Math101 · Independent Ontario learning library
Math101learn.math101.caRational Numbers
Rational numbers are ratios of integers and include fractions, integers, terminating decimals, and repeating decimals.
A rational number is any number that can be written as $a/b$, where $a$ and $b$ are integers and $b\ne0$.
What belongs in the set
Fractions are the obvious rational numbers, but the set is much larger. Every integer is rational because $-7=-7/1$. A terminating decimal such as $0.35$ is rational because $0.35=35/100=7/20$. A repeating decimal is also rational.
The denominator cannot be zero. The expression $4/0$ is undefined, not rational, irrational, or infinite.
Equivalent forms
Multiplying numerator and denominator by the same nonzero number preserves value:
Simplest form removes common factors from numerator and denominator. Equivalent forms are useful for different jobs: decimals support measurement, fractions preserve exactness, and percent communicates a comparison to $100$.
Decimal behaviour
A rational number’s decimal expansion terminates or eventually repeats. In lowest terms, a fraction has a terminating decimal exactly when its positive denominator contains no prime factors other than $2$ and $5$.
For example, $7/40$ terminates because $40=2^3\cdot5$. The fraction $2/3$ repeats because its denominator includes the factor $3$.
Turning a repeating decimal into a fraction
The multiplication by $100$ aligns the repeating blocks so subtraction removes the infinite tail.
Comparing rational numbers
Use a common representation. To compare $5/8$ and $0.61$, write $5/8=0.625$, so $5/8>0.61$. Fractions can also be compared with a common denominator or, when denominators are positive, cross products.
On a number line, greater numbers lie farther right. Remember that among negative values, the number closer to zero is greater: $-2/3>-3/4$ because approximately $-0.667>-0.75$.
Operations
Rational numbers are closed under addition, subtraction, and multiplication. They are also closed under division when the divisor is not zero.
Use a common denominator only for addition and subtraction. Multiplication and division follow different structures.
Rational numbers in context
Rational values describe exact shares, rates, probabilities, financial amounts, and measurements. Context may restrict which rational values make sense. A probability lies from $0$ to $1$, and a count of people usually cannot be $4.5$, even though $4.5$ is rational.
Common mistakes
Believing rational means positive. Negative fractions and zero are rational.
Treating every long decimal as irrational. A decimal may repeat later; a displayed calculator approximation may also represent an exact rational value.
Adding denominators. $1/2+1/3$ is not $2/5$ because halves and thirds are different-sized parts.
Dividing by zero. A zero numerator is allowed, but a zero denominator is not.
Quick self-check
- Can I express the number as a fraction of integers?
- Does its decimal terminate or repeat?
- Have I kept exact form until a decimal is useful?
- Does the final value make sense in its context?
Related topics
Explore the idea
Number model
Change one quantity at a time and connect what moves to Rational Numbers.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Why is 0.272727… rational?
- The block 27 repeats forever.
- Every repeating decimal can be converted to a fraction.
- Therefore the number is rational.
End of lesson
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