Rational or Irrational? Deciding Quickly

A brief note on Rational or Irrational? Deciding Quickly. Read it before a test. Try each question at the end before you open its answer.

The two kinds

• Form: rational: $\dfrac{p}{q}$, $q \neq 0$; irrational: not a fraction
• Decimal: rational: ends or repeats; irrational: never ends, never repeats
• Examples: rational: $-3$, $0.75$, $\dfrac{2}{7}$, $\sqrt{49}$; irrational: $\sqrt{2}$, $\sqrt{10}$, $\pi$

Together they make the real numbers, and each one is a point on the number line.

Quick tests

• A fraction. Reduce it. It ends exactly when the bottom has no prime factor other than $2$ and $5$.
• A square root of a whole number. Rational only if the number is a perfect square.
• A decimal. Look for one block that repeats for ever. A pattern that keeps growing is not a repeating block.
• A sum or product. Rational plus irrational is irrational. Two irrational numbers can give either kind.

Between two numbers

There are always infinitely many rational numbers between two numbers. The average is one of them. The square root spiral and a compass place $\sqrt{2}$, $\sqrt{3}$ and the rest exactly on the line.

A worked example

Decide quickly: $0.142857\,142857\ldots$, $\sqrt{1.44}$, $3 + \sqrt{2}$, $\dfrac{17}{80}$.

1. $0.142857\,142857\ldots$ repeats the block $142857$: rational. It is $\dfrac{1}{7}$.
2. $\sqrt{1.44}$ $= 1.2$, since $1.2^2 = 1.44$: rational.
3. $3 + \sqrt{2}$ is rational plus irrational: irrational.
4. $\dfrac{17}{80}$: $80 = 2^4 \times 5$, so it ends: $0.2125$, rational.

Practice with new numbers

A program to try

The program runs the two quick tests on a few numbers.

Mistakes to avoid

• Calling every long decimal irrational. $0.123123\ldots$ repeats, so it is rational.
• Calling every square root irrational. $\sqrt{0.25} = 0.5$.
• Taking $\dfrac{22}{7}$ as $\pi$. It is only close.
• Looking at the bottom before reducing. $\dfrac{21}{30} = \dfrac{7}{10}$ ends, though $30$ has a factor $3$.

Check yourself

1. Is $0.125125125\ldots$ rational?

2. Does the decimal of $\dfrac{9}{150}$ end?

3. Is $\sqrt{1.21}$ rational?

4. Is $7 - \sqrt{3}$ rational?