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Revision:Numbers

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Numbers

1 Types of numbers
2 LCM and HCF
3 Rounding Numbers
4 Approximations
5 BODMAS
6 Comments

Types of numbers

Integers

Integers are whole numbers (both positive and negative). Zero is usually classed as an integer.

Natural Numbers

Natural numbers are positive integers.

Rational Numbers

A rational number is a number which can be written as a fraction where numerator and denominator are integers (where the top and bottom of the fraction are whole numbers). For example 1/2, 4, 1.75 (7/4). A rational number can be written as an exact or recurring decimal. For example 0.175 is rational since it is an exact decimal. 0.345345345... is rational since it is a recurring decimal.

Irrational Numbers

Irrational numbers are numbers which cannot be written as fractions, such as the square root of 2, $\pi$ and the $\sqrt[3]{5}$ . In decimal form irrational numbers go on forever and the same pattern of digits are not repeated.

Surds

Surds are expressions which contain irrational square roots such as $\sqrt{2} + \sqrt{3}$ and $\sqrt{5}\sqrt{3} - \sqrt{2}\sqrt{3}$ .

There are some rules to learn about them which will help answering lots of questions, for example, when needing to give 'exact answers' or solving quaratic equations with the quadratic formula.

$\displaystyle \sqrt{a} \times \sqrt{b} = \sqrt{ab}$ ,

$\displaystyle \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ ,

$\displaystyle (a + \sqrt{b})^2 = (a + \sqrt{b})(a + \sqrt{b}) = a^2 + 2a\sqrt{b} + b$ ,

$\displaystyle (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$ .

Square Numbers

Square numbers are formed by multiplying a whole number by itself. E.g. 36 is a square number because it is 6 x 6 .

What are the first 15 sqaure numbers?

Triangle Numbers

Triangle numbers associated with a triangular sequence of dots. Each pattern has an extra row of dots.

The difference pattern of triangular numbers (ie 'what you add on to get to the next number')is simply 2, 3, 4, 5, 6...

Prime

Prime numbers are numbers above 1 which cannot be divided by anything, other than 1 and itself, to give an integer. The first 8 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

Real

Real numbers are all the numbers which you will have come across (i.e. all the rational and irrational numbers).

Fractors

Multiples

LCM and HCF

The lowest common multiple (LCM) of two or more numbers is the smallest number into which they evenly divide. For example, the LCM of 2, 3, 4, 6 and 9 is 36.

The highest common factor (HCF) of two or more numbers is the highest number which will divide into them both. Therefore the HCF of 6 and 9 is 3.

Rounding Numbers

If the answer to a question was 0.00256023164, you would not write this down. Instead, you would 'round off' the answer. There are two ways to do this, you can round off to a certain number of decimal places or a certain number of significant figures.

The above number, rounded off to 5 decimal places (d.p.) is 0.00256 . You write down the 5 numbers after the decimal point. To round the number to 5 significant figures, you write down 5 numbers. However, you do not count any zeros at the beginning. So to 5 s.f. (significant figures), the number is 0.0025602 (5 numbers after the first non-zero number appears).

From what we know so far, if you rounded 4.909 to 2 decimal places, the answer would be 4.90 . However, the number is closer to 4.91 than 4.90, because the next number is a 9. Therefore, the rule is: if you are rounding a number, if the number after the place you stop is 5 or above, you add one to the last number you write. So 3.486 to 3s.f. is 3.49 0.0096 to 3d.p. is 0.010 (This is because you add 1 to the 9, making it 10.

When rounding to a number of decimal places, always write any zeros at the end of the number. If you say 3d.p., write 3 decimal places, even if the last digit is a zero).

Approximations

If the side of a square field is given as 90m, correct to the nearest 10m, what is a) the smallest possible actual length b) the largest possible length c) the smallest possible area d) the largest possible area?

a) The smallest value the actual length could be is 85m (since this is the lowest value which, to the nearest 10m, would be rounded up to 90m).

b) The largest value is 95m (since this is the highest values which, to the nearest 10m, would be rounded down to 90m).

c) The smallest possible area is $85 \times 85 = 7725m^2$ .

d) The largest possible area is $95 \times 95 = 9025m^2$ .

BODMAS

When simplifying an expression such as 3 + 4 × 5 - 4(3 + 2), remember to work it out in the following order: brackets, order, division, multiplication, addition, subtraction. Order refers to powers, e.g. 5³. I.e.

$3 + 4 \times 5 - 4 \times 5$