← Irrational numbers and index laws

Maths reference: Irrational numbers and index laws

The words and methods from Irrational numbers and index laws, each with an example and, where it helps, a picture of the board.

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Recurring decimals

38=0.37517=0.142857‾\tfrac{3}{8} = 0.375 \qquad \tfrac{1}{7} = 0.\overline{142857}
terminating and recurring

In long division a remainder of 0 means the decimal stops; a remainder that comes back means the digits repeat. A bar (or dots) marks the repeating block.

1 ÷ 7: the remainders come round again
1 ÷ 7: the remainders come round again
simplest form, denominator only 2s and 5s⇒terminates\text{simplest form, denominator only 2s and 5s} \Rightarrow \text{terminates}
which decimals stop

Simplify first, then break the denominator into primes. Any prime other than 2 or 5 makes it recur, because 10 = 2 × 5.

Irrational numbers

rational: abirrational: 2, π\text{rational: } \tfrac{a}{b} \qquad \text{irrational: } \sqrt{2},\ \pi
rational and irrational

Rational numbers are fractions: their decimals stop or repeat. Irrational numbers never stop and never repeat. √16 = 4 is rational.

A number line to zoom in on √2
A number line to zoom in on √2
25<30<36⇒5<30<625 < 30 < 36 \Rightarrow 5 < \sqrt{30} < 6
placing a square root

Find the perfect squares either side, then square decimals to get closer. A rounded value is an approximation.

Index laws

am×an=am+nam÷an=am−n(am)n=amna^m \times a^n = a^{m+n} \quad a^m \div a^n = a^{m-n} \quad (a^m)^n = a^{mn}
index laws

Write the powers out as factors to see why. Only powers of the same base combine.

3⁵ ÷ 3²: pairs of 3s cancel
3⁵ ÷ 3²: pairs of 3s cancel
a0=1(a≠0)a^0 = 1 \quad (a \ne 0)
zero index

aⁿ ÷ aⁿ = 1, and the quotient law calls it a⁰.