Multiplication and Division of Directed Numbers

In mathematics, understanding the rules of multiplication and division of directed numbers is crucial. Directed numbers are numbers that have a sign attached to them, either positive (+) or negative (-). The rules for multiplying and dividing these numbers are straightforward and consistent.

Rule for Multiplication and Division of Directed Numbers

The rule for multiplication and division of directed numbers is as follows:

When multiplying or dividing two numbers of the same sign, the result is always positive.
When multiplying or dividing two numbers of different signs, the result is always negative.

This can be mathematically represented as follows:

For any two positive integers $a$ and $b$ :

$a \times b = ab$ if both $a$ and $b$ are positive.
$-a \times -b = ab$ if both $a$ and $b$ are negative.
$-a \times b = -ab$ if $a$ is negative and $b$ is positive.
$a \times -b = -ab$ if $a$ is positive and $b$ is negative.

The same rules apply to division:

$a \div b = \frac{a}{b}$ if both $a$ and $b$ are positive.
$-a \div -b = \frac{a}{b}$ if both $a$ and $b$ are negative.
$-a \div b = -\frac{a}{b}$ if $a$ is negative and $b$ is positive.
$a \div -b = -\frac{a}{b}$ if $a$ is positive and $b$ is negative.

Practical Examples

Let’s illustrate these rules with some examples:

$-2 \times -7 = 14$ (Both numbers are negative, so the result is positive)
$2 \times -10 = -20$ (The numbers have different signs, so the result is negative)
$14 \div -2 = -7$ (The numbers have different signs, so the result is negative)
$20 \div -4 = -5$ (The numbers have different signs, so the result is negative)
$-100 \div 5 = -20$ (The numbers have different signs, so the result is negative)
$-13 \times -3 = 39$ (Both numbers are negative, so the result is positive)
$-400 \div -2 = 200$ (Both numbers are negative, so the result is positive)
$-100 \times -3 = 300$ (Both numbers are negative, so the result is positive)

In conclusion, the rules for multiplication and division of directed numbers are consistent and straightforward. Understanding these rules is fundamental in mathematics, as they apply to various mathematical operations and concepts.