Multiple and dividing integers

Multiplying Integers

1. Definition

An integer is any whole number that can be positive, negative, or zero (\dots, -3, -2, -1, 0, 1, 2, 3, \dots).

2. The Sign Rules

When multiplying two integers, determine the sign of the product using these two rules:

Like Signs \rightarrow Positive Product

Examples: 6 \times 3 = 18 | -6 \times (-3) = 18

Unlike Signs \rightarrow Negative Product

Examples: 6 \times (-3) = -18 | -6 \times 3 = -18

3. Step-by-Step Method

1. Multiply the absolute values: Ignore any negative signs and multiply the numbers normally.

2. Apply the sign rule: Count the total number of negative factors.

Even number of negatives \rightarrow Positive answer.

Odd number of negatives \rightarrow Negative answer.

4. Special Properties

Multiplication by Zero: Any integer multiplied by 0 equals 0.

a \times 0 = 0 (e.g., -15 \times 0 = 0)

Multiplication by One: Any integer multiplied by 1 keeps its identity.

a \times 1 = a (e.g., -8 \times 1 = -8)

Multiplication by -1: Multiplied by -1 gives the opposite sign.

a \times (-1) = -a (e.g., 7 \times (-1) = -7)

Dividing integers follows the exact same sign rules as multiplication: divide the absolute values normally, then apply the sign based on whether the original numbers match.

Sign Rules

Same signs = Positive quotient:

(+) \div (+) = + (e.g., 12 \div 3 = 4)

(-) \div (-) = + (e.g., -12 \div -3 = 4)

Different signs = Negative quotient:

(+) \div (-) = - (e.g., 12 \div -3 = -4)

(-) \div (+) = - (e.g., -12 \div 3 = -4)

Step-by-Step Method

1. Divide the numbers: Ignore the signs and carry out the division on the absolute values.

2. Determine the sign: Count the negative signs involved in the division.

1 negative \rightarrow result is negative.

0 or 2 negatives \rightarrow result is positive.

Rules for Zero

Zero divided by any non-zero integer is zero: 0 \div a = 0 (e.g., 0 \div -7 = 0).

Division by zero is undefined: a \div 0 has no mathematical meaning (e.g., -7 \div 0 is undefined).