Rational Number
In short: Rational number names any value that equals one integer divided by another nonzero integer, which covers 3/4, -2.5, 6 and 0.125 alike.
6th GradeNumbers and Operations6.NS.C.6
What is Rational Number?
Whenever a value can be written as one whole number over another, it is a rational number. Fractions, mixed numbers, whole numbers, negatives and decimals that end or repeat all qualify, because each can be rewritten in that form.
Precisely, \(x\) is rational when \(x = \frac{p}{q}\) for integers \(p\) and \(q\) with \(q \neq 0\). For example \(-2.5 = -\frac{5}{2}\), \(6 = \frac{6}{1}\) and \(0.\overline{3} = \frac{1}{3}\).
How to spot it
- A story measures with halves, quarters or decimals of dollars, inches or degrees, often with a negative sign.
- A decimal either stops, like 0.125, or repeats a block forever, like 0.272727.
- You are asked to place a fraction or decimal between two integers on a number line.
- A question asks whether a given number can be written as one integer over another.
The method
- Try to write the value as a fraction: a decimal that stops uses a power of 10 on the bottom.
- Simplify the fraction by dividing top and bottom by their greatest common factor.
- Locate it on the number line by splitting each unit into equal parts matching the bottom.
- Compare two rationals by placing them on the line or by turning both into decimals.
Rational Number: solved example
Problem
A weather station in Fargo records changes of \(-3.6\), \(0.45\) and \(\frac{7}{8}\) degrees. Rewrite each one as an integer over an integer, reduced.
Solution
\(-3.6 = -\frac{36}{10} = -\frac{18}{5}\). Next, \(0.45 = \frac{45}{100} = \frac{9}{20}\). The third value is already \(\frac{7}{8}\).
All three fit the form \(\frac{p}{q}\), so each one is a rational number.
Try it
Question 1
Turn 0.875 into a reduced fraction.
Show the answer
\(\frac{875}{1000} = \frac{7}{8}\), after dividing by 125.
Question 2
Is 0.3333… (the 3 repeats) rational? Give the fraction.
Show the answer
Yes. A repeating decimal like this equals \(\frac{1}{3}\).
Rational Number vs. Integer
Integers are only the whole-number marks: \(-4, 0, 9\). Rational numbers include all of those and every fraction or terminating or repeating decimal between them, such as \(\frac{9}{4}\) or \(-0.35\). So the integers sit inside the rational numbers, not the other way around.
Keep going
Lessons
Worksheets and quizzes
- Integers and Rational Numbers on the Number Line – Worksheet
- Integers and Rational Numbers on the Number Line – Quiz
- Ordering Rational Numbers and Absolute Value – Worksheet
- Ordering Rational Numbers and Absolute Value – Quiz
Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations