Multiplication and Division of Positive and Negative Numbers

This lesson covers how to multiply and divide positive and negative numbers. The key idea is the sign rule: deal with the signs first, then do the ordinary arithmetic.

The sign rules for multiplication

For any two numbers, the sign of the product depends only on their signs:

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

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

In words: when the two numbers have the same sign the product is positive, and when they have different signs the product is negative.

The sign rules for division

Division follows exactly the same pattern:

$(+)\div(+)=(+) \qquad (-)\div(-)=(+)$

$(+)\div(-)=(-) \qquad (-)\div(+)=(-)$

Same signs give a positive answer, different signs give a negative answer.

Multiplication examples

$(+8)\times(+3)=+24$

$(-3)\times(-8)=+24$

$(+10)\times(-5)=-50$

$(-2)\times(+15)=-30$

In each case we first fix the sign from the rule, then multiply the digits.

Multiplication with brackets

When an expression has brackets, work out each bracket first and then combine.

$(+2\times-8)\times(-2\times-2)=(-16)\times(+4)=-64$

$(-12\times-2)+(2\times0)=(+24)+0=+24$

$(-8\times-2)+(-3\times-2)=(+16)+(+6)=+22$

Division examples

$(+10)\div(+5)=+2$

$(-20)\div(-10)=+2$

$(+7)\div(+7)=+1$

$(-8)\div(-1)=+8$

Division with brackets

Again, evaluate each bracket first.

$(-12\div-3)\times(+8\div-2)=(+4)\times(-4)=-16$

$(+20\div-10)\div(-2\div-2)=(-2)\div(+1)=-2$

$(-100\div-2)\times(-50\div-1)=(+50)\times(+50)=+2500$

$(-25\div-1)+(+8\times-2)=(+25)+(-16)=+9$

$(0\times8)-(-2\div-2)=0-(+1)=-1$

Key takeaways

• When two numbers share the same sign, both their product and their quotient are positive.
• When two numbers have different signs, both their product and their quotient are negative.
• For expressions with brackets, simplify each bracket first, then apply the sign rule to combine the results.