 is undefined. __, __, The square root of 0 is 0, since 0√0 = 0. Zero is neither positive nor negative but we define √0 = 0., , NUMBER & ALGEBRA, , In general:, __, • √x is undefined for x < 0, __, • √x = 0 for x = 0, __, • √x is the positive square root of x when x > 0., __, • −√x is the negative square root of x when x > 0., , EXAMPLE 2, Find the following, where possible., ___, , a √36, , ___, , a √36 = 6, , ___, , b − √36, , ___, , b − √36 = −6, , ____, , __, , c √−36, , d √0, , c undefined, , d 0, , Chapter 11 Surds and indices, , 3

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2 Find the following, where possible., ____, _____, a √100, b √−100, ___, , ___, , ____, , f − √16, , ____, , g √−16, ____, l √−81, , ____, , k √−25, , __, , c √−4, , h √−49, ____, m √−64, , ___, , d √0, __, i − √1, , e √16, __, j √1, , n −√100, , o √−1, , ____, , ___, , B Recurring decimals, 3, , 1, , As a decimal, _8 = 0.375 and _3 = 0.333 33 …, 3, , • When converted to a decimal, the fraction _8 terminates. That, is, the digits after the decimal point stop after 3 places have, been filled. We call this a terminating decimal., , When converted to a decimal, all, fractions either terminate or recur., , 1, , • When the fraction _3 is converted to a decimal, the digits after the decimal point keep repeating or recurring., We call this a recurring decimal., ., 0.3333… is written 0. 3., We call this dot notation., The dot above the 3 indicates that this digit recurs., , S, F, O, O, R, , EXAMPLE 1, Write the following recurring decimals using dot notation., a 0.4444 …, b 0.411 11 …, d 0.415 415 415 …, e 0.415 341 534 153 …, ., , a 0. 4, . ., d 0. 41 5, , P, E, AG, , P. ., c 0. 4 1, D, E, T, C, , ., , b 0.4 1, . ., e 0. 415 3, , RE, R, O, , c, , 0.414 141 …, , The dots are put above the first and last, digits of the group of digits that repeat., , Exercise 11B UNC, , 1 Write the following recurring decimals using dot notation., a 0.7777 …, b 0.355 55 …, c 0.282 828 …, e 0.678 467 846 784 … f 1.4444 …, g 6.922 22 …, i 0.234 234 234 …, j 0.033 33 …, k 0.909 090 …, m 0.217 77 …, , 11005 Photo of an aerial view of an, , NUMBER & ALGEBRA, , iron ore train or similar (say Pilbra), , 4, , Insight Maths 9 Australian Curriculum, , d 0.325 325 325 …, h 0.494 949 …, l 0.536 666 …

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EXAMPLE 2, Use your calculator to convert the following fractions to decimals., , a, , 5, _, 8, , b, , 2, _, 3, , a By calculating 5 ÷ 8 or using the fraction key, _58 = 0.625., b By calculating 2 ÷ 3 or using the fraction key, the display could show 0.666666666 or 0.666666667,, depending on the calculator used., ., Both are approximations for the recurring decimal 0. 6. In the first case the calculator has truncated the, answer (because of the limitations of the display), and in the second case the calculator has automatically, rounded up to the last decimal place., ., 2, Hence _3 = 0. 6., , 2 Convert the following fractions to decimals., a, , 7, _, 8, , 5, _, 9, 11, __, 18, , b, , f 1_23, , g, , c, h, , 1, _, 6, 22, __, 33, , 2, d ___, 11, , 5, e 1__, 12, , 13, i 1__, 22, , j, , S, F, O, O, R c 0.148, , EXAMPLE 3, Convert the following decimals to fractions., a 0.8, b 0.63, 8, 4, _, a 0.8 = __, 10 = 5, , E, T, C, , RE, R, O, , C, N, U, , P, E, AG, , DP, , 63, b 0.63 = ___, 100, , 3 Convert the following decimals to fractions., a 0.6, b 0.78, c, , 11, __, 24, , 148, c 0.148 = ____, 1000, 37, , = ___, 250, , d 0.08, , 0.125, , e 0.256, , EXAMPLE 4, Convert the following recurring decimals to fractions., ., .., a 0. 4, b 0. 5 7, ., Let n = 0. 4., n = 0.4444 …, Then, 10n = 4.4444 …, By subtraction, 9n = 4, 4, Hence, n = __, 9, ., 4, ∴ 0. 4 = _9, , b, , .., Let n = 0. 5 7, n = 0.575757 …, Then, 100n = 57.575757 …, By subtraction, 99n = 57, 57 19, Hence, n = ___ = ___, 99 33, .., 19, __, ∴ 0. 5 7 = 33, , 4 Convert the following recurring decimals to fractions., ., ., ., a 0. 2, b 0. 3, c 0. 5, , ., , d 0. 8, , NUMBER & ALGEBRA, , a, , Before subtracting, multiply by the, power of 10 that makes the decimal, parts the same., , ., , e 0. 7, , ., , 5 Convert 0. 9 to a fraction. Discuss the result with your class., , Chapter 11 Surds and indices, , 5

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6 Convert the following recurring decimals to fractions., .., .., .., a 0. 4 6, b 0. 9 1, c 0. 3 0, , .., , .., , d 0. 6 3, , e 0. 9 8, , 7 Convert the following recurring decimals to fractions., . ., . ., . ., a 0. 58 6, b 0. 23 9, c 0. 85 2, . ., . ., d 0. 42 3, e 0. 61 5, , Hint: Before subtracting, multiply by 1000., , EXAMPLE 5, Convert the following recurring decimals to fractions., ., ., a 0.3 5, b 0.512, , a, , ., Let n = 0.3 5., n = 0.35555 …, Then, 10n = 3.5555 …, and, 100n = 35.5555 …, By subtraction, 90n = 32, 32 16, Hence, n = ___ = ___, 90 45, ., 16, __, ∴ 0.35 = 45, , b, , Make the decimal parts the, same before subtraction., , ., Let n = 0.512, n = 0.512222 …, Then, 100n = 51.2222 …, and, 1000n = 512.2222 …, By subtraction, 900n = 461, 461, Hence, n = ____, 900, ., 461, ∴ 0.512 = ___, 900, , 8 Convert the following recurring decimals to fractions., ., ., ., a 0.3 8, b 0.65, c 0.92, , ., P, E, AG, , P, D, E, T, C, , 9 Convert the following recurring decimals to fractions., ., ., ., a 0.546, b 0.723, c 0.762, , S, F, O, O, R, , d 0.1 6, , ., , d 0.905, , ., , e 0.0 9, ., , e 0.049, , RE, R, O, , In general:, Step 1: Let n equal the recurring decimal., Step 2: Multiply n by the positive power of the decimal place before the first repeating digit., Step 3: Multiply n by the positive power of the decimal place of the last repeating digit., Step 4: Subtract Step 2 from Step 3., Step 5: Solve for n as a fraction., , C, N, U, , 10 Convert. .the following recurring decimals to fractions, using the. .multiples of n given., a 0.057 (10n and 1000n), b 0.3045 (100n and 10 000n), . ., . ., c 0.2205 (n and 10 000n), d 0.1275 (10n and 10 000n), , NUMBER & ALGEBRA, , C Real number system, Real numbers are those that can be represented by points on a number line. Real numbers are either rational, or irrational., • A rational number is a real number that can be expressed, a, as the ratio __ of two integers, where b ≠ 0., b, • An irrational number is a real number that is not rational., , 6, , Insight Maths 9 Australian Curriculum, , Think: rational means 'ratio-nal'.

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EXAMPLE 1, Show that the following are rational numbers., , a 2_34, , b 0.637, , ., , c 3, , e −3.1, , d 0. 4, , Any terminating or recurring decimal is, 11, a rational number., 4, a, 3, __, _, This is in the form , where a and b are integesr; hence 2 4 is a rational number., b, , a 2_34 = ___, , 637, b 0.637 = ____, 1000, , a, This is in the form __, where a and b are integers; hence 0.637 is a rational number., b, , c 3 = _31, , a, This is in the form __, where a and b are integers; hence 3 is a rational number., b, ., d 0. 4 = _49, ., a, This is in the form __, where a and b are integers; hence 0. 4 is a rational number., b, 1, e −3.1 = −3__, 10, 31, , = −__, 10, , S, F, O, O, R, , a, This is in the form __, where a and b are integers; hence −3.1 is a rational number., b, , P, E, AG, , Exercise 11C, 2, _, 3, , ___, , 2, , a, , 3, _, 5, , d, , 1, _, 7, , 1, _, 2, , converted to a decimal, the decimal, either terminates or recurs., , b 1_58, , c 4_23, e, , __, , 5, __, 12, , f 69%, , g 6.5%, , h 17_23 %, , EXAMPLE 2, Convert the following real numbers to decimals and discuss whether they are rational or irrational., __, , a √2, , Using a calculator:, __, , a √2 = 1.414 213 562 …, , __, , b √5, , __, , NUMBER & ALGEBRA, , 1, , P, a, D, Show that the following are rational numbersE, by expressing them in the form __., b, T, ., C, a 4, b 0.91, c, 5, d 0.7, e −5, f 2.84, E, R, .., 4, R i √16, g 0.53, h −2.6, j √__, k 30%, l 7.3%, O, 9, C, UN rational numbers to decimals., Convert the following, When a rational number is, , b √5 = 2.236 067 978 …, , Since neither decimal terminates or recurs (although we can show answers to only 9 decimal places,, the limit of the calculator display) these numbers cannot be expressed as the ratio of two integers and, hence are not rational. They are irrational numbers., , Chapter 11 Surds and indices, , 7

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EXAMPLE 3, Determine whether the following real numbers are rational or irrational., ___, , __, , a √6, , √___49, 16, , b, , __, , a √6 = 2.449 897 43 …, Since__the decimal neither terminates nor recurs, it cannot be expressed as the ratio of two integers,, so √6 is an irrational number., ___, , b, , √___49 = __7 (since __7 × __7 = ___49), 16, , 4, , 4, , 16, , 4, , ___, , √, , 16, a, This is in the form __, where a and b are integers, so ___ is a rational number., 49, b, , 3 Determine whether the following real numbers are rational or irrational, __, , __, , a √8, , ___, , ___, , b √9, , c √11, , d, , ___, , √, , 4, ___, 25, , e, , √___16, 5, , EXAMPLE 4, , S, F, O, O, R, , __, , Using a calculator √6 = 2.449 489 743 … Write true or false for the following statements and discuss., __, , __, , __, , a √6 = 2.44, , b √6 = 2.449, , c √6 = 2.4494, , a 2.442 = 5.9536, b 2.4492 = 5.997 601, c 2.44942 = 5.999 560 36, , The statement is false., The statement is false., The statement is false., , P, E, AG, , P, D, E as a decimal., written, Because √6 is irrational, its exact value cannot be, T, ECapproximations for √6 ., The values given in parts a, b and c are rational, R, R, O, C, Using a calculator, √2 = N, 1.414 213 562 … Write true or false for the following statements., U, a √2 = 1.41, b √2 = 1.414, c √2 = 1.4142, __, , __, , 4, , __, , __, , __, , __, , 5 Write rational approximations, correct to 3 decimal places, for the following., ___, ___, ___, ___, a √11, b √15, c √37, d √99, , ____, , e √151, , 6 a Using a ruler and set square, copy the diagram., P1, , 1, O, NUMBER & ALGEBRA, , −3, , −2, , −1, , 0, , 1, , 2, , 3, , b Use Pythagoras’s rule to calculate the length of the interval OP1., __, c Using a pair of compasses, with point at O, accurately mark the position of √2 on the number line., , 8, , Insight Maths 9 Australian Curriculum

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7 a Extend the diagram in question 6 as shown., P2, 1, P1, 2, 1, O, −3, , −2, , −1, , 0, , 1, , 2, , 3, , b Calculate the length of OP2., __, c Mark the position of √3 on the number line., __, , __, , __, , 8 Extend the diagram in question 7 to show the positions on the number line of √4 , √5 , √6 , …, Extension, , __, , 9 Proof that √2 is irrational, , __, , In Example 2 we cannot be certain that the decimal form, of √2 does not terminate or recur after some large, __, number of decimal places; hence it is not a proof that √2 is irrational. Work through thesg proof with your, teacher., __, __, a, Assume that √2 is rational. That is, assume that √2 can be written in the form __, where a and b are integers, b, and the fraction is written in its simplest form (that is, a and b have no common factors)., __, a, If, √2 = __, b, a2, then, squaring both sides,, 2 = __2, b, and, a 2 = 2b 2 …. (1), , S, F, O, O, R, , P, E, AG, , P, D, E, T, C, , Hence a 2 is even (any multiple of 2 is even) and therefore a is even., If a is even, then a may be written in the form 2k, where k is an integer., a = 2k, a 2 = 4k 2, Substituting a 2 = 4k 2 into (1), 4k 2 = 2b 2, b 2 = 2k 2, , RE, R, O, , C, N, U, , Hence b 2 is even and therefore b is even., , a, But if a and b are both even, the fraction __ cannot be in its simplest form, which contradicts our original, b, statement., __, a, Therefore, √2 cannot be written in the form __, where a and b are integers with no common factor., __, b, Hence √2 is not rational; it is irrational., , D Properties of surds, __, , __, , The set of irrational numbers contains numbers such as √2 , √2 , π, etc., 3, , A set is a group of objects, (numbers, letters, names, etc.)., , NUMBER & ALGEBRA, , In section C we distinguished between rational and irrational numbers., __, , Irrational numbers that contain the radical sign √8 are called surds., When working with surds we may use the following properties:, If x > 0 and y > 0,, __, , __, , (√x ) = x = √x, 2, , 2, , __, , __, , __, , √xy = √x × √y, , __, , √, , __, , x__, x__ √, ___, y = √y, , Chapter 11 Surds and indices, , 9

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Some properties of real numbers, , I, , Exercise 11I, 1 Use the fraction key on your calculator to evaluate (as fractions) the following., a, , −1, , (__52), , b, , a, 2 Show that __b, , −1, , (), , −1, , (__65), , c, , −1, , (__97), , d, , (__23), , −1, , e, , −1, , (__34), , b, = __, a., , 3 If a and b are real numbers, determine whether the following, , A counter example is an example that, demonstrates the statement is false., , statements are true or false. If false, give a counter example., (Hint: Try several different pairs of real numbers to test the truth of the statements.), a i a + b is a real number., ii a − b is a real number., iii a × b is a real number., iv a ÷ b is a real number., b i a+b=b+a, ii a − b = b − a, iii a × b = b × a, iv a ÷ b = b ÷ a, c i (a + b) + c = a + (b + c), ii (a − b) − c = a − (b − c), iii (a × b) × c = a × (b × c), iv (a ÷ b) ÷ c = a ÷ (b ÷ c), d i a×0=0, ii a + 0 = a, e i a×1=a, ii a ÷ a = 1, , S, F, O, O, R, , 4, , 5, , P, E, Gstatements are true or false. If false give, If m and n are rational numbers, determine whether the following, A, P, a counter example., ED, a m + n is always rational., b m − n is always rational., T, C, c m × n is always rational., d m ÷ n is always rational., E, R, Rcondition., Find a pair of surds that satisfy each, O, Remember:, C, ×, gives, product., a The product of the surds, is, irrational., N, U, ÷, gives, quotient., b The product of the surds is rational., c The quotient of the surds is irrational., d The quotient of the surds is rational., , 6 a Write three consecutive integers starting with y., b Hence show that the sum of any three consecutive, , Consecutive means in order, and without gaps., , integers is divisible by 3., , NUMBER & ALGEBRA, , 7 a Show that any even real number can be written in the form 2k, where k is an integer., b Show that any odd real number can be written in the form 2k + 1, where k is an integer., c Hence prove the following properties of real numbers., i The sum of any two even numbers is even., ii The sum of any two odd numbers is even., iii The sum of an even number and an odd number is odd., iv The product of two even numbers is even., v The product of an odd number and an even number is even., vi The product of two odd numbers is odd., , 22, , Insight Maths 9 Australian Curriculum

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Language in mathematics, 1 a Explain, in words, how to calculate:, i the square of a number, ii the square root of a number., b Write down the:, i square of 9, ii square root(s) of 9., c Which of the following numbers are perfect squares: 1, 4, 12, 25, 27, 200?, d Write in words:, 1, __, __, __, i m2, ii √m, iii −√m, iv m2, 2 What is a:, a real number?, , b rational number?, , c irrational number?, , 3 Use words from the list below to complete the following statements., a, b, c, d, , radical, terminating, false, rationalising, recurring, Every rational number can be expressed as a _____ or _____ decimal., Converting the denominator, of a fraction into a rational number is called _____ the denominator., __, The mathematical sign √ is called the _____ sign., A counter example is an example that demonstrates that a statement is _____., , S, F, O, O, R, , 4 Write down an example of three consecutive integers., 5 a Find the sum and the product of 5 and 9., b What is the quotient when 13 is divided by 5?, , P, E, 6 Use a dictionary to write down two meanings of each word., AG, P, a general, b property, c index, D, E, T been spelt incorrectly. Rewrite them with the correct spelling., 7 Three of the words in the following listC, have, E intejer, ratio, convurt, indixes, recur,R, fractional,, R, O, C, Terms, N, U, approximation, binomial expansion, conjugate, consecutive, convert, counter, fractional, like surds, radical sign, square, , example, general, perfect square, ratio, square root, , denominator, index law, power, rational, surd, , distributive law, indices, product, rationalise, terminating, , dot notation, integer, property, real number, undefined, , expansion, irrational, quotient, recurring, , Check your skills, ___, , C −√25 = −5, , ____, , D √−25 = −5, , NUMBER & ALGEBRA, , 1 Which of the following statements is not correct?, ___, A the square root of 25 is 5 or −5, B √25 = 5, ..., , 2 Convert 0. 312 to a fraction., 78, A ___, 25, , 103, B ____, 300, , 39, C ____, 125, , 103, D ____, 330, , C 3_34, , D −21, , 3 Which of the following is not a ___, rational number?, ___, , A √17, , B, , √___16, 9, , Chapter 11 Surds and indices, , 23