Integers Chapter 2, Page 23

The number line

Addition of integers Adding a positive integer to a positive integer gives a sum that is bigger than both the integers E.g 1: 3 + 8 = 11 The answer will always be a positive integer Adding positive numbers on the number line is indicated by a shift to the right

Adding a negative integer to a positive integer (or vice versa) gives a sum that is smaller than the positive integer and bigger than the negative integer The sum can be positive or negative E.g 2: 8 + (-3) = 5 E.g 3: 7 + (-9) = -2 Adding negative numbers on the number line or subtracting numbers is indicated on the number line by a shift to the left

Subtraction of integers A number and its additive inverse always add up to zero E.g 1: (-4) + 4 = 0 Therefore the additive inverse of (-4) is 4 Subtraction can be considered to be the addition of the additive inverse E.g 2: 6 – (+3) = 6 + (-3) E.g 3: 7 – (-4) = 7 + (4)

To do: Ex 2.1 pg 23 # 1; 2; 4a-d, i; 5a, c; 6a,c; 7a-c; 8a,g,h; 9e-g Ex 2.2 pg 26 # 1a-f; 2a-f Ex 2.3 pg 27 # 1RHS Additive inverse Ex 2.4 pg 28 #1RHS

Exercise 2.1: answers 1) -7; -5; -1; 0; 2; 4; 7 2) 24; 15; 3; -9; -21; -40 4a) 8 < 12 b) -15 < 8 c) 4 > -5 d) -3 < 6 i) 0 > -17 5a) -4; -3; -2; -1; 0; 1 5c) -6; -5 6a) 8˚C – 12 = - 4˚C 6c) 25˚C – (-7˚C) = 32˚C

Exercise 2.1: answers 7a) 3 + 4 = 7 b) -5 + 4 = -1 c) -11 +4 = -7 8a) Level 1 to level 5: Up 4 levels 8g) Level -2 to level 5: Up 7 levels 8h) Level -4 to level -3: Up 1 level 9e) - R150 – R250 = - R400 9f) - R150 + R70 = - R80 9g) - R150 + R400 – R300 = - R50

Exercise 2.2: answers 1 a) - 3 b) -13 c) 3 d) 5 e) -5 f) -19 2 a) 6 b) 18 c) -18 d) 0 e) -30 f) -11

EXERCISE 2.3: ANSWERS b) (-5) + 6 + (-11) + (-2) + 4 = -8 c) 6 + (-9) + 10 + (-3) + (-4) = 0 f) 10 + 4 + (-8) + 3 + (-6) = 3 h) (-3) + (-4) + 12 + (-2) + (-6) = -3 j) -15 +14 - 25 +25 - 5 = -6 l) 12 + 3 – 7 + 15 – 4 – 5 = 14

Exercise 2.4: answers c) 4 – 10 = 4 + (-10) = - 6 f) 6 – (-11) = 6 + (+11) = 17 i) 16 – (-8) = 16 + (+8) = 24 l) - 7 – (-9) = - 7 + (+9) = 2 o) 3 – (-18) = 3 + (+18) = 21 r) 11 – 15 = 11 + (-15) = - 4 u) - 2 – (-5) – 8 = -2 + (+5) + (-8) = -5

MULTIPLICATION AND DIVISION OF INTEGERS PAGE 29 - 32

Multiplication of integers

The multiplication sign is not needed when brackets are used: (-3) x (2) can be written as (-3)(2) As well as: -3(2) or even: (-3)2 E.g 1: (+3) x (+3) = 9 E.g 2: 3(-3) = -9 E.g 3: (-3)3 = -9 E.g 4: (-3)(-3) = 9 E.g 5: 3(-3)(-3) = 27 E.g 6: -3(-3)(-3) = -27 Even number of negatives, answer is positive. Odd number of negatives, answer is negative.

Division of integers

The rules for division work the same as for multiplication E.g 1: 8 2 = 4 E.g 2: 5 −5 = -1 E. g 3: −6 2 = -3 E.g 4: −9 −3 = 3 E.g 5: (−3)(−4) 3 = 12 3 = 4 E.g 6: −12 +8 4 = −4 4 = -1

FOR YOU TO DO: Multiplication: Exercise 2.6 pg. 31 ALL Multiplication, Addition and Subtraction: Exercise 2.7 pg. 31 #1h – n #2 ALL Division and Addition: Exercise 2.8 pg. 32 ALL Remember BODMAS

EXERCISE 2.6: ANSWERS 1a) (-2) x (-6) = 12 b) (-1) x (+12) = -12 c) 3(-5) = -15 d) (-5) x 3 = -15 e) (-9)2 = -18 f) (-11)(-7) = 77 g) 9 x (-3) = -27 h) 4(-4) = -16 i) (7)(-4) = -28 j) -2(-3)(-5) = -30 k) (-2)(-3)(-4) = -24 l) -1(-2)(-3)3 = -18

Exercise 2.7: answers 1 h) 6 + (-4) = 2 i) 6 – (-4) = 10 j) (-4)(-12) = 48 k) (-4) + (-12) = -16 l) (-4) – (-12) = 8 m) 4 – (-12) = 16 n) 4 + (-12) = -8 2 a) -2(5) = -10 b) -3(5-2) = -9 c) 5 – 2(-3) = 11 d) (-5)(-5) x 3 = 75 e) (-5)(-5) x(-5) = -125 f) -5 + (-5) x (-2) = 5

Exercise 2.8: answers a) −12 4 = -3 b) −24 −6 = 4 c) −14 7 = -2 d) −15 3 = -5 e) −12 −4 = 3 f) −25 5 = -5 g) −32 −8 = 4 h) (−5)(−4) −10 = -2 i) −5 +(−4) −3 = 3 j) 4 −8 −4 = - 𝟏 𝟑 k) −12+4 4 = -2 l) (−3)(−6) 9 = 2

SQUARES AND CUBES PAGE 33

SQUARES Eg. 1: 32 = 3 x 3 = 9 Eg. 2: (-4)2 = = Eg. 3: -62 – 6 = Eg. 4: -3(-2)2 = Eg. 5: -(7)2 - 32 = (-4)(-4) 16 - 36 - 6 - 42 - 3(-2)(-2) - 3(4) - 12 - (7)(7) – 9 - 49 – 9 - 58

cubes Eg 1: (-2)3 = = Eg 2: 23 – (-3)3 = DO: Exercise 2.9 pg. 33 # 1g – o AND: Exercise 2.10 pg. 34 ALL (-2)(-2)(-2) - 8 8 – (-3)(-3)(-3) 8 – (- 27) 8 + 27 35

Exercise 2.9: answers g) 32 + (-5)2 = 9 + (-5)(-5) = 9 + 25 = 34 h) 62 + (-1)2 = 36 + (-1)(-1) = 36 + 1 = 37 i) 42 + (-2)2 = 16 + (-2)(-2) = 16 + 4 = 20

Exercise 2.9: answers j) (-7)2 + (-3)2 = (-7)(-7) + (-3)(-3) = 49 + 9 = 58 k) (-1)2 + (-2)2 = (-1)(-1) + (-2)(-2) = 1 + 4 = 5 l) (-8)2 – (-10)2 = (-8)(-8) – (-10)(-10) = 64 – 100 = -36

Exercise 2.9: answers m) 22 – 42 = 4 – 16 = -12 n) (-2)3 + 42 = (-2)(-2)(-2) + 16 = -8 + 16 = 8 o) (-2)3 + (-4)3 = (-2)(-2)(-2) + (-4)(-4)(-4) = - 8 + (-64) = - 8 – 64 = -72

Exercise 2.10: answers a) (-3)(2) = -6 b) (-7)(5) = -35 c) 15 – (-3) = 15 + 3 d) (-12)+(-8)= -12 - 8 = 18 = - 20 e) -22 -15 = - 37 f) -13 –(-19) = - 13 + 19 = 6 g) (-11) x 5 = -55 h) -6 – 15 = -21

Exercise 2.10: answers i) 7 + (-17) = 7 - 17 j) -10 – (-14) = - 10 + 14 = -10 = 4 k) (-5)2 = (-5)(-5) l) - 8 + (-18) = -8 -18 = 25 = - 26 m) (-9)(-5) = 45 n) 13 – (-6) = 13 + 6 = 19 o) -10 – (-15) = -10 + 15 p) - 10 (-15) = 150 = 5

Exercise 2.10: answers q) - 10 - 15 = -25 r) 19 - 24 = -5 s) (-3)3 = (-3)(-3)(-3) t) -92 = -81 = -27 u) (-9)2 = (-9)(-9) v) (−2)(−3) 6 = 6 6 = 81 = 1 w) −2(8) (−4) = −16 −4 x) 3(−5)(−2) (−6) = 30 −6 = 4 = -5 y) (−3)(−4) (−2) = 12 −2 z) −3 +(−4) (−7) = −3 −4 −7 = -6 = 1

Exercise 2.11 ANSWERS a) 2(-4) + 12 – (-3) = -8 + 12 + 3 = 7 b) (-5)2 + 4(-3) – (-2) = (-5)(-5) -12 + 2 = 25 – 10 = 15 c) - 7 + (-2)(-7) – 5 = -7 + 14 - 5 = 2 d) 2(-8) + 16 + (-6) = -16 + 16 - 6 = -6

Exercise 2.11 ANSWERS e) (-3)3 – 4(-3) + (-2)2 = (-3)(-3)(-3) + 12 + (-2)(-2) = -27 + 12 + 4 = - 11 f) (9)(-3) – 6 – 10 = -27 – 6 – 10 = - 43 g) -(-3) – 6 + (-1)(5) = 3 – 6 - 5 = -8 h) 2(8) – 4(3) + (-15) = 16 – 12 - 15 = -11

Exercise 2.11: answers i) (-1)3 – 2(-3)2 + (-2)(+2) = (-1)(-1)(-1) -2(-3)(-3) – 4 = -1 - 2(9) – 4 = - 1 – 18 – 4 = -23 j) (-7)(-4) – 12 + (-16) = 28 – 12 - 16 = 0