Rational and Irrational Numbers

Use the definition before judging the appearance

A real number is rational when it can be written as

Integers are rational because $n=n/1$. So are terminating decimals: $0.125=125/1000=1/8$. A radical symbol does not automatically mean irrational; $\sqrt{49/9}=7/3$ is rational.

The rational and irrational numbers partition the real numbers. A non-real complex number such as $i$ belongs to neither group. Saying “anything that is not rational is irrational” without restricting the discussion to real numbers is incomplete.

Explain terminating decimals through prime factors

A fraction in lowest terms has a terminating base-ten decimal exactly when its positive denominator has no prime factors except 2 and 5. The reason is that a decimal with $k$ places is an integer divided by $10^k=2^k5^k$.

For example, $7/40$ terminates because $40=2^3\times5$. Multiplying numerator and denominator by 25 gives

By contrast, the reduced denominator of $1/6$ contains a factor 3, so no power of ten can supply it. Its decimal does not terminate.

Reduction matters. The fraction $3/6$ has a denominator containing 3 as written, but it reduces to $1/2=0.5$. Apply the criterion to the reduced denominator, not an arbitrary equivalent fraction.

Convert repeating decimals exactly

Let $x=0.\overline{27}$, meaning $0.272727\ldots$ with the two-digit block repeated forever. Multiply by 100 to align the repeating tails:

Therefore $99x=27$ and

A decimal can have a non-repeating beginning. For $x=0.1\overline6$, use $10x=1.\overline6$ and $100x=16.\overline6$. Subtraction gives $90x=15$, so $x=1/6$.

For any rational fraction, long division has finitely many possible remainder states. It either reaches zero or repeats a state, so the decimal terminates or eventually repeats. Conversely, aligning and subtracting repeated tails turns any eventually repeating decimal into a rational equation. A written finite string followed by dots is not enough to specify a decimal unless the rule for continuing it is clear.

Prove that a familiar square root is irrational

Suppose $\sqrt2=p/q$ in lowest terms, with positive integers $p$ and $q$. Squaring gives $p^2=2q^2$. Hence $p^2$ is even, so $p$ is even: an odd integer has an odd square.

Write $p=2k$. Substitution yields $4k^2=2q^2$, or $q^2=2k^2$. Thus $q$ is even as well. Both numerator and denominator have a factor 2, contradicting the claim that the fraction was reduced. Therefore $\sqrt2$ is irrational.

This is a general proof, not a conclusion from a calculator displaying many digits. Prime factorization supplies another viewpoint: the exponent of 2 in $p^2$ would be even, while its exponent in $2q^2$ would be odd.

Separate exact values from approximations

The number $1.414$ is rational because it equals $707/500$. It is a useful approximation to $\sqrt2$, but the two numbers are not equal:

Since both endpoints are positive,

Narrowing rational bounds locates an irrational value without turning it into a rational number. Every finite decimal approximation remains rational, regardless of how accurately it approximates the target.

An infinite recurring decimal can also have a terminating representation. In particular, $0.\overline9=1$. One justification uses the finite sums $0.9,0.99,\ldots$: the value after $n$ digits is $1-10^{-n}$, which approaches 1. There is no positive “last-digit gap” after infinitely many nines.

Check what operations preserve

Rational numbers are closed under addition, subtraction and multiplication, and under division by a non-zero rational number. Adding a rational number to an irrational number must produce an irrational number: otherwise subtracting the rational addend would make the original irrational number rational.

Multiplication by a non-zero rational number also preserves irrationality. The non-zero restriction matters because multiplying anything by zero produces rational zero.

Two irrational numbers can have a rational or irrational sum or product. For example,

There is no blanket “irrational plus irrational” rule. Exact algebra, not the label of each input alone, determines the result.

Find numbers between numbers

Between any two different rational numbers $a<b$, their midpoint $(a+b)/2$ is another rational number. Repeating the process creates indefinitely many rational values between them.

There are also irrational numbers arbitrarily close to any real value. The number line is not divided into a rational region and an irrational region. The intervals in the diagram contain both types, even though a finite diagram cannot mark every point. Properties of Real Numbers distinguishes this closeness property from completeness.

Related topics

Properties of Real Numbers develops the arithmetic and order structure shared by rational and irrational values. Complex Numbers explains why some equations require numbers outside the real line.