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-1- CHAPTER 1: REAL NUMBERS 1) The Pythagorean Theorem a) The square Area of a square: A = s2 Where, A = area and s = side length E.g. A = (5 cm)2 = 25 cm2 5 cm A = (b cm)2 = b2 cm2 b cm When the area of a square is known, how can you find the side length? » by taking the square root of the area s= A -2- E.g. s= 81cm2 = 9 cm d cm2 s= d 2 cm2 = d cm 17 cm2 s= 17cm2 = 81 cm2 2 17 17 cm ≈ 4.123105626… If A is not a perfect square number (1, 4, 9, 16, 25, 36, 49…) it is preferable to leave the answer in radical form. Squaring a number and taking the square root are INVERSE operations. » Taking the square root of a number means finding the side length of a square whose area is that number. E.g. Calculate 23 ≈ ______________ 23 units2 each side ≈ 4.7958… -3- b) Triangles Equilateral: all sides/angles are equal Isosceles: two sides/angles are equal Right: contains a right angle Scalene: no sides equal Isosceles right: Area of a triangle A = base x height = b x h 2 2 h h b c) Disk Area of a disk = r2 b -4- Examples Find the area of the following figures: 1. 3 cm 8 cm A=bxh 2 A=8x3 2 A = 12 cm2 2. r r = 5 cm A = r2 A = 52 (use key on calculator) A ≈ 78.54 cm2 (78.539816339744830961566084581988…) -5- d) Right triangles The longest side of a right triangle is called the HYPOTENUSE; the two sides that form the right angle are called the LEGS. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other sides. » c2 = a2 + b2 With this formula (The Pythagorean Theorem) you can calculate the length of any one side of a right triangle when the other two lengths are known. » c2 = a2 + b2 c= a 2 b2 » a2 = c2 – b2 a= c2 b2 » b2 = c2 – a2 b= c2 a 2 -6- c= a 2 b2 c= 32 4 2 c= 9 16 c= 25 = 5 b= c2 a 2 b= 132 112 b= 169 121 b= 48 ≈ 6.93 A triangle with sides that satisfy the relation c2 = a2 + b2 must be a right triangle. E.g. Would a triangle with sides measuring 9 cm, 12 cm and 15 cm be a right triangle? ? a2 + b2 9 12 = c2 ? 2 + 2 = 152 ? 81 + 144 225 = = 225 225 YES! It is a right triangle -7- E.g. Would a triangle with sides measuring 7 cm, 24 cm and 26 cm be a right triangle? ? a 2 + b = 2 c2 ? 72 + 242 = 262 ? 49 + Sets of Pythagorean triples. 576 625 = 676 ≠ 676 NO! It is not a right triangle. three numbers that Theorem are called satisfy the Pythagorean E.g. (9, 12, 15) form a Pythagorean triple because 92 81 + + 122 144 225 = 152 = 225 = 225 Complete the following Pythagorean triple: (15, 20, ?) 152 225 + + 202 400 625 = ?2 = ?2 = ?2 625 = 25 » (15, 20, 25) -8- e) The Pythagorean Theorem and the Cartesian plane 2 points on a grid represent the end points of the hypotenuse of a right triangle. The length of the hypotenuse is calculated using the Pythagorean Theorem. E.g. Workout p. 184 nos. 1-3, 6, 8, 9, 13, 15, 17, 20, 22 to 25, 27 to 29, 33, 34, 36, 37, 39, 41, 46, 49, 50, 51 Workbook, p. 4 act. 1 a, b, nos. 1 to 16, act. 2 a, b, nos. 17 to 19 -9- 2) Rational numbers a) Sets of numbers : the set of natural numbers = {0, 1, 2, 3, 4 …} * : the set of natural numbers excluding zero. : the set of integers = {… -3, -2, -1, 0, 1, 2, 3, 4 …} +: the set of positive integers = {0, 1, 2, 3, 4 …} -: the set of negative integers = {… -3, -2, -1, 0} * : the set of integers excluding zero. Other types of numbers: - Fractions: ½ ¾ -2/3 … - Decimals: 0.2 0.375 -2.25 … x 100% DECIMAL NUMBER PERCENT n% n ÷100 a÷b FRACTION a/b reduction of n 100 - 10 - When you solve an equation, the solution can be shown on a number line, or by using a Venn diagram. E.g. 3x + 6 = -3 3x + 6 – 6 = -3 – 6 3x = -9 3x = -9 3 3 x = -3 -4 -3 -2 -1 0 1 2 3 4 5 6 We can write that –3 Workbook, p. 9 act. 1a) – d), act. 2a) – d) b) Rational numbers ( ) - - 11 - They are numbers that can be written as a fraction a/b, where a and b *. E.g. 5= -2 = 0.3 = the set of rational numbers + = the set of positive rational numbers - = the set of negative rational numbers * = the set of rational numbers excluding zero » Natural numbers are included in integers which are included in rational numbers. - 12 - E.g. Complete the following by using , , or : 5 ____ ____ 2/3 ____ - ____ -4 ____ * _____ 0.68 ____ _____ + + Workbook, p. 10, nos. 1 to 3 + p.11, act.3 a)-c) Rational numbers can also be defined this way: » numbers with repeating decimals whose period can be zero or non-zero. A bar is placed over the digit(s) of the first period after the decimal point. _ E.g. 4 = 0.4444… = 0.4 9 __ 7 = 0.63636363… = 0.63 11 _ 16 = 0.6400000… = 0.640 = 0.64 25 _ 5 = 5.0000… = 5.0 = 5 1 - 13 - » It is not necessary for these numbers to be written in the form a/b for them to be rational, but it MUST be possible to write them in this form. Workbook, p. 11, nos. 4 a) – g), act.4 a) – b) c) Method to determine the corresponding fraction of a rational number __ METHOD a = 1.12 EXAMPLE b 1. Write an equation expressing the number in decimal notation 2. Multiply both sides of the equation by the power of 10 that moves the decimal point: - after one period - before the first period 3. Subtract the second equation from the first 4. Solve the equation to find the fraction n = 1.121212… 100n = 112.1212… n= 1.1212… - 100n = 112.1212… n= 1.1212… 99n = 111 99n = 111 99 99 n = 111 = 37 99 33 - 14 - E.g. 1. _ 6.16 n = 6.166666… 2. 100n 10n = 616.66666… = 61.66666… 3. 90n = 555 4. 90n = 555 90 90 = 37 6 ____ 0.629 1. n = 0.629629629… 2. 1000n = 629.629629… n = 0.629629… 3. 999n = 629 4. 999n = 629 999 999 Workbook, p. 13, nos. 5 to 8 = 17 27 - 15 - d) Representing a rational number on the number line A rational number can be represented approximately on the number line by rounding up or down the decimal form of the number. E.g. Show 726/160 on the number line. The decimal form is 726 ÷ 160 = 4.5375 If we round (up or down) to the nearest tenth we get: 4.5 < 4.5375 < 4.6 4.0 5.0 We could decide to round to the nearest hundredth: 4.53 < 4.5375 < 4.54 4.50 Workbook, p.13-14, act.5 4.55 4.60 - 16 - To represent a rational number precisely on the number line (for example 11/6), follow these steps: 1. We have: 11/6 = 1 + 5/6 2. Knowing that 1 < 11/6 < 2, draw a half-line starting at 1 on the number line (the angle between this half-line and the number line is not important). 0 1 2 3 3. Scale this line using 6 sections, and complete a triangle by connecting the last section (the 6th one) to the 2, on the number line. 0 1 2 6 3 - 17 - 4. From the 5th section on the half-line, draw a line parallel to the line that was last drawn to complete your triangle. The intersection of this line and the original number line is the point corresponding to 11/6. 11/6 0 1 2 5 6 E.g. Locate 12/7 on the number line. Workbook, p.14, act.6 1)-4), nos. 9 - 12 3 - 18 - 3) Irrational numbers a) The set of irrational numbers ( | ) It is possible to imagine numbers that have nonperiodic decimals, and that are non-terminating: E.g. » » » 0.123456789101112131415… impossible to cancel the period impossible to write it as a fraction (a/b) it is not a rational number! Classic examples: ≈ 3.1415926535… (et pis après…) 1 1 2 1 2 ≈ 1.4142135623… 1 » there is no rational number whose square is 2 or » 2 , no period appears the decimal form is only an approximate value of 2 - 19 - Numbers that have non-terminating, repeating decimals are irrational numbers. non- These numbers are the result of square roots, cube roots, etc. which cannot be taken precisely. Workbook, p.16, act.1 a)-d), p.17, nos. 1-2 b) Locating an irrational number on the number line Gaps in the number line correspond to irrational numbers. Each point on the line corresponds to one, and only one, rational or irrational number. E.g. Locate 13 on the number line Use the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, … Find two perfect squares that when added or subtracted will give you 13. 13 = 49 13 = 2 2 32 This means that the legs of the right triangle are 2 and 3, and the hypotenuse is 13 . - 20 - 13 0 1 2 2 3 4 5 13 E.g. Locate 21 on the number line Use the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, … Find two perfect squares that when added or subtracted will give you 21. 21 = 25 4 21 = 52 2 2 This means that the legs of the right triangle are 2 and 21 , and the hypotenuse is 5. - 21 - 0 1 2 Workbook, p.17, act.2 3)-4), p.17, nos. 3-10 3 4 5 - 22 - 4) Real numbers ( ) a) The set of real numbers ( ) The set of all rational and irrational numbers is called the set of real numbers. | -9/10 2 -15 17/5 -1 ½ 0 8 -6 0.57 1 5 -2/3 - 3 2 5 -2.54 = the set of all positive real numbers - = the set of all negative real numbers * = the set of real numbers excluding zero + Workbook, p.21, act.1, p.22, nos. 1-3 b) Real number line Every real number corresponds to a unique point on the number line. This is why the number line is called the real number line. Workbook, p.23, act.2 a)-b), nos. 4-3 - 23 - 5) Intervals a) Intervals as line segments Any set of real numbers that is represented by a segment on the real number line is called an interval. E.g. [2, 5[ lower limit upper limit 0 1 2 3 This interval is closed on the left, and open on the right (which means that 2 is included but not 5). 4 5 6 7 This interval can also be shown using the setbuilder notation: {x 2 x 5} - 24 - Intervals 1) Closed 0 1 2) Closed on the left, open on the right 0 1 3) Open on the left, closed on the right 0 {x [2, 5] 1 4) Open 2 3 4 {x [2, 5[ 2 3 4 3 4 1 2 3 5 {x ]2, 5[ 0 5 {x ]2, 5] 2 5 4 5 2 x 5} 6 7 2 x 5} 6 7 2 x 5} 6 7 2 x 5} 6 7 - 25 - b) Interval represented by a ray Any set of real numbers that is represented on the real number line by a ray is called an interval. E.g. Represent on the number line the set of numbers that are greater than or equal to 2. [2, +∞ [ lower limit plus infinity 0 1 2 3 This interval is closed on the left, and there is no upper limit. 4 5 6 7 This interval can also be shown using the setbuilder notation: {x x 2} - 26 - Intervals 1) Unlimited on the left, and closed on the right -2 -1 0 2) Unlimited on the left, and open on the right -2 -1 0 3) Closed on the left, and unlimited on the right -2 -1 -1 1 2 3 1 2 0 1 2 3 1 2 4 {x 3 4 {x ]2, +∞[ 0 4 {x ]- ∞, 2[ [2, +∞[ 4) Open on the left, and unlimited on the right -2 {x ]- ∞, 2] 3 4 x 2} 5 6 x 2} 5 6 x 2} 5 6 x 2} 5 6 - 27 - By convention, the empty set (), any single element, and the set of real numbers ( , are all considered intervals. Workbook, p.24, act.1, + p.25, act.2, p.26, nos. 1-7 - 28 - 6) Natural number exponents a) Power of a real number Expressions such as 219 or ma are called exponential expressions. Such expressions are written according to convention. When a number is written in exponential notation it is in exponent form. (base)(exponent) = (power) E.g. exponent 23 = 8 base power Some exponent rules: 1. For any base m and any integral exponent a > 1 ma = m • m • m … • m (a times) 73 = 7 • 7 • 7 x4 = x • x • x • x (-5)2 = (-5) • (-5) = 25 (-5)3 = (-5) • (-5) • (-5) = -125 » ma is negative when the base “m” is negative and the exponent “a” is odd. - 29 - 2. For base m and exponent 1 m1 = m 91 = 9 y1 = y 3. For base m ≠ 0 and exponent 0 m0 = 1 Workbook, p.28, act.1, nos. 1-11 50 = 1 n0 = 1 (n ≠ 0) - 30 - Workbook, p.30, act.2 a), b), c) b) Multiplying two powers with the same base The exponent of the product of two powers with identical bases is the sum of the powers’ exponents. The base m is not 0 and the exponents a and b are integers. ma • m b = ma + b E.g. 5 2 • 53 = 52 + 3 = 55 Same base Add exponents 3 4 • 35 = 34 + 5 = 39 24 • 2-2 = 24 + -2 = 22 10-2 • 10-3 = 10-2 + -3 = 10-5 IMPORTANT AVOID THESE COMMON MISTAKES! 52 + 53 ≠ 55 24 • 23 ≠ 212 42 • 53 ≠ 205 Workbook, p.30, nos. 1 to 3 » not a multiplication! » do not multiply the exponents! » not the same base! - 31 - Workbook, p.31, act.3 a), b), c) c) Dividing two powers with the same base The exponent of the quotient (÷) of two powers with the same base is the difference between the powers’ exponents. The base m is not 0 and the exponents a and b are integers. ma ÷ mb = ma – b or ma = ma – b mb E.g. 55 ÷ 52 = 55 - 2 = 53 Same base Subtract exponents 34 ÷ 31 = 34 - 1 = 33 26 ÷ 2-3 = 26 - -3 = 29 10-3 ÷ 10-1 = 10-3 - -1 = 10-2 (explain***) 105 = 105 - 3 = 102 103 IMPORTANT AVOID THESE COMMON MISTAKES! 56 – 52 ≠ 54» not a division! 56 ÷ 52 ≠ 53 » do not divide the exponents! 42 ÷ 22 ≠ 22 » not the same base! Workbook, p.32, nos. 4 to 6 - 32 - Workbook, p.32, act.4, a), b), c) d) Power of a power To find the power of a power, you need to multiply the exponents. (ma)b = ma x b E.g. (55)2 = 55 x 2 = 510 Multiply exponents (34)3 = 34 x 3 = 312 ((-4)3)5 = (-4)3 x 5 = (-4)15 ((103)2)4 = 103 x 2 x 4 = 1024 Workbook, p.33, nos. 7 to 10 - 33 - Workbook, p. 33, act.5, a), b) e) Power of a product To find the power of a product, you need to distribute the exponent. (m • n)a = ma • na E.g. (5 • 2)2 = (5 • 2) • (5 • 2) = 5 • 5 • 2 • 2 = 52 • 22 (3 • 4)5 = 35 • 45 (4x)2 = 42 • x2 = 16x2 (3y2)4 = 34 • y8 = 81y8 Workbook, p.34, nos. 11 to 13 - 34 Workbook, p.34, act.6, a), b) f) Power of a quotient To find the power of a quotient, you need to distribute the exponent. a m m a n n a 4 E.g. c4 c 4 d d 3 3 3 x 27 x 4 64 3 3x 27 x 6 3 8y9 2y 2 4x 3y 2 3 2 3x 64 x 6 9 x 4 576x10 16x10 . 3 . 6 3 9 27 y 4 y 108 y 3y 9 2y Workbook, p.35, nos. 14 to 17 2 - 35 - Workbook, p.36, act.1 a) – i) 7. Negative exponents For any base m ≠ 0 and exponent a m-a = 1 ma E.g. 1 (t ≠ 0) t2 t-2 = 4-3 = 1 = 1 43 64 (-4)-2 = 1 =1 (-4)2 16 (-5)-3 = 1 =-1 (-5)3 125 2 42 32 9 4 2 2 3 4 16 3 2 2a 2x 2 2 x 2 32 y 2 9 y 2 2 2 2 2 2 3 y 2 x 4x 3y Workbook, page 36, nos. 1 to 13 3 3 2 3 a 9 1 1 23 a 9 8a 9 Workbook, page 39, act. 1 a) – c) - 36 - 8. Scientific notation a) Product of a real number by a power of 10 Given a real number “a” and a natural number “n”: 1- To determine the product (a x 10n) the decimal point is moved “n” decimal places to the right. 2- To determine the product (a x 10-n) the decimal point is moved “n” decimal places to the left. E.g. 3.459 x 102 = 345.9 125.24 x 10-2 = 1.2524 0.025493 x 105 = 2549.3 12.87 x 10-4 = 0.001287 Workbook, page 39, nos. 1 to 3 - 37 - b) Scientific Notation Scientific notation is an easy and practical shortcut to writing very large or very small numbers. Scientific notation uses the powers of 10. A positive number in scientific notation is in the form a x 10n where 1 ≤ a < 10 and n is an integer. E.g. 45 000 = 4.5 x 104 235 = 2.35 x 102 1 220 000 = 1.22 x 106 0.025 = 2.5 x 10-2 0.000 045 = 4.5 x 10-5 - 2 315 = -2.315 x 103 -0.0069 = -6.9 x 10-3 DECIMAL NOTATION SCIENTIFIC NOTATION - 38 - ** Calculators (Learn how to use your calculator) Key for scientific notation » E or EE or EXP To enter 2.5 x 103 » 2.5 KEY 3 Display: 2.5 E 3 or 2.5 03 or or 2.5 03 or 2.5 1003 Calculator display 2.5 EE 03 Answer 3.4575 1003 3.4575 x 103 2.154 -04 2.154 x 10-4 Prefixes Deca Hecto Kilo Mega Giga Tera 101 102 103 106 109 1012 Workbook, page 40, act.2 + nos. 4 to 10 Deci Centi Milli Micro Nano Pico 10-1 10-2 10-3 10-6 10-9 10-12 - 39 - 9. Rational number exponents a) nth root of a real number The nth root of the real number “a” written as is the unique real number “b” such that bn = a. E.g. N.B. 2 36 6 because 62 = 36 3 64 4 because 43 = 64 3 125 5 4 81 3 4 16 n n a because (-5)3 = -125 because 34 = 81 does not exist in a does not exist when the index “n” is even and “a” is negative. Workbook, page 42, act.1 and 2 + nos. 1, 2 - 40 - b) Rational number exponents If “a” is a real number and “n” is a natural number, we have: 1 n a n a E.g. 1 2 49 2 49 7 1 3 1000 3 1000 10 1 3 ( 343) 3 343 7 1 4 ( 625) 4 625 does not exist in N.B. 1 n a does not exist when the “n” is even and the base “a” is negative. Workbook, page 43, act.3 + nos. 3 to 6 Workbook, page 44, Evaluation 1, nos. 1 to 15 Workbook, page 3, Challenge 1, nos. 1 to 7