Number systems
Rational and irrational numbers, surds and rational powers.
Learn with Kutty
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Some numbers can’t be written as a fraction. Their decimals go on for ever and never repeat.
Look: Two sides of 1, so the long side is √2 (by Pythagoras: 1 + 1 = 2).
✏️ Your turn!
Type your answer and press Check. Kutty will tell you if it is right.
1Write 0.333… as a fraction.
2Is √49 rational?
3Is √8 rational?
! Careful!
✗π = 22/7 exactly.
✓22/7 is only close to π.
✗√9 + √16 = √25.
✓√9 + √16 = 3 + 4 = 7, but √25 = 5.
See the whole lesson on one page
Some numbers can’t be written as a fraction. Their decimals go on for ever and never repeat.
Fractions give decimals that stop (0.25) or repeat (0.333…). Numbers like √2 and π do neither. They are called irrational.
Think of it like thisDraw a square with 1 cm sides. Its diagonal is exactly √2 cm long. You can draw it perfectly, but its decimal 1.41421356… never ends or repeats.
- Rational
- Can be a fraction: ¾, 0.5, 7.
- Irrational
- Cannot be a fraction: √2, π.
Solved together
Write 0.4747… (47 repeating) as a fraction.
Call it x. Two digits repeat, so multiply by 100.
x = 0.4747…100x = 47.4747…Take x away. The repeating tail disappears!
99x = 47Divide by 99.
x = 47/99
Answer0.4747… = 47/99.
For older learners and grown-ups: a little more
To rationalise 1/(√5 − 2), multiply top and bottom by √5 + 2: (√5 + 2)/(5 − 4) = √5 + 2. Proof that √2 is irrational: if √2 = p/q in lowest terms, then p² = 2q², so p and q are both even — a contradiction.
Now practise it
Kanakku makes new questions on number systems every time. It shows the working and gives a hint when you are stuck.