Which fractions make decimals that stop and which repeat, numbers whose digits never stop or repeat (square roots and π), and the laws for multiplying and dividing powers.
Long division shows why a fraction's decimal stops or repeats, and the denominator's prime factors predict which.
Dividing a fraction out gives a decimal that either stops (terminates) or repeats a block of digits forever (recurs). In long division, a remainder of 0 means it stops; a remainder that comes back means it repeats. A bar or dots mark the repeating digits.
Write the fraction in simplest form and break the denominator into primes. If its only prime factors are 2 and 5, the decimal terminates; any other prime factor makes it recur. It works because 10 = 2 × 5.
Numbers like √2 and π that are not fractions: their digits never stop or repeat, and where they sit on the number line.
A rational number is a fraction: its decimal stops or repeats. An irrational number cannot be written as a fraction: its decimal never stops and never repeats. √2, √3 and π are irrational; √16 = 4 is not.
Every point on the number line is a real number, rational or irrational. To place √30, find the perfect squares either side: 25 < 30 < 36, so √30 is between 5 and 6, nearer 5. A rounded value like 5.48 is close, not exact.
Multiplying, dividing and raising powers of the same base, and why a⁰ = 1.
Write powers out as factors and the laws appear: multiplying powers of the same base adds the indices, dividing subtracts them, and a power of a power multiplies them. Anything (except 0) to the power 0 is 1.
Simplify one step at a time: powers of powers first, then multiply and divide from left to right, keeping each base. The laws only join powers of the same base: 2³ × 3² cannot become one power. They also make scientific notation quick.
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