Real numbers: primes, irrationals and what this chapter does

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When Srinivasa Ramanujan was ill in England, the mathematician G. H. Hardy came to visit him in a taxi numbered 1729, and remarked that it seemed a dull number.

Ramanujan replied at once that it was very interesting: it is the smallest number that can be written as the sum of two cubes in two different ways, $1^3 + 12^3$ and $9^3 + 10^3$ .

Break 1729 down and you find $1729 = 7 \times 13 \times 19$ , three primes multiplied together.

Every whole number bigger than 1 can be broken down like this, into primes, and this chapter is about what that fact lets you do.

The real number family

You met these number sets in Class IX.

Each new set grows out of the one before it.

Natural numbers are the counting numbers 1, 2, 3, 4, ...

Whole numbers are the natural numbers together with 0.

Integers add the negative whole numbers: ..., −3, −2, −1, 0, 1, 2, 3, ...

Rational numbers are all the numbers that can be written as a fraction $\dfrac{p}{q}$ of two integers, with $q \neq 0$ . Every integer is rational, since $-7 = \dfrac{-7}{1}$ . Terminating decimals such as 0.6 and recurring decimals such as $0.\overline{3} = \dfrac{1}{3}$ are rational too.

Irrational numbers are the numbers on the number line that cannot be written as such a fraction. Their decimals never end and never settle into a repeating block. $\sqrt{2}$ , $\sqrt{5}$ and $\pi$ are irrational.

Real numbers are the rational and irrational numbers taken together. Every point on the number line is a real number.

The figure below shows how the sets sit inside one another.

Nested boxes showing the real number family. The large outer box is the real numbers. On its left is the box of rational numbers, which holds the integers, which hold the whole numbers, which hold the natural numbers. On its right is the box of irrational numbers. Sample numbers are written in each box: 1, 2, 3 as natural numbers; 0 as a whole number; minus 7 as an integer; 3/4 and minus 0.6 as rational numbers; and √2, √5 and π as irrational numbers.

A number belongs to every box it sits inside: 3 is natural, whole, an integer, rational and real.

But no number is both rational and irrational.

The two boxes on the right and left never overlap, and together they fill the whole of the real numbers.

Primes, composites and the number 1

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, ...

A composite number is a natural number greater than 1 with at least one more factor: 4, 6, 9, 91 (which is $7 \times 13$ ).

The number 1 has only one factor, so it is neither prime nor composite.

And 2 is the only even prime, because every other even number has 2 as an extra factor.

Primes are the building blocks of multiplication.

A composite number can always be split into smaller factors, and those into smaller ones, until only primes are left.

Primes cannot be split any further, so they are the "atoms" every other natural number is built from.

What this chapter does

This chapter uses prime factors for three jobs.

1. The Fundamental Theorem of Arithmetic. Every composite number is a product of primes, and only one set of primes works. This uniqueness settles questions such as "can a power of 35 ever end in 0?"

2. HCF and LCM. Once two numbers are written as products of primes, their highest common factor and lowest common multiple can be read straight off. Word problems about packing, tiling and bells ringing together use them.

3. Proving numbers irrational. You were told in Class IX that $\sqrt{2}$ is irrational, but it was never proved. Here you prove that $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ and numbers such as $3 + 2\sqrt{5}$ are irrational. The proof rests on prime factors too, which is why the two ideas share a chapter.

Older papers sometimes asked about decimal expansions of rational numbers or used Euclid's division lemma to find an HCF; neither is in your syllabus now.

Key terms

Prime number: a natural number greater than 1 whose only factors are 1 and the number itself.

Composite number: a natural number greater than 1 that has at least one factor other than 1 and itself.

Rational number: a number that can be written in the form $\dfrac{p}{q}$ , where p and q are integers and $q \neq 0$ .

Irrational number: a number ‘s’ is called irrational if it cannot be written in the form $\dfrac{p}{q}$ , where p and q are integers and $q \neq 0$ .

Co-prime numbers: two positive integers whose only common factor is 1, such as 8 and 15.

How the marks work

Answers in Sections B to E are marked in ½-mark steps that add up to the question's marks. Each correct line of working earns its own share.

In Sections B to D a bare answer with no working earns little or nothing. Every prime factorisation line, for example, carries its own ½ before the HCF or LCM earns a mark.

An error is penalised only once. If you carry a wrong value forward correctly, your later steps still score.

Any correct method earns full marks, even one that differs from the method in these notes.

In a proof, the first line (what you assume) and the last line (the conclusion, such as "Hence $\sqrt{5}$ is irrational") each earn ½, so never leave them out.

Multiple-choice and Assertion-Reason questions carry 1 mark each, with no part marks.

How this chapter is examined

This chapter carries 6 of the 80 marks in the board examination, in both Mathematics Standard and Mathematics Basic.

Expect one to three multiple-choice questions (HCF and LCM, prime factorisation, the last digit of a power), sometimes an Assertion-Reason item, a 2-mark question and a 3-mark question.

Every paper has one irrationality proof: either $\sqrt{p}$ from first principles or a number such as $a + b\sqrt{p}$ "given that $\sqrt{p}$ is irrational".

Worked example 1: prime, composite or neither?

Sort the numbers 1, 2, 51, 53, 91 and 0 into prime, composite, or neither.

1. 2 has exactly two factors, 1 and 2, so it is prime.

2. 51 = 3 × 17, so it is composite. (Its digits add to 6, which shows 3 is a factor.)

3. 53 is not divisible by 2, 3, 5 or 7. Since $7^2 = 49$ and $8^2 = 64$ , any factor pair of 53 would need a factor of 7 or less, so 53 is prime.

4. 91 = 7 × 13, so it is composite, even though it looks prime.

5. 1 has only one factor, and 0 is not a natural number, so both are neither prime nor composite.

Answer: prime: 2, 53; composite: 51, 91; neither: 1, 0.

This is a Basic-paper style.

To test whether a number n is prime, you only need to try the primes up to $\sqrt{n}$ .

If n had a factor bigger than $\sqrt{n}$ , its partner factor would be smaller than $\sqrt{n}$ , and you would already have found it.

Worked example 2: rational or irrational?

Which of these are rational, and which are irrational?

$\sqrt{64}$ , $\sqrt{48}$ , $\dfrac{22}{7}$ , $\pi$ , $2 + \sqrt{9}$ , $\sqrt{12} \times \sqrt{3}$ .

1. $\sqrt{64} = 8 = \dfrac{8}{1}$ : rational.

2. $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$ : irrational, because 48 is not a perfect square.

3. $\dfrac{22}{7}$ is a fraction of two integers: rational. It is only an approximate value of $\pi$ .

4. $\pi$ is irrational.

5. $2 + \sqrt{9} = 2 + 3 = 5$ : rational.

6. $\sqrt{12} \times \sqrt{3} = \sqrt{36} = 6$ : rational.

Answer: rational: $\sqrt{64}$ , $\dfrac{22}{7}$ , $2 + \sqrt{9}$ , $\sqrt{12} \times \sqrt{3}$ ; irrational: $\sqrt{48}$ , $\pi$ .

Both papers set this.

The lesson of worked example 2 is to simplify before you decide.

A square-root sign does not make a number irrational: the square root of a natural number is rational only when that number is a perfect square, and irrational otherwise.

Worked example 3: can a product of two primes be prime?

Is it possible for the product of two prime numbers to be a prime number?

Justify your answer.

1. Let the primes be p and q, and let n = pq.

2. Then p is a factor of n, and p is neither 1 (a prime is greater than 1) nor n (since q > 1, n is bigger than p).

3. So n has a factor other than 1 and itself, which makes n composite.

Answer: no. The product of two primes always has a third factor besides 1 and itself, so it is always composite.

For example, 3 × 5 = 15 has the factors 1, 3, 5 and 15.

This is a Standard-paper style.

Exam tip

The number 1 is neither prime nor composite, and multiple-choice questions use it as a trap.

If an option says "1 is the smallest prime" or counts 1 among the prime factors of a number, it is wrong: the smallest prime is 2.

Common mistakes

Counting 1 as a prime. A prime needs exactly two different factors; 1 has only one. Never write 1 in a prime factorisation.

Thinking every odd number is prime. 9, 15, 51 and 91 are odd and composite. Test by dividing by the primes up to the square root.

Thinking every square root is irrational. $\sqrt{49} = 7$ and $\sqrt{0.25} = 0.5$ are rational. Simplify first, then decide.

Treating $\dfrac{22}{7}$ as equal to $\pi$ . $\dfrac{22}{7}$ is rational and $\pi$ is irrational. You use 22/7 in place of $\pi$ only because the question tells you to.

Check your understanding

1. (Basic and Standard) Which of the following is a composite number?

(A) 2

(B) 1

(D) 47

[1]

2. (Basic) Which of the following is an irrational number?

(A) $\sqrt{0.36}$

(B) $\dfrac{22}{7}$

(D) $0.\overline{6}$

[1]

3. (Basic and Standard) Assertion (A): The number 1 is not a prime number.

Reason (R): A prime number has exactly two distinct factors, 1 and the number itself.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(D) Assertion (A) is false but Reason (R) is true.

[1]

4. (Standard) Assertion (A): $\sqrt{36}$ is a rational number.

Reason (R): The square root of every composite number is a rational number.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(D) Assertion (A) is false but Reason (R) is true.

[1]

5. (Basic and Standard) Check whether 221 is a prime number or a composite number. Justify your answer.

[2]

6. (Standard) Is it possible for the sum of two odd prime numbers to be a prime number? Justify your answer.

[2]

Answers

1. (C) 87, since 87 = 3 × 29. [1]

2. (C) $\sqrt{72}$ , since $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ . The others are 0.6, a fraction, and the recurring decimal $\dfrac{2}{3}$ . [1]

3. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). 1 has only one factor, so it fails the rule in R. [1]

4. (C) Assertion (A) is true but Reason (R) is false. $\sqrt{36} = 6$ , so A is true. R is false: 12 is composite, but $\sqrt{12} = 2\sqrt{3}$ is irrational. [1]

5. 221 is not divisible by 2, 3, 5, 7 or 11 ........ ½

221 = 13 × 17 ........

So 221 has factors other than 1 and itself, and it is composite ........

½ [2]

6. Every odd prime is an odd number, and odd + odd = even ........ ½

The smallest odd prime is 3, so the sum is at least 3 + 3 = 6 ........

An even number greater than 2 has 2 as a factor, so it is composite ........

So it is not possible ........

½ [2]

Real numbers: primes, irrationals and what this chapter does

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