Numbers

Types of numbers Integers are whole numbers (both positive and negative). Zero is usually classed as an integer. Natural numbers are positive integers. 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 are numbers which cannot be written as fractions, such as pi and Ö2. In decimal form these numbers go on forever and the same pattern of digits are not repeated. Square numbers are numbers which can be obtained by multiplying another number by itself. E.g. 36 is a square number because it is 6 x 6 . Surds are numbers left written as Ön , where n is positive but not a square number. E.g. Ö2. 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 numbers are all the numbers which you will have come across (i.e. all the rational and irrational numbers).  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: 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). The largest value is 95m. Using inequalities, 85£ length <95. Sometimes you will be asked the upper and lower bounds of the area. The area will be smallest when the side of the square is 85m. In this case, the area will be 7725m². The largest possible area is 9025m² (when the length of the sides are 95m).  BODMAS When simplifying an expression such as 3 + 4 × 5 - 4(3 + 2), remember to work it out in the following order: brackets, of, division, multiplication, addition, subtraction. So do the thing in the brackets first, then any division, followed by multiplication and so on. The above is: 3 + 20 - 4 × 5 = 3 + 20 - 20 = 3. You mustn't just work out the sum in the order that it is written down