How to Solve the Absolute Value of Rational Numbers?

How to solve the absolute value of rational numbers, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

You already know that the absolute value of an integer \(\color{blue}{\text{ like } -7}\) is 7. The same idea extends perfectly to rational numbers, fractions, decimals, and mixed numbers. Whether the number is \(\color{blue}{-\frac{3}{4}}\) or \(\color{blue}{-2.85}\), the absolute value simply “removes the negative sign” and gives the distance from zero. This lesson shows you exactly how to handle every case.

What Is the Absolute Value of a Rational Number?

The absolute value of a rational number is its distance from zero on the number line. Because distance is always non-negative:

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• The absolute value of any positive rational number is the number itself.
• The absolute value of any negative rational number is its positive counterpart.
• The absolute value of zero is zero.

In symbols: \(\color{blue}{|r| = r}\) if \(\color{blue}{r \ge 0}\), and \(\color{blue}{|r| = -r}\) if \(\color{blue}{r < 0}\).

How to Find the Absolute Value of Rational Numbers

Fractions

Apply the absolute value to the entire fraction. The result has the same numerator and denominator, but is positive.

• \(\color{blue}{|\frac{3}{4}| = \frac{3}{4}}\)
• \(\color{blue}{|-\frac{5}{8}| = \frac{5}{8}}\)
• \(\color{blue}{|-\frac{7}{3}| = \frac{7}{3}}\)

Decimals

The absolute value of a decimal is simply the decimal with no negative sign.

• \(\color{blue}{|-0.6| = 0.6}\)
• \(\color{blue}{|-3.14| = 3.14}\)

Mixed Numbers

The absolute value of a mixed number removes the negative sign from the entire number.

• \(\color{blue}{|-2 \frac{1}{4}| = 2 \frac{1}{4}}\)
• \(\color{blue}{|3 \frac{7}{10}| = 3 \frac{7}{10}}\)

Comparing Absolute Values

When comparing \(\color{blue}{|a|}\) and \(\color{blue}{|b|}\), first evaluate each absolute value, then compare the resulting non-negative numbers.

\(\color{blue}{|-\frac{5}{4}|}\) vs \(\color{blue}{|\frac{7}{8}|}\): \(\color{blue}{\frac{5}{4} = 1.25}\) and \(\color{blue}{\frac{7}{8} = 0.875}\). Since \(\color{blue}{1.25 > 0.875}\), we have \(\color{blue}{|-\frac{5}{4}| > |\frac{7}{8}|}\).

Step-by-Step Summary

1. Identify whether the rational number is positive, negative, or zero.
2. If positive or zero: the absolute value equals the number.
3. If negative: the absolute value is the positive version of the number.
4. For comparisons: evaluate each absolute value, then compare the non-negative results.

Watch: Interpreting Absolute Value (Video Lesson)

Khan Academy explains what absolute value means in real-world contexts for 6th-grade level learners:

Worked Examples

Example 1: Find \(\color{blue}{|-\frac{9}{10}|}\)

The number is negative, so the absolute value is its positive counterpart: \(\color{blue}{|-\frac{9}{10}| = \frac{9}{10}}\).

Example 2: Compare \(\color{blue}{|-\frac{3}{5}|}\) and \(\color{blue}{|\frac{1}{2}|}\).

\(\color{blue}{|-\frac{3}{5}| = \frac{3}{5} = 0.6}\) and \(\color{blue}{|\frac{1}{2}| = 0.5}\). Since \(\color{blue}{0.6 > 0.5}\), we have \(\color{blue}{|-\frac{3}{5}| > |\frac{1}{2}|}\).

Example 3: Find \(\color{blue}{|-4.75|}\)

\(\color{blue}{|-4.75| = 4.75}\)

Example 4: Order from least to greatest by absolute value: \(\color{blue}{-1 \frac{1}{2}, \frac{2}{3}, -\frac{5}{4}, 0.1}\)

Absolute values: \(\color{blue}{\frac{3}{2} = 1.5}\), \(\color{blue}{\frac{2}{3} &\text{ approx }; 0.667}\), \(\color{blue}{\frac{5}{4} = 1.25}\), \(\color{blue}{0.1}\). Ordering: \(\color{blue}{0.1 < \frac{2}{3} < \frac{5}{4} < \frac{3}{2}}\).

More Practice: What Is Absolute Value? Video

Math with Mr. J covers absolute value with clear examples that extend to rational numbers:

Exercises

1. Find \(\color{blue}{|-\frac{7}{8}|}\)
2. Find \(\color{blue}{|-0.45|}\)
3. Compare: \(\color{blue}{|-2 \frac{3}{4}|}\) ___ \(\color{blue}{|2 \frac{1}{2}|}\)
4. Order by absolute value from least to greatest: \(\color{blue}{-0.9, \frac{3}{4}, -\frac{1}{8}, 1.1}\)
5. Which has the greater absolute value: \(\color{blue}{-\frac{4}{3}}\) or \(\color{blue}{\frac{5}{4}}\)?
6. Find \(\color{blue}{|-\frac{3}{5}| + |-\frac{7}{10}|}\)

Answers

1. \(\color{blue}{\frac{7}{8}}\)
2. \(\color{blue}{0.45}\)
3. \(\color{blue}{|-2 \frac{3}{4}| = 2.75 > 2.5 = |2 \frac{1}{2}|}\), so \(\color{blue}{>}\)
4. Absolute values: \(\color{blue}{0.9, 0.75, 0.125, 1.1}\). Order: \(\color{blue}{-\frac{1}{8} < \frac{3}{4} < -0.9 < 1.1}\)
5. \(\color{blue}{|-\frac{4}{3}| = \frac{4}{3} &\text{ approx }; 1.333}\) and \(\color{blue}{|\frac{5}{4}| = 1.25}\). So \(\color{blue}{-\frac{4}{3}}\) has the greater absolute value.
6. \(\color{blue}{\frac{3}{5} + \frac{7}{10} = \frac{6}{10} + \frac{7}{10} = \frac{13}{10} = 1.3}\)

Frequently Asked Questions

What is absolute value in rational numbers?

For a rational number, absolute value is its distance from zero on the number line, exactly as with integers. A rational number is any number you can write as a fraction of two integers, which includes fractions, terminating or repeating decimals and mixed numbers. So |−3/4| = 3/4, because −3/4 sits three quarters of a unit from zero, and |0.6| = 0.6.

How do you find the absolute value of rational numbers?

If the number is positive or zero, it stays the same; if it is negative, take its opposite by dropping the negative sign. Apply this to the whole number, not just part of it: |−5/8| = 5/8, |−3.14| = 3.14 and |−2 1/2| = 2 1/2. When there is an expression inside the bars, simplify first: |1/2 − 3/4| = |−1/4| = 1/4.

How is absolute value related to integers and rational numbers?

Every integer is also a rational number, since 7 can be written as 7/1, so one absolute value rule covers both. |−7| = 7 and |−7/2| = 7/2, because each measures distance from zero. This also helps you compare negative values: |−1.5| = 1.5 is greater than |−1 1/4| = 1.25, so −1.5 is farther from zero and is the smaller number.

Is absolute value rational or irrational?

It depends on the number inside the bars. The absolute value of a rational number is always rational, for example |−2/3| = 2/3. The absolute value of an irrational number stays irrational, so |−√2| = √2 and |−π| = π. Absolute value can only change the sign, never whether the number can be written as a fraction.

Is the absolute value of a rational number always positive?

It is always positive or zero, never negative. Any nonzero rational number, such as −0.45 or 9/10, has a positive absolute value, and |0| = 0. That is why an answer like |−4/9| = −4/9 is always wrong. Opposite numbers such as 4/9 and −4/9 always share the same absolute value, since they are the same distance from zero.

Related Topics

• Absolute Value Definition
• Solve Word Problems of Absolute Value and Integers
• Using Number Lines to Compare and Order Rational Numbers
• Convert Rational Numbers to a Fraction
• Opposite Integers