Multiplying and Dividing with Positives, Negatives, One, and Zero

Rules and Formulas

Rules for signs:

Same sign always results in a positive $$(+)$$ outcome:

$$(+) \times (+) = (+)$$

$$(-) \times (-) = (+)$$

$$(+)$$: $$(+) = (+)$$

$$(-)$$: $$(-) = (+)$$

Different signs always result in a negative $$(-)$$ outcome:

$$(+) \times (-) = (-)$$

$$(-) \times (+) = (-)$$

$$(+)$$: $$(-) = (-)$$

$$(-)$$: $$(+) = (-)$$

Special Numbers:

Multiplication by Zero: Any number multiplied by 0 equals 0. $$(a \times 0 = 0)$$

Division by Zero: 0 divided by any number (other than 0) equals 0. $$(\frac{0}{a} = 0)$$

Note: It is impossible to divide a number by 0 (meaningless).

Property of Number 1: Any number multiplied or divided by 1 equals itself. ($$a \times 1 = a$$ and $$\frac{a}{1} = a$$)

Example

Calculate:

1) $$(-6) \times (-4)$$

2) $$\frac{(-15)}{3}$$

3) $$12 \times 0$$

Show solution

Step 1: For $$(-6) \times (-4)$$, same signs (negative × negative) = +24.

Step 2: For $$\frac{(-15)}{3}$$, different signs (negative / positive) = -5.

Step 3: For $$12 \times 0$$, any number multiplied by zero = 0.

Practice problems

Problem 1
Calculate the value of: $$(-8) \times (-7)$$

Show solution

Answer: 56

Step 1: Identify signs: negative × negative.
Step 2: Same signs give a positive result.
Step 3: Calculate: $$8 \times 7 = 56$$
Problem 2
Calculate the value of: $$\frac{(-36)}{4} \times (-2)$$

Show solution

Answer: 18

Step 1: Work from left to right: $$\frac{(-36)}{4} = -9$$ (different signs -> negative).
Step 2: Multiply by next term: $$(-9) \times (-2)$$.
Step 3: Same signs (negative × negative) -> +18.
Problem 3
Evaluate the expression: $$[(-5) \times 6 + 0] / (-3) + (\frac{15}{1})$$

Show solution

Answer: 25

Step 1: Inside brackets: $$(-5) \times 6 = -30$$, and $$-30 + 0 = -30$$.
Step 2: Divide: $$\frac{(-30)}{(-3)} = 10$$ (negative / negative = positive).
Step 3: Identity rule: $$\frac{15}{1} = 15$$.
Step 4: Final addition: $$10 + 15 = 25$$

Remember

Division by zero is undefined (impossible in math)! Also, count negative factors in multiplication: an EVEN number of negative factors gives a positive result, while an ODD number gives a negative result.