Rational or Irrational? Approximating Square Roots

A rational number is a number you can write as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$. As a decimal, a rational number either terminates (stops), like $0.75$, or repeats a block of digits forever, like $\frac{1}{3} = 0.\overline{3}$, where the bar marks the digit that repeats.

An irrational number is a real number that is not rational. No fraction of integers equals it, and its decimal neither terminates nor repeats.

So for $n \ge 0$, asking whether $\sqrt{n}$ is rational is really asking: is there a rational number whose square is $n$?

When $n$ is a whole number, there is a quick test. A perfect square is the square of a whole number, such as $49 = 7^2$.

• If $n$ is a perfect square, $\sqrt{n}$ is a whole number, so it is rational: $\sqrt{49} = 7 = \frac{7}{1}$.
• If $n$ is not a perfect square, $\sqrt{n}$ is irrational.

Why? A positive fraction in lowest terms has a numerator and a denominator with no common factor greater than $1$. Squaring it squares both parts, as in $\left(\frac{7}{2}\right)^2 = \frac{49}{4}$. Squaring repeats the prime factors each part already has and adds no new ones, so the square is in lowest terms too. If the denominator was greater than $1$, it still is, so the square is not a whole number. The only positive rational numbers whose squares are whole numbers are therefore the whole numbers themselves. Take $n = 50$: no whole number has square $50$, since $7^2 = 49$ and $8^2 = 64$ skip over it. So no rational number has square $50$, and $\sqrt{50}$ is irrational.

The number $10$ is not a perfect square, so $\sqrt{10}$ is irrational: you cannot write it exactly as a fraction, or as a decimal that ends or repeats. But you can trap it between two rational numbers, one below it and one above it. The ends of the trap, called its lower and upper bounds, are rational approximations of the root: rational numbers that are close to it but never equal to it.

You build the trap by comparing squares. Among numbers that are not negative, the larger number has the larger square. Here is why: take numbers $a$ and $b$ with $0 \le a < b$. Multiplying $a < b$ by $a$ (not negative) gives $a^2 \le ab$, and multiplying it by $b$ (positive) gives $ab < b^2$, so $a^2 < b^2$. It works backwards too: the larger square must come from the larger number, because an equal or smaller number would give an equal or smaller square.

So from $9 < 10 < 16$ you can conclude $3 < \sqrt{10} < 4$.

The condition matters. With negative numbers the order can flip: $-4 < 3$, but $(-4)^2 = 16$ is greater than $3^2 = 9$. That is why both ends of a trap must be $0$ or greater.

Often you want the nearest whole number or the nearest tenth. Suppose the ends of your trap are neighbors, such as the whole numbers $6$ and $7$ or the tenths $6.4$ and $6.5$. The midpoint of the trap is the number exactly halfway between its ends. It is the dividing line: inside the trap, a number below the midpoint is closer to the lower end, and a number above it is closer to the upper end.

So square the midpoint $m$ and compare $n$ with $m^2$:

• if $n < m^2$, then $\sqrt{n} < m$, so round to the lower end;
• if $n > m^2$, then $\sqrt{n} > m$, so round to the upper end.

This is the same rule about squares, and it applies because $m$ and $\sqrt{n}$ are not negative. When $\sqrt{n}$ is irrational, $n$ can never equal $m^2$, because that would make $\sqrt{n}$ equal to the rational number $m$.

Knowing $3 < \sqrt{10} < 4$ leaves a trap that is $1$ wide. The width of a trap is its upper end minus its lower end. Because $\sqrt{10}$ is inside the trap, each end is less than one width away from it. So the narrower the trap, the smaller the possible error.

To improve an approximation, narrow the trap:

1. Cut the trap into ten equal parts. Between $3$ and $4$ the cut points are $3.1, 3.2, \dots, 3.9$.
2. Square the cut points in order until a square passes $10$: $3.1^2 = 9.61$ is below $10$, and $3.2^2 = 10.24$ is above it.
3. Keep those two neighbors as the new trap: $3.1 < \sqrt{10} < 3.2$.

The new trap is $0.1$ wide, ten times narrower. Repeat the steps with hundredths, then thousandths, as the table shows. Each round fixes one more decimal digit of $\sqrt{10}$.

Takeaway. The square root of a whole number is rational only when that number is a perfect square. Otherwise it is irrational: you cannot write it exactly as a fraction, and its decimal never ends or repeats. But you can trap it between two rational numbers that are not negative by comparing squares, round it by checking the square of the midpoint, and cut the trap into ten parts again and again to get as close as you need. Next, you will use these traps to locate irrational numbers on a number line.