How to Solve the Absolute Value of Rational Numbers?
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:
- 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
- Identify whether the rational number is positive, negative, or zero.
- If positive or zero: the absolute value equals the number.
- If negative: the absolute value is the positive version of the number.
- 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
- Find \(\color{blue}{|-\frac{7}{8}|}\)
- Find \(\color{blue}{|-0.45|}\)
- Compare: \(\color{blue}{|-2 \frac{3}{4}|}\) ___ \(\color{blue}{|2 \frac{1}{2}|}\)
- Order by absolute value from least to greatest: \(\color{blue}{-0.9, \frac{3}{4}, -\frac{1}{8}, 1.1}\)
- Which has the greater absolute value: \(\color{blue}{-\frac{4}{3}}\) or \(\color{blue}{\frac{5}{4}}\)?
- Find \(\color{blue}{|-\frac{3}{5}| + |-\frac{7}{10}|}\)
Answers
- \(\color{blue}{\frac{7}{8}}\)
- \(\color{blue}{0.45}\)
- \(\color{blue}{|-2 \frac{3}{4}| = 2.75 > 2.5 = |2 \frac{1}{2}|}\), so \(\color{blue}{>}\)
- Absolute values: \(\color{blue}{0.9, 0.75, 0.125, 1.1}\). Order: \(\color{blue}{-\frac{1}{8} < \frac{3}{4} < -0.9 < 1.1}\)
- \(\color{blue}{|-\frac{4}{3}| = \frac{4}{3} &\text{ approx }; 1.333}\) and \(\color{blue}{|\frac{5}{4}| = 1.25}\). So \(\color{blue}{-\frac{4}{3}}\) has the greater absolute value.
- \(\color{blue}{\frac{3}{5} + \frac{7}{10} = \frac{6}{10} + \frac{7}{10} = \frac{13}{10} = 1.3}\)
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Frequently Asked Questions
Is |−a| always equal to |a|?
Yes. Since absolute value measures distance from zero, and both \(\color{blue}{a}\) and \(\color{blue}{-a}\) are the same distance from zero (just on opposite sides), \(\color{blue}{|a| = |-a|}\) for any number \(\color{blue}{a}\).
How do absolute values of fractions compare to those of whole numbers?
The same rules apply. \(\color{blue}{|-\frac{1}{2}| = \frac{1}{2} = 0.5}\), which is less than \(\color{blue}{|3| = 3}\). Just convert fractions to decimals when a numeric comparison is needed.
Can the absolute value of a rational number be irrational?
No. The absolute value of any rational number is a rational number (the non-negative version of that rational number).
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