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Real Numbers
Introduction
The Fundamental Theorem of Arithmetic
HCF and LCM, the Prime Way
Practice: Factors, HCF and LCM
Revisiting Irrational Numbers
Practice: Proving Irrationality
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Real NumbersIntroduction

Reading time: ~10 min

Everything around you — water, air, your phone, you — is made from about a hundred kinds of atoms. Keep breaking matter into smaller and smaller pieces, and eventually you reach building blocks that cannot be broken any further.

Whole numbers work in exactly the same way. Their atoms are the prime numbers — numbers like 2, 3, 5 and 7 that have exactly factors: 1 and the number itself.

In this chapter you will discover the one law that controls how every number is built from primes — the Fundamental Theorem of Arithmetic. Then you will put it to work on two very different jobs: finding the HCF and LCM of numbers in a few strokes, and proving that numbers like \sqrt{2} can never be written as a fraction.

Warm-up: Factor or Multiple?

A quick reminder of two words we will use all chapter. Since 3 × 7 = 21, we say that 3 and 7 are factors of 21, and that 21 is a multiple of 3 and of 7.

Speed round: get five right to move on.

${x}is a?of${y}

Primes Refuse to Be Rectangles

Here is a way to see factors. Lay a number out as tiles and try to arrange them into a rectangle. Every rectangle you can make is a factor pair. A prime is so stubborn that it allows only rectangle — a single long row.

Drag the black tab to reshape the tiles. Every time the tiles form a full rectangle, you have found a factor pair.

24 — composite

24 rearranges in many ways — 1 × 24, 2 × 12, 3 × 8 and 4 × — so it has plenty of factors. Numbers like this, with more than two factors, are called composite.

23 — prime

23 refuses every arrangement except . With no other factor pair, it cannot be broken into smaller pieces — it is one of the "atoms".

Hunting for Primes

How can we find all the primes up to 100 without testing each number one by one? More than 2,000 years ago, the Greek scholar Eratosthenes invented a clever filter called a sieve.

1 is not a prime, so it is already greyed out. 2 is the first prime — but every other multiple of 2 has 2 as a factor, so none of them can be prime. The same goes for multiples of 3, 5 and 7.

Press the four buttons to sieve out the multiples of 2, 3, 5 and 7.

Every number still standing is a prime. (Why can we stop at 7? Because the next prime, 11, has 11 × 11 = 121 — bigger than 100. Any composite number up to 100 must already have a factor of 2, 3, 5 or 7.) Count the survivors: there are primes below 100.

Sina