What is an irrational number?
An irrational number is a number that cannot be written as p/q (p, q integers, q ≠ 0).
Its decimal never ends and never repeats a block. Examples: √2 = 1.41421356…, √3 = 1.7320508…, π = 3.14159265…, and 0.101001000100001… (the zeros keep growing, so no block repeats).
Careful: not every square root is irrational. √4 = 2, √9 = 3 and √(9/16) = 3/4 are rational. √n of a whole number n is irrational when n is not a perfect square.
Showing √2 on the number line
Draw a square with side 1 on the number line, with one corner at 0. By Pythagoras, its diagonal is √(1² + 1²) = √2.
Put the compass point at 0 and open it to the other end of the diagonal. Swing it down onto the line. The mark is exactly √2, between 1.41 and 1.42.
For √3: at the point √2 draw a line of length 1 at a right angle. The new long side is √(2 + 1) = √3. Swing it down the same way.
Proof that √2 is irrational
We use proof by contradiction: pretend the opposite is true and show it leads to nonsense.
- Assume √2 = p/q, with p and q having no common factor (lowest terms).
- Square: 2 = p²/q², so p² = 2q². So p² is even.
- If p were odd, p² would be odd (odd × odd = odd). So p is even. Write p = 2c.
- Then 4c² = 2q², so q² = 2c². So q² is even, and q is even.
- Shock: p and q both have the factor 2. But we said they have no common factor. Contradiction!
So our assumption was wrong: √2 is irrational.
Proof that √3 is irrational
- Assume √3 = p/q in lowest terms.
- Squaring: p² = 3q². So 3 divides p².
- Since 3 is a prime number, if 3 divides p², then 3 divides p. (Any number not a multiple of 3 leaves remainder 1 or 2 when divided by 3; its square leaves remainder 1, never 0.) Write p = 3c.
- 9c² = 3q², so q² = 3c². So 3 divides q² and hence q.
- 3 divides both p and q, which contradicts lowest terms.
So √3 is irrational. The same steps work for √5, √7 or √ of any prime.
The square root spiral
Start at a point O. Draw OA = 1 and, at A, a line AB = 1 at a right angle. OB = √2.
At B draw BC = 1 at a right angle to OB. Then OC = √(2 + 1) = √3. At C draw CD = 1 at a right angle to OC: OD = √4 = 2. Keep going.
Every new long side follows the rule (√n)² + 1² = n + 1. The triangles curl round into a snail-shell shape: the square root spiral. It shows √2, √3, √5, √6 … as real lengths you can draw, even though their decimals never end.
Real numbers and recurring (cyclic) decimals
Real numbers = all rational numbers + all irrational numbers. Every point on the number line is a real number, and every real number is a point on the line.
- Terminating decimal (0.375) → rational.
- Non-terminating repeating decimal (0.4747…) → rational.
- Non-terminating non-repeating decimal (1.4142…) → irrational.
Turning a repeating decimal into p/q: let x = 0.4747… The block has 2 digits, so multiply by 100: 100x = 47.4747… Subtract: 100x − x = 47, so 99x = 47 and x = 47/99.
A decimal like 1/7 = 0.142857 is called cyclic: 2/7, 3/7 … 6/7 use the same six digits in the same circle, just starting at a different place (2/7 = 0.285714…).
Also: rational + irrational is irrational (2 + √3), and a non-zero rational × irrational is irrational (5√2). But irrational + irrational can be rational: √2 + (−√2) = 0.
Try it: make a square root spiral
You need paper, a ruler, a set square (or a notebook corner) and a pencil. Take 1 unit = 2 cm.
- Draw OA = 2 cm. At A draw AB = 2 cm at a right angle. Join OB.
- At B draw BC = 2 cm at a right angle to OB. Join OC.
- Repeat 8 times. Measure OB, OC, OD … and divide by 2. Do you get about 1.41, 1.73, 2, 2.24?
Predict, then check: guess which long sides will be whole numbers, then move the slider in the 3D free-play step. (Only √4, √9, √16, √25.)
Key formulas and definitions
- Irrational: cannot be written as p/q; decimal never ends, never repeats
- Diagonal of a unit square = √(1² + 1²) = √2
- Spiral rule: (√n)² + 1² = (√(n + 1))²
- Prime p divides a² ⇒ p divides a
- 0.abab… = ab/99; 0.aaa… = a/9
- Real numbers = rational numbers + irrational numbers
Worked examples
1. Classify as rational or irrational: √25, √12, 0.3333…, 7.070070007…
√25 = 5, rational. √12: 12 is not a perfect square, irrational. 0.3333… repeats, rational (= 1/3). 7.070070007…: the zeros keep growing, it never repeats, irrational.
2. Locate √3 on the number line.
Step 1: draw a unit square on 0 to 1; its diagonal OB = √2. Step 2: swing OB onto the line to mark √2. Step 3: at √2 draw a line of length 1 upward at a right angle. Step 4: join 0 to its top: length √(2 + 1) = √3. Step 5: swing this down with a compass. The mark is √3 ≈ 1.732.
3. Prove that √5 is irrational.
Assume √5 = p/q in lowest terms. Square: p² = 5q², so 5 divides p². 5 is prime, so 5 divides p; let p = 5c. Then 25c² = 5q², so q² = 5c², and 5 divides q. Now 5 divides both p and q, which contradicts lowest terms. So √5 is irrational.
4. Express 0.6666… in the form p/q.
Let x = 0.6666… One digit repeats, so multiply by 10: 10x = 6.6666… Subtract: 10x − x = 6, so 9x = 6 and x = 6/9 = 2/3.
5. Express 0.4747… (0.47 bar) in the form p/q.
Let x = 0.4747… Two digits repeat, so multiply by 100: 100x = 47.4747… Subtract: 99x = 47, so x = 47/99.
6. Express 0.2353535… (only 35 repeats) in the form p/q.
Let x = 0.2353535… First move the non-repeating 2 past the point: 10x = 2.353535… Then 1000x = 235.353535… Subtract: 1000x − 10x = 235 − 2 = 233, so 990x = 233 and x = 233/990.
7. Is 0.9999… equal to 1?
Let x = 0.9999… Then 10x = 9.9999… Subtract: 9x = 9, so x = 1. Yes, 0.9999… and 1 are the same real number, written two ways.
8. Find one irrational number between 1/7 and 2/7.
1/7 = 0.142857… and 2/7 = 0.285714… Pick a decimal between them that never ends and never repeats, like 0.150150015000150000… It lies between them and is irrational.
Common mistakes
- Saying every square root is irrational. √16 = 4 and √(4/9) = 2/3 are rational.
- Calling 22/7 or 3.14 equal to π. They are only close; π is irrational, 22/7 is rational.
- In the proof, forgetting to say p and q have no common factor. Without it there is no contradiction.
- Multiplying by 10 when two digits repeat. For 0.4747… you must use 100x, or the repeating parts do not cancel.