Finding the Opposite of an Integer

Hello! In the previous lesson, you plotted integers by starting at $0$, using the sign to choose a direction, and counting equal intervals according to the number’s size. That picture of integers on a number line now helps us identify opposites.

In this lesson, you will learn to recognize the opposite of any integer, including integers described in everyday situations. The central idea is symmetry around zero: opposites are equally far from $0$, but placed in different directions.

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Opposites on a number line

Consider $2$ and $-2$. Both are two intervals from $0$, but $2$ is to the right and $-2$ is to the left. They are opposites.

An opposite does not change how far a number is from zero. It changes the number’s direction, or sign.

• The opposite of $6$ is $-6$.
• The opposite of $-6$ is $6$.
• The opposite of $125$ is $-125$.
• The opposite of $-125$ is $125$.

Positive numbers often appear without a plus sign. So $6$ really means $+6$. Its opposite is $-6$.

The formal definition is:

The opposite of a nonzero number is the number at the same distance from zero on the other side of zero.

A number line is useful because it lets you see why the sign changes while the digits stay the same.

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A reliable method: keep the number, switch the sign

To find the opposite of an integer, use this short process:

1. Identify the integer’s sign.
2. Keep the same digits.
3. Replace the sign with its opposite.

Given integer	Opposite
$+18$, or simply $18$	$-18$
$-47$	$+47$, or simply $47$
$0$	$0$
$-1,205$	$+1,205$

For example, the opposite of $-47$ is $47$. On the number line, $-47$ is 47 intervals left of zero, while $47$ is 47 intervals right of zero.

Be careful not to change the digits:

• The opposite of $-8$ is not $8$ with a different size such as $9$.
• The opposite of $4$ is not $-4$ plus or minus another number.
• The opposite of $-12$ is $12$, not $-12$.

The pair must have the same distance from zero. Therefore, $4$ and $-5$ are not opposites: they are on opposite sides of zero, but they are different distances away.

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Zero is its own opposite

Zero is a special case. It is neither positive nor negative, and it has no “other side” of zero. The opposite of $0$ is still $0$.

$$\text{opposite of }0=0$$

One way to check opposites is to add the two numbers. A number and its opposite combine to make zero:

$$9+(-9)=0$$ $$-14+14=0$$

For zero, this is also true:

$$0+0=0$$

You do not yet need to use integer addition rules to find opposites. The number-line definition and sign-switching method are enough. The “sum is zero” idea is simply a useful confirmation.

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Opposites in everyday language

Often, a problem gives a situation rather than a signed number. First identify the reference point and direction, then reverse that direction while keeping the amount unchanged.

Situation	Integer	Its opposite	Meaning of the opposite
A loss of $34$ points	$-34$	$+34$	a gain of $34$ points
$500$ feet above sea level	$+500$	$-500$	$500$ feet below sea level
Owing $20$ dollars	$-20$	$+20$	having or gaining $20$ dollars
A rise of $7$ degrees	$+7$	$-7$	a drop of $7$ degrees

The word opposite means the situation reverses around the same reference point:

• above sea level and below sea level use sea level as $0$;
• a gain and a loss use no change as $0$;
• a rise and a drop use the starting level as $0$.

So the opposite of “a loss of $34$” is “a gain of $34$,” not a gain of some different amount.

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Reading opposite notation

Sometimes math writes “the opposite of a number” with a negative sign outside parentheses.

For example:

$$-(-8)=8$$

Read this as: “the opposite of negative eight is positive eight.”

The first negative sign is outside the parentheses, so it tells you to find the opposite of the entire number inside. Since the number inside is $-8$, its opposite is $+8$.

Likewise,

$$-(5)=-5$$

This says: “the opposite of positive five is negative five.”

At this stage, focus on the meaning of the notation rather than treating it as subtraction. There is no number before the outside minus sign, so it is not asking you to subtract. It is asking for an opposite.

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Key takeaways

An integer’s opposite:

• has the same digits;
• has the opposite sign;
• lies the same distance from $0$ on the other side of a number line;
• reverses the direction of a real-world situation, such as gain versus loss or above versus below;
• makes a sum of $0$ when added to the original number.

The special case is $0$: its opposite is $0$.

Next, you will build on the “same distance from zero” idea to determine an integer’s absolute value.