Irrational Numbers

Irrational numbers are real numbers that cannot be written as a fraction of two integers, so their decimal expansions never end and never repeat. In brief: √2 ≈ 1.41421356… and π ≈ 3.14159265… are irrational, while 0.75 = 3/4 and 0.333… = 1/3 are rational. A number is irrational if no fraction a/b with whole numbers a and b equals it exactly.

Irrational numbers fill the gaps on the number line between the rational numbers. They appear as soon as you measure the diagonal of a square or the circumference of a circle. This guide explains how to tell rational from irrational numbers, proves that √2 is irrational and lists the most important examples with their decimal values.

What is an irrational number?

The rational numbers are all numbers that can be written as a/b with integers a and b and b ≠ 0. Together with the irrational numbers they make up the real numbers. An irrational number is any real number that is not rational.

The decimal expansion of an irrational number goes on forever without a repeating block. This is different from a rational number, whose decimal either stops, like 0.125, or repeats a block, like 0.272727… = 3/11.

How can you tell rational and irrational numbers apart?

1. If the decimal stops or repeats, the number is rational.
2. The square root of a positive integer is rational only if the integer is a perfect square. So √16 = 4 is rational, but √17 is irrational.
3. In general, the nth root of an integer is either an integer or irrational.
4. Well-known constants such as π and e are irrational, and this has been proved.

Be careful with the calculator display. The value 3.14159265 is only a rounded approximation of π, and 22/7 = 3.142857… is a rational approximation, not π itself.

How do you prove that √2 is irrational?

The classical proof is by contradiction.

1. Assume √2 = a/b, where a and b are integers with no common factor.
2. Squaring gives 2 = a²/b², so a² = 2b². Hence a² is even, which forces a to be even. Write a = 2k.
3. Then 4k² = 2b², so b² = 2k². This means b² is even and b is even.
4. Now both a and b are even, which contradicts the assumption that they have no common factor.

Therefore √2 cannot be written as a fraction, and it is irrational. The same style of argument works for √3, √5 and the square root of any non-square integer.

Table of important irrational numbers

What happens when you add or multiply irrational numbers?

The sum of a rational and an irrational number is always irrational. For example, 3 + √2 is irrational, because otherwise √2 would equal a rational number minus 3. The product of a nonzero rational number and an irrational number is also irrational, so 5√3 is irrational.

Two irrational numbers can have a rational result. For example, √2 · √2 = 2, and (3 + √5) + (3 − √5) = 6. So the set of irrational numbers is not closed under addition or multiplication. It is useful to check the result in each case, instead of assuming it.

Where does the idea come from?

The discovery that the diagonal and side of a square have no common measure is traditionally linked to the Pythagorean school in ancient Greece. Later stories name Hippasus of Metapontum as the person who revealed it, and say he was punished for it. These stories are told in sources written long after the events, so historians treat them with caution. What is certain is that Greek mathematicians understood incommensurable lengths well, and Euclid treats them in the Elements.

The transcendence of e was proved by Charles Hermite in 1873 and that of π by Ferdinand von Lindemann in 1882, which also settled the old problem of squaring the circle with compass and straightedge.

Where are irrational numbers used today?

Almost every geometric formula that involves circles or diagonals contains an irrational number. The area of a circle with radius 5 is 25π ≈ 78.54 square units. A standard A4 sheet has sides in a ratio close to √2 : 1, which allows folding the sheet in half to get a similar rectangle. Exponential growth and decay use the irrational number e, and music theory uses the 12th root of 2 to define the equal-tempered semitone.

Which mistakes are common?

Calling a rounded decimal exact

3.14 and 1.414 are approximations. Use the symbol, such as π or √2, if you need an exact answer.

Assuming every long decimal is irrational

1/7 = 0.142857142857… has a long decimal but a repeating block, so it is rational.

Thinking √ of any number is irrational

√49 = 7 and √(9/4) = 3/2 are rational. Only roots of non-perfect powers are irrational.

Forgetting that irrational numbers are real

Irrational numbers are real numbers with a place on the number line.