Irrational numbers (intro)
The idea
Some lengths refuse to be fractions. The diagonal of a 1-by-1 square is √2, and no fraction of whole numbers, however clever, equals it exactly — its decimal runs forever without ever settling into a repeating pattern. Numbers like √2, √40, and π are called irrational, and together with the rationals you already know — fractions, decimals that end or repeat, and their negatives — they fill in the complete number line with no gaps.
You cannot write an irrational number down exactly in decimal form, but you can trap it as tightly as you like between rationals, and that squeeze is the working skill. A common myth is that π equals 22/7 or 3.14 — both are merely convenient approximations, since a true fraction would make π rational, which it is not. So treat values like √40 as exact names, and squeeze out a decimal estimate only when a measurement or comparison demands one.
Worked example
Without a calculator, estimate √40 to one decimal place.
- Trap 40 between neighboring perfect squares: 36 = 6² and 49 = 7², so √40 lies between 6 and 7, and nearer to 6 because 40 sits close to 36.
- Test a candidate in the gap: 6.3² = 39.69, which is just under 40, so √40 is a bit more than 6.3.
- Test the next tenth: 6.4² = 40.96, which overshoots 40, so √40 is trapped between 6.3 and 6.4.
- Decide which tenth is closer: 40 − 39.69 = 0.31 while 40.96 − 40 = 0.96, so 40 sits much nearer to 6.3², making 6.3 the better one-decimal estimate.
Answer. √40 ≈ 6.3, since it is trapped between 6.3 and 6.4 and lies much closer to 6.3.
Check your understanding
- Why does squeezing a square root between two perfect squares work, and how could you push the estimate to two decimal places?
- What is the difference between a decimal that never ends but repeats and one that never ends and never repeats?
- Why are 3.14 and 22/7 not the same thing as π, and when does that distinction actually matter?
- Which square roots of whole numbers are rational, and how can you spot them before reaching for an estimate?
Build the foundations first
Irrational numbers (intro) builds on these concepts. If any feel shaky, start there.