Cambridge International Mathematics

2 Assumed Knowledge (Number)

The set ofirrationalnumbers includes all real numbers which cannot be written in the form

a

b

whereaandbare integers and b 6 =0.

For example: ¼,

p

2 and

p

3 are all irrational.

p

9 and

p

1 : 21 are rational since

p

9=3=^31 and

p

1 :21 = 1:1=^1110 :

PRIMES AND COMPOSITES

Thefactorsof a positive integer are the positive integers which divide exactly into it, leaving no remainder.

For example, the factors of 10 are: 1 , 2 , 5 and 10.

A positive integer is aprimenumber if it has exactly two factors, 1 and itself.

A positive integer is acompositenumber if it has more than two factors.

For example: 3 is prime as it has two factors: 1 and 3.

6 is composite as it has four factors: 1 , 2 , 3 and 6.

1 is neither prime nor composite.

If we are given a positive integer, we can use the following procedure to see if it is prime:

Step 1: Find the square root of the number.

Step 2: Divide the whole number in turn by all known primes less than its square root.

If the division is never exact, the number is a prime.

The first few prime numbers are 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , ......

Example 1 Self Tutor

Is 131 a prime number?

p

131 = 11: 445 ......, so we divide 131 by 2 , 3 , 5 , 7 and 11.

131 ¥2=65: 5

131 ¥3=43: 66 :::::

131 ¥5=26: 2

131 ¥7=18: 7142 :::::

131 ¥11 = 11: 9090 :::::

None of the divisions is exact, so 131 is a prime number.

OTHER CLASSIFICATIONS

Aperfect squareorsquare numberis an integer which can be written as the square of another integer.

For example, 4 and 25 are perfect squares since 4=2^2 and 25 = 5^2.

Aperfect cubeis an integer which can be written as the cube of another integer.

For example, 8 and¡ 125 are perfect cubes since 8=2^3 and ¡125 = (¡5)^3.

IGCSE01

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