**1.1 Directed Numbers**

**1.1.1 Multiplication and Division of Integers**

**1.**Multiplication and division of

**like signs**gives (

**+**)

$\overline{)\begin{array}{l}\text{}(+)\times (+)=+\text{}(+)\xf7(+)=+\text{}\\ \text{}(-)\times (-)=+\text{}(-)\xf7(-)=+\end{array}}$

**Multiplication and division of**

2.

2.

**unlike signs**gives (

**–**)

$\overline{)\begin{array}{l}\text{}(+)\times (-)=-\text{}(+)\xf7(-)=-\text{}\\ \text{}(-)\times (+)=-\text{}(-)\xf7(+)=-\end{array}}$

*Example:* (a) –25 ÷ 5 = –5

(b) 8 × (–5) = –40

**Multiplication of**

3.

3.

**3**integers.

$\overline{)\begin{array}{l}\text{}(+)\times (+)\times (+)=(+)\text{}\\ \text{}(+)\times (+)\times (-)=(-)\\ \text{}(+)\times (-)\times (-)=(+)\text{}\end{array}}$

**Division of**

4.

4.

**3**integers.

$\overline{)\begin{array}{l}\text{}(+)\xf7(+)\xf7(+)=(+)\text{}\\ \text{}(+)\xf7(+)\xf7(-)=(-)\\ \text{}(+)\xf7(-)\xf7(-)=(+)\text{}\end{array}}$

**The product of an integer and zero is always zero.**

5.

5.

*Example:*–5 × 0 = 0

**When zero is divided by any integer except zero, the quotient is zero. Any integer divided by zero is undefined.**

6.

6.

*Example:* (a) 0 ÷ 9 = 0

(b) –6 ÷ 0 is undefined

**1.1.2 Combined Operations of Integers**

**1.**

**BODMAS**→

**(Brackets of Division, Multiplication, Addition and Subtraction)**

brackets should be carried out first. 1. Operations in the × or ÷ from left to right. 2. Followed by + or – from left to right. 3. Followed by |

**Example 1:**

(a) –52 ÷ 13 – 15 × 4

(b) 63 ÷ (16 – 7) × (–2)

(c) –30 + 9 × 7 – 16

Solution:Solution:

**(a)**

–52 ÷ 13 – 15 × 4

= (–52 ÷ 13) – (15 × 4) ← (calculate from left to right; ÷ and × are done first)

= –4 – 60

= –64

(b)

(b)

63 ÷ (16 – 7) × (–2)

= 63 ÷ 9 × (–2) ← (bracket is done first, then work from left to right)

= 7 × (–2)

= –14

(c)

(c)

–30 + 9 × 7 – 16

= –30 + (9 × 7) – 16 ← ( multiply first)

= –30 + 63 – 16

= 17

**1.1.3 Fractions**

**1.**

**Positive fractions**are fractions with the positive sign (+), and their values are greater than 0.

**2.**

**Negative fractions**are fractions with the negative sign (–), and their values are less than 0.

**1.1.4 Decimals**

**1.**There are positive decimals and negative decimals.

**2.**Positive decimals are greater than zero and negative decimals are less than zero.

**1.1.5 Directed Numbers**

**1.**Integers, fractions and decimals are

**directed numbers**.

**2.**The computations of directed numbers is the same as that for whole numbers.

Example 2:

Example 2:

$-\frac{1}{2}+\left(-0.37\right)-\left(-5\right)$
(a)

(b) [(–28) – (–4)] ÷ (–5.147 – 0.853)

Solution:Solution:

**(a)**

$\begin{array}{l}-\frac{1}{2}+\left(-0.37\right)-\left(-5\right)\\ =-0.5-0.37+5\\ =-0.87+5\\ =4.13\end{array}$

(b)

(b)

[(–28) – (–4)] ÷ (–5.147 – 0.853)

= [–28 + 4] ÷ (–6)

= (–24) ÷ (–6)

= 4