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Finding Opposites

Finding Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers \(2\) and \(-2\) are the same distance from zero, they are called opposites. The opposite of \(2\) is \(-2,\) and the opposite of \(-2\) is \(2\) as shown in figure (a) below. Similarly, \(3\) and \(-3\) are opposites as shown in figure (b) below.

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.

Opposite:

The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero.

Example

Find the opposite of each number:

  1. \(\phantom{\rule{0.2em}{0ex}}7\)
  2. \(\phantom{\rule{0.2em}{0ex}}-10\)

Solution

  1. The number \(-7\) is the same distance from \(0\) as \(7,\) but on the opposite side of \(0.\) So \(-7\) is the opposite of \(7\) as shown in the figure below.
    This figure is a number line. The points negative 7 and 7 are labeled. Above the line it is shown the distance from 0 to negative 7 and the distance from 0 to 7 are both 7.
  2. The number \(10\) is the same distance from \(0\) as \(-10\), but on the opposite side of \(0.\) So \(10\) is the opposite of \(-10\) as shown in the figure below.
    This figure is a number line. The points negative 10 and 10 are labeled. Above the line it is shown the distance from 0 to negative 10 and the distance from 0 to 10 are both 10.

Opposite Notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol \(\text{“−”,}\) in three different ways.

\(10-4\) Between two numbers, the symbol indicates the operation of subtraction.
We read \(10-4\) as 10 minus \(4\).
\(-8\) In front of a number, the symbol indicates a negative number.
We read \(-8\) as negative eight.
\(-x\) In front of a variable or a number, it indicates the opposite.
We read\(-x\) as the opposite of \(x\).
\(-\left(-2\right)\) Here we have two signs. The sign in the parentheses indicates that the number is negative 2.
The sign outside the parentheses indicates the opposite. We read \(-\left(-2\right)\) as the opposite of \(-2.\)

Opposite Notation:

\(-a\) means the opposite of the number \(a\)

The notation \(-a\) is read the opposite of \(a.\)

Example

Simplify: \(-\left(-6\right).\)

\(-\left(-6\right)\)
The opposite of \(-6\) is \(6.\) \(6\)

Integers

The set of counting numbers, their opposites, and \(0\) is the set of integers.

Integers

Integers are counting numbers, their opposites, and zero.

\(\text{…}\phantom{\rule{0.2em}{0ex}}-3,-2,-1,0,1,2,3\text{…}\)

We must be very careful with the signs when evaluating the opposite of a variable.

Example

Evaluate \(-x:\)

  1. when \(x=8\)
  2. when \(x=-8.\)

Solution

To evaluate \(-x\) when \(x=8\), substitute \(8\) for \(x\).
\(-x\)
. .
Simplify. \(-8\)
To evaluate \(-x\) when \(x=-8\), substitute \(-8\) for \(x\).
\(-x\)
. .
Simplify. \(8\)

Optional Video: Simplifying the Opposites of Negatives Integers by Mathispower4u

This lesson is part of:

Introducing Integers

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