Learn ยท Mathematics ยท Class 8

Rational & Real Numbers

The number families, how they fit together, and why some numbers never end.

10 parts
Scroll to begin โ†“
๐Ÿ“˜ DefinitionThe families of numbersPart 1 of 10
๐Ÿงฉ The partsWhat counts as rationalPart 2 of 10
๐Ÿ” In detailWhy โˆš2 cannot be a fractionPart 3 of 10
โœ๏ธ Worked exampleConverting a recurring decimal to a fractionPart 4 of 10
๐Ÿ” In detailOperations and closurePart 5 of 10
โœ๏ธ Worked exampleSurdsPart 6 of 10
๐ŸŒ In the real worldWhere these matterPart 7 of 10
โš ๏ธ Common mistakesWhere it usually goes wrongPart 8 of 10
โœ… SummaryWhat to rememberPart 9 of 10
โœ๏ธ Worked exampleWatch one worked throughPart 10 of 10
1๐Ÿ“˜ DefinitionPart 1 / 10

The families of numbers

Numbers are grouped into nested sets, each containing the one before it.

  • Natural: 1, 2, 3, โ€ฆ
  • Whole: the naturals plus 0.
  • Integers: whole numbers and their negatives.
  • Rational: anything writable as a fraction p/q where q โ‰  0.
  • Irrational: cannot be written as a fraction.
  • Real: rationals and irrationals together.

Each set is contained in the next, so every integer is also rational.

2๐Ÿงฉ The partsPart 2 / 10

What counts as rational

More things than you might expect.

  • Integers: 5 is 5/1.
  • Terminating decimals: 0.75 is 3/4.
  • Recurring decimals: 0.333โ€ฆ is 1/3.

The test: a decimal is rational if it either stops or repeats a pattern for ever.

Irrational numbers do neither. They continue for ever with no repeating pattern.

โˆš2 = 1.41421356โ€ฆ
ฯ€ = 3.14159265โ€ฆ
3๐Ÿ” In detailPart 3 / 10

Why โˆš2 cannot be a fraction

This was proved over two thousand years ago and the argument is short enough to follow.

Suppose it could be. Then โˆš2 = p/q, in lowest terms, so p and q share no common factor.

Squaring: 2 = pยฒ/qยฒ, so pยฒ = 2qยฒ.

That means pยฒ is even, so p must be even (an odd number squared is always odd).

If p is even, write p = 2k. Then pยฒ = 4kยฒ, so 4kยฒ = 2qยฒ, giving qยฒ = 2kยฒ. So q is even too.

But we assumed they shared no common factor: and now both are even. That is a contradiction.

So the original assumption was false: โˆš2 cannot be written as a fraction. This is proof by contradiction, and it is one of the most powerful ideas in mathematics.

4โœ๏ธ Worked examplePart 4 / 10

Converting a recurring decimal to a fraction

Question. Write 0.363636โ€ฆ as a fraction.

Step 1: name it. Let x = 0.363636โ€ฆ

Step 2: multiply so the repeating block lines up. The block "36" is two digits, so multiply by 100.

100x = 36.363636โ€ฆ

Step 3: subtract the first from the second. The infinite tails are identical and cancel exactly.

100x โˆ’ x = 36.3636โ€ฆ โˆ’ 0.3636โ€ฆ
99x = 36

Step 4: solve and simplify.

x = 36/99 = 4/11

Check: 4 รท 11 = 0.3636โ€ฆ โœ… This is why every recurring decimal is rational, the method always works.

5๐Ÿ” In detailPart 5 / 10

Operations and closure

A set is closed under an operation if the answer always stays in that set.

  • Integers are closed under +, โˆ’ and ร—, but not รท. 3 รท 2 leaves the integers.
  • Rationals are closed under all four (excluding division by zero).

Irrationals are not closed under anything. โˆš2 ร— โˆš2 = 2, which is rational. This is why "irrational" is not a well-behaved family. It is defined by what it is not.

Ordering rationals: to compare 3/7 and 5/11, cross-multiply: 3 ร— 11 = 33 and 5 ร— 7 = 35, so 3/7 < 5/11.

6โœ๏ธ Worked examplePart 6 / 10

Surds

A surd is a root left in exact form rather than as a rounded decimal.

Why keep them: โˆš2 is exact; 1.414 is not. In a long calculation, rounding early accumulates error.

Simplifying: take out any square factor.

โˆš50 = โˆš(25 ร— 2) = 5โˆš2

Multiplying:

โˆš3 ร— โˆš12 = โˆš36 = 6

Adding: only like surds combine, exactly like algebraic terms.

3โˆš2 + 5โˆš2 = 8โˆš2 โœ…
โˆš2 + โˆš3  , cannot be combined

The common error: โˆš2 + โˆš3 is not โˆš5. Check with a calculator: 1.414 + 1.732 = 3.146, while โˆš5 = 2.236.

7๐ŸŒ In the real worldPart 7 / 10

Where these matter

Measurement. The diagonal of a 1 m square is exactly โˆš2 m, an irrational length that exists physically. Irrational numbers are not abstractions.

Circles. ฯ€ appears in every circle calculation, and it is irrational, so no circle measurement is ever exactly expressible as a fraction.

Computing. Computers store irrationals as approximations, which is why floating-point arithmetic occasionally gives 0.1 + 0.2 = 0.30000000000000004.

Engineering. Knowing when to keep an exact surd and when to round is a real skill, rounding too early is a genuine source of error in construction.

8โš ๏ธ Common mistakesPart 8 / 10

Where it usually goes wrong

  • "Recurring decimals are irrational." They repeat, so they are rational.
  • "ฯ€ = 22/7." That is an approximation; ฯ€ is irrational.
  • "โˆš2 + โˆš3 = โˆš5."
  • "All roots are irrational." โˆš9 = 3.
  • "Integers are not rational." Every integer is n/1.
9โœ… SummaryPart 9 / 10

What to remember

  • Rational = writable as p/q; decimals that stop or repeat.
  • Irrational never ends and never repeats, โˆš2 proved by contradiction.
  • Recurring โ†’ fraction by multiply and subtract.
  • Only like surds add.
10โœ๏ธ Worked examplePart 10 / 10

Watch one worked through

Here is one done slowly, with the reason for every move. Then one for you.

Watch one first

Is 0.333... (recurring) rational or irrational? Prove your answer.

  1. Recall the test: a rational number can be written as one whole number over another.
    rational โ‡” a/b (b โ‰  0)
    Decide by the definition, not by appearance. A long decimal is not automatically irrational.
  2. Let x be the number, then multiply by 10.
    x = 0.333... 10x = 3.333...
    Multiplying by 10 shifts the digits one place, and because the pattern repeats the tail is now identical.
  3. Subtract the first line from the second. The repeating tails cancel.
    10x โˆ’ x = 3.333... โˆ’ 0.333... 9x = 3
    The whole trick is making the two tails match so they vanish. That only works because the pattern never changes.
  4. Solve for x.
    x = 3/9 = 1/3 โ†’ rational
    Every recurring decimal can be written this way, so every recurring decimal is rational. Only non-repeating, non-ending decimals like ฯ€ are not.

Answer: Rational, because it equals 1/3

Now you try

Is 0.25 rational or irrational? One word.

Show the answer

rational

Practise Rational & Real Numbers

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๐Ÿง  5 quizzes on Rational & Real Numbers (2 easy, 2 medium, 1 hard)

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๐Ÿ–จ๏ธ Printable worksheet

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