SM1.1 Real numbers
You can break any whole number into its prime factors, use that to find the HCF and LCM of two or three numbers, and explain why a number like the square root of 2 can never be written as a fraction.
Topics shown in module order.
Every whole number is a product of primes in exactly one way, and that single fact does surprising work: it finds the HCF and LCM of large numbers without long division, and it proves that some numbers are not fractions at all. Those numbers, the irrationals, sit on the number line beside the fractions, and together they are the real numbers this chapter is named for.
Work through these
Factorise three large numbers into primes, and find the HCF and LCM of a pair from the factorisations alone
The factor tree is the tool the whole chapter rests on. Reading HCF and LCM straight off the shared and combined factors, without any division, is the skill the method exists to give you.
Check that HCF times LCM equals the product of the two numbers, and say why that must happen
The check works because every prime factor lands in exactly one of the two answers. Seeing why it must be true, rather than treating it as a formula, is what lets you use it to find a missing value.
Follow the proof that the square root of 2 is irrational until you can retell it in your own words
This is most people's first proof by contradiction: assume the opposite, follow it honestly, and watch it break. Retelling it without the book is the test of whether you followed the argument or the notation.
Decide from the denominator alone whether a fraction's decimal form ends or repeats forever
A fraction ends only when its bottom is built from 2s and 5s, because those are the primes of 10. The rule turns a long division you might get wrong into a factorisation you can check in seconds.
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