Lesson 4: Negation

Property

To find the negation of any number $a$, ask: "What number adds to $a$ to give zero?" The answer is $-a$, where $a + (-a) = 0$.

Examples

Property

The opposite of a negative number is a positive number. We write the opposite of $-6$ as $-(-6)$, so

Examples

• The opposite of $-25$ is written as $-(-25)$, which simplifies to $25$.
• If you cancel a debt of 30 dollars, your financial change is $-(-30)$ dollars, which is a gain of $30$ dollars.
• On a number line, the opposite of $-8.2$ is found by reflecting it over zero, resulting in $8.2$.

Explanation

Taking the 'opposite' of a number flips it across zero on the number line. If you flip a negative number, it lands on the positive side. So, a double negative like $-(-6)$ becomes a positive $6$.

Property

Multiplying a number by $-1$ gives its opposite.

Examples

• To multiply $-1 \cdot 9$, you get the opposite of $9$, which is $-9$.
• To multiply $-1(-15)$, you get the opposite of $-15$, which is $15$.
• If you have the number $x$, multiplying it by $-1$ gives you $-x$.

Explanation

Multiplying any number by $-1$ is a shortcut to find its opposite.
It flips the number to the other side of zero on the number line, changing its sign from positive to negative or from negative to positive.