Computation and Directed Numbers

Quick Review

1. Divisibility

When a number (first number) is divided by another (second number) and no remainder is obtained, we say that the first number is divisible by the second number.

A. Divisibility test of 4

Check whether the last two digits of a natural number form a number divisible by 4.

B. Divisibility test of 8

Check whether the last three digits of a natural number form a number divisible by 8.

C. Divisibility test of 3

Check whether the sum of all digits of a natural number is divisible by 3.

D. Divisibility test of 9

Check whether the sum of all digits of a natural number is divisible by 9.

E. Divisibility test of 6

Check whether a natural number is divisible by both 2 and 3.

2. Index Notation

Repeated multiplication of a number can be represented by index notation（指數記數法）. For example, 54 _{= 5 × 5 × 5 × 5}

3. Prime Factorization

Prime factorization（質因數分解）is the process of expressing a number as a product of prime factors.

Determine whether each of the following is true (1 – 6).

1. 7 × 7 × 7 × 7 × 7 can be written as 75_.

T / F

2. A number divisible by 4 must be

divisible by 8. T / F

3. The sum of three negative numbers

can be a positive number. T / F

4. -11 is larger than -15. T / F

5. The product of a positive number and

a negative number must be a negative number. T / F 6. - 34 is smaller than -43. T / F 7. (a) (-4.5) + (-2.5) = (b) (-4) × (-2) = Ans: (a) (b) Concept Check

1

Basic Computation and

Directed Numbers

Basic Computation and Directed Numbers ₃

7. Arithmetic Operations of Directed Numbers

A. Addition and Subtraction of Directed Numbers

(a) +(+a) = +a (b) -(+a) = -a

(c) +(-a) = -a (d) -(-a) = +a

E.g., +5 + (+3) + (-2) = + 5 + 3 - 2 = 6, -4 - (+3) - (-2) = - 4 - 3 + 2 = -5 B. Multiplication and Division of Directed Numbers

(a) (+a) × (+b) = +(a × b) (b) (+a) × (-b) = -(a × b) (c) (-a) × (+b) = -(a × b) (d) (-a) × (-b) = +(a × b) (e) +₊a = + b ab (f) + −ab = −ab (g) −₊a_b = −a_b (h) −₋a_b = +_ba E.g., +5 × (+8) = 40, +2 × (-3) = -6, −₊9 = − 3 3, −−2814 = 2

Mistake Hunt

We should apply the rules of removing brackets to simplify + (-3). Hence the correct solution is as follows:

-5 + (-3) = -5 - 3 = -8

!!

We should do the operation inside the square brackets first. Hence the correct solution is as follows:

-3 × [-2 + (-1)] = -3 × (-2 - 1) = -3 × (-3) = 9

!!

Study each of the following and determine whether the solution of each question is correct. If it is not, correct the solution.

1. Calculate (–6) – (–5). Solution: (-6) - (-5) = +(6 + 5) = 11 2. Calculate -4 × [-1 + (-7)]. Solution: -4 × [-1 + (-7)] = (-4) × (-1) + (-7) = +4 - 7 = –3

Level-up Practice

Multiple-choice Questions

1. Which of the following numbers is NOT divisible by 8?

A. 2832 B. 5736 C. 6524 D. 8496 2. The L.C.M. of 23 _{× 3 × 5}2 _{and 2}2 _{× 3}3 _{× 7 is} A. 22 _{× 3. B. 2}3 _{× 3}3_. C. 22 _{× 3}3 _{× 5}2 _{× 7. D. 2}3 _{× 3}3 _{× 5}2 _{× 7.} 3.

Referring to the above number line, A - B =

A. -8. B. -4.

C. 4. D. 8.

4. Which of the following expressions gives a positive number as the answer?

A. (-2) + (-3) B. (-3) - (-2) C. − × −2 ( 3)₋₄ D. − + −2 ( 3)₋₄

5. Which of the following negative numbers is the smallest?

A. - 2

3 B. - 25

C. - 2

7 D. - 2₉

6. Peter uses a positive number to represent an amount deposited in a bank and a negative

number to represent an amount withdrawn from the bank. Use a directed number to represent the deposit of $1000 and then a withdrawal of $200.

A. +1000 - (-200) B. +1000 + (-200) C. -1000 - (-200) D. -1000 + (-200)

7. In a shooting game, a player scores 2 points when hitting the target and scores -1 point when

missing the target. A player shoots for 10 shots, but the target is hit 3 times only. What is his score?

A. 13 points B. 6 points C. -1 point D. -13 points

Basic Computation and Directed Numbers ₅

Basic Questions

8. Determine whether 6713 is divisible by 4.

9. Determine whether 4104 is divisible by 6.

10. Determine whether 7240 is divisible by 8.

11. Determine whether 4005 is divisible by 9.

12. Use index notation to represent each of the following expressions.

(a) 2 × 2 × 2 × 2 × 2 (b) 7 × 3 × 7 × 7 × 3 × 3 × 3 × 3

13. Express 88 as a product of prime factors. (Give your answer in index notation.)

14. Find the H.C.F. and L.C.M. of 48 and 54 by prime factorization.

Calculate each of the following expressions (15 – 16).

15. (6 + 20 ÷ 4) × 3 16. (175 - 55) ÷ (12 × 2)

Self-assessment

1. Arrange the numbers -3, 1.5, - 5₂ and 2 in ascending order.

2. Calculate the following expressions.

(a) (-4) + (-3) - (-2)(-1) (b) [(-2)(-3) - (+3)(+1)] ÷ (-3) Calculate each of the following expressions (33 - 37).

33. 125 - [2 + 5 × (3 + 15)] 34. 650 ÷ [(13 + 27) ÷ 8] × (17 - 13)

35. [-5 - (+2)] ÷ [(-4)(+2) - (-1)] 36. −₋ − + − − + ( 32)( 9)

( 36) ( 4)( 6)( 2) 37. ( 3)( 3)( 4)( 4)− − + +( 6)( 6)− −

38. The h ighest and the lowest temperatures of a city yesterday were 2°C and -8°C respectively.

The highest temperature increases by 3°C and the lowest temperature decreases by 3°C today.

(a) Find the highest and the lowest temperature today.

(b) Find the difference between the highest and the lowest temperature yesterday. (c) Find the difference between the highest and the lowest temperature today.

39. In Beijing, the temperature in the morning is 5°C. In the afternoon, the temperature rises

by 2°C. At night, the temperature falls down to -10°C. Find the difference between the temperatures in the afternoon and at night.

Train Up Mathematics 1 (S1 to S2) 8

Internet Adventure

Knowledge Extension Corner

3. Ann has an Octopus card with an amount of $13. She buys some fast food which costs $22.

Then she re-loads the card with $100. What is the amount on the card now?

There are two common ways to do the arithmetic operations for fractions of directed numbers. Consider −4 1₄ +2 5₆ − −_ 12_{3 .}_ Method 1: − + − −_ _ = − − + + + + = − + + + − 4 1₄ 2 5₆ 12₃ 4 1₄ 2 5₆ 1 2₃ 4 2 1 1 ( ) ₄₄ 5₆ 2₃ 1 ₁₂3 10_{12 12}8 1 15₁₂ + +    = − + −_ + + _ = − + = −11 1 1₄ 1 4 + =

Work it out

Calculate the following using the above two methods (1 – 2).

1. −11₅ +2 2₅ 2. −3 1₂ +4 1₃+ −_ 2 3_4 Method 2: − + − −_ _ = − + + = − + + = − 4 1₄ 2 5₆ 12₃ 4 1₄ 2 5₆ 12₃ 17 4 176 53 511 12 3412 2012 3 12 1 4 + + = = Separate the integers and fractions. Convert to improper fractions. Number Game

Let’s try the following number game to master your techniques in doing operations of directed numbers.

1. Which of the following statements is

INCORRECT?

A. If the sum of all digits of an even number is divisible by 9, then it must be divisible by 6. B. If the tens digit of an even

number is an odd number, then it must not be divisible by 4.

C. A number divisible by 9 must be divisible by 3.

D. An even number divisible by 5 must be divisible by 10.

2. How many common multiples of 12, 15

and 20 lie between 500 and 700? A. 2 B. 3 C. 4 D. 5 3. (–3) + (–6) – (–9) + (–12) = A. –30 B. –12 C. 0 D. 24 4. If a = 3 and b = 4 , then ab−₊ = b 2 2 2 A. 5₄ . B. ₄3. 6

Section A

Choose the most suitable answer from the four choices.

5. Solve −3₅x = +x 4 . A. x = –10 B. x = –5 C. x = − 5 2 D. x = − 1₂

6. When a number is decreased by 20%, it

becomes 100. What is the number? A. 80

B. 120 C. 125 D. 150

7. A buys an article at $50 and sells it to B

with a 40% profit. B sells it to C at a loss of 10%. At what price does C buy the article?

A. $80 B. $70 C. $63 D. $60

8. Round off 1357.94 cm to the nearest m.

A. 13.5794 m B. 1357 m C. 13.57 m D. 14 m

Train Up Mathematics 1 (S1 to S2) 70

A

Acute angle（銳角）: An angle with size between 0° to 90° (not including 0° and 90°) p. 28

B

Bar chart（棒形圖）: A statistical graph using bars to denote the frequencies of the data p. 55 Broken line graph（折線圖）: A statistical graph that shows the change in frequencies of the data over a period of time p. 56

C

Cost（成本）: An amount needed to get a product p. 18

D

Directed numbers（有向數）: A set of numbers including positive and negative numbers, e.g., 4, -7

p. 2 Discount（折扣）: An amount decreased from the original price p. 18

E

Equation（方程）: A mathematical expression containing an equal sign, e.g., 3x + 7 = 4 p. 9

F

Formula（公式）: A symbolic expression that usually represent a rule or principle in mathematics,

e.g., Area(A) = b × h p. 9

G

General term（通項）: An expression that can represent the nth term of a sequence p. 10

H

Highest common factor（最大公因數）: The largest common factor of two or more natural numbers

p. 2

I

Index notation（指數記數法）: A notation to represent repeated multiplication of a number p. 1

L

Least common multiple（最小公倍數）: The smallest common multiple of two or more natural

numbers p. 2

Loss（虧蝕）: An amount lost from selling a product p. 18

M

Marked price（標價）: A price marked originally p. 18

N

Negative number（負數）: Numbers which are smaller than 0, e.g., -2, -4.6 p. 2 Number line（數線）: A straight line with numbers marked at equal distance p. 2

O

Obtuse angle（鈍角）: An angle with size between 90° to 180° (not including 90° and 180°) p. 28

P

Parallel lines（平行線）: Two (or more) lines on the same plane do not intersect no matter how far they

extend p. 28

Percentage（百分數）: A fraction with 100 as its denominator, e.g., 25%, 3.12% p. 17 Perpendicular lines（垂直線）: Two lines on the same plane meet at a right angle p. 28 Pictogram（象形圖）: A statistical graph using pictures to denote the frequencies of the data p. 55 Pie chart（圓形圖）: A statistical graph in the shape

of a circle p. 55

Positive number（正數）: Numbers which are larger than 0, e.g., 1, 5.5 p. 2

Glossary