Comparing Absolute Values: |-1/3| and |3| - Are They Opposite Numbers?

Q: What does absolute value actually do to a number?

Absolute value gives you the distance from zero on the number line. It makes negative numbers positive and leaves positive numbers unchanged. So $ \left|-\frac{1}{3}\right| = \frac{1}{3} $ and $ \left|3\right| = 3 $.

Q: How do I know if two numbers are opposites?

Two numbers are opposites if they add up to zero. For example, 5 and -5 are opposites because 5 + (-5) = 0. The numbers must be the same distance from zero but on opposite sides.

Q: Why aren't 1/3 and 3 opposite numbers?

Because they don't add up to zero! $ \frac{1}{3} + 3 = \frac{1}{3} + \frac{9}{3} = \frac{10}{3} $, which is not zero. Opposite numbers must have equal distances from zero but opposite signs.

Q: What would be the opposite of 1/3?

The opposite of $ \frac{1}{3} $ is $ -\frac{1}{3} $ because $ \frac{1}{3} + \left(-\frac{1}{3}\right) = 0 $. Opposites are the same number with different signs.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Are the numbers opposite?

$\left|-\frac{1}{3}\right|,\left|\frac{3}{1}\right|$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Compute the absolute value of each given number.
Step 2: Determine if the numbers are opposites based on their absolute values.

Now, let's proceed step-by-step:

Step 1: Calculate the absolute values.
The absolute value of a number gives its distance from zero on the number line, disregarding its sign.

$\left|-\frac{1}{3}\right|$
The negative sign is removed by the absolute value, so:
$\left|-\frac{1}{3}\right| = \frac{1}{3}$ $\left|\frac{3}{1}\right|$
The absolute value remains the same as there is no negative sign:
$\left|\frac{3}{1}\right| = 3$

Step 2: Determine if the numbers are opposites.

Two numbers are considered opposites if their sum equals zero. In this case, we compare the results:

The absolute values we found were $\frac{1}{3}$ and $3$ . These numbers do not meet the condition for being opposites since their sum $\frac{1}{3} + 3 = \frac{10}{3}$ is not equal to zero.

Therefore, the numbers are not opposites.

Therefore, the correct answer is No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value removes negative signs, making all results positive
Technique: Calculate $\left|-\frac{1}{3}\right| = \frac{1}{3}$ and $\left|3\right| = 3$
Check: Opposite numbers sum to zero: $\frac{1}{3} + 3 = \frac{10}{3} \neq 0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing absolute values with original numbers
Don't think that since -1/3 and 3 have opposite signs they must be opposites = wrong conclusion! Absolute value removes all negative signs first. Always compute the absolute values first, then check if those results sum to zero.

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

FAQ

Everything you need to know about this question

What does absolute value actually do to a number?

+

Absolute value gives you the distance from zero on the number line. It makes negative numbers positive and leaves positive numbers unchanged. So $\left|-\frac{1}{3}\right| = \frac{1}{3}$ and $\left|3\right| = 3$ .

How do I know if two numbers are opposites?

+

Two numbers are opposites if they add up to zero. For example, 5 and -5 are opposites because 5 + (-5) = 0. The numbers must be the same distance from zero but on opposite sides.

Why aren't 1/3 and 3 opposite numbers?

+

Because they don't add up to zero! $\frac{1}{3} + 3 = \frac{1}{3} + \frac{9}{3} = \frac{10}{3}$ , which is not zero. Opposite numbers must have equal distances from zero but opposite signs.

What would be the opposite of 1/3?

+

The opposite of $\frac{1}{3}$ is $-\frac{1}{3}$ because $\frac{1}{3} + \left(-\frac{1}{3}\right) = 0$ . Opposites are the same number with different signs.

Can absolute values ever be opposites?

+

Only if both absolute values equal zero! Since absolute values are always positive or zero, the only way two absolute values can be opposites is if they're both zero: $|0| = 0$ and $|0| = 0$ .

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Comparing Absolute Values: |-1/3| and |3| - Are They Opposite Numbers?

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