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Directed Numbers Bookmark this page Numbers can either be positive or negative. Often brackets are put around negative numbers to make them easier to read, e.g. (-2). If a number is positive, the + is usually missed out before the number. So 3 is really (+3). Adding and multiplying combinations of positive and negative numbers can cause confusion and so care must be taken. Addition and Subtraction Two 'pluses' make a plus, two 'minuses' make a plus. A plus and a minus make a minus. Example: 3 + (-2) A plus and a minus make a minus, so this is the same as 3 - 2 = 1 Example: (-2) + (-5) = -7 This is the same as (-2) - 5 = -7 Multiplication and Division If two positive numbers are multiplied together or divided, the answer is positive. If two negative numbers are multiplied together or divided, the answer is positive. If a positive and a negative number are multiplied or divided, the answer is negative. Examples: (-2) ÷ (-4) = ½ (8) ÷ (-2) = (-4) 2 × (-3) = (-6) (-2) × (-2) = 4
Fractions Introduction 1/2 means 1 divided by 2. If you try this on a calculator, you will get an answer of 0.5 . 3/6 means 3 divided by 6. Using a calculator, you will find that this too gives an answer of 0.5 . That is because 1/2 = 3/6 = 0.5 . Fractions such as 3/6 can be cancelled. You can divide the top and bottom of the fraction by 3 to get 1/2 . With fractions, you are allowed to multiply or divide the top and bottom of the fraction by some number, as long as you multiply (or divide) everything on the top and everything on the bottom by that number. So 5/12 = 10/24 (multiplying top and bottom by 2). Adding and subtracting fractions To add two fractions, the bottom (denominator) of the two fractions must be the same. 1/2 + 3/2 = 4/2 ; 1/10 + 3/10 + 5/10 = 9/10 . If the denominators are not the same, multiply or divide the top and bottom of one of the fractions by a number to make the denominator the same as the other. Example: 5 + 2 = 5 + 4 = 9 = 3 6 3 6 6 6 2 The same is true when subtracting fractions. Multiplying fractions This is simple, just multiply the two numerators (top bits) together, and the two denominators together: 2 × 5 = 10 = 5 3 8 24 12 Dividing Fractions If A, B, C and D are any numbers, A divided by C = A multiplied by D B D B C So: 1 ÷ 2 = 1 × 3 = 3 2 3 2 2 4 Harder examples These rules work even when the fractions involve algebra. 2x ÷ x = 2x × 3 = 6x = 6 (the x's cancel) 5 3 5 x 5x 5 (See algebraic fractions section for harder examples) A note on cancelling Fractions, of course, can often be 'cancelled down' to make them simpler. For example, 4/6 = 2/3. You can divide or multiply the top and bottom of any fraction by any number, as long as you do it to both the top and bottom. However, when there is more than one term on the top and/or bottom, to cancel you must divide every term in the top and bottom by that number. Examples: 2 + x 2 In this example, some people might try to cancel the 2s, but you cannot do this. You would have to divide the x by 2 also, to get 1 + ½x . 2(x + 4) = (x + 4) 4 2 Here there is only one term in the numerator (top) and denominator (bottom) of the fraction, so you can divide top and bottom by 2. Example: Simplify the expression 2x² - 5x + 2 x² - 4 In questions such as this, it is often useful to factorise. (2x - 1)(x - 2) (x + 2)(x - 2) Factorising means that there is now only one term in the numerator and denominator, whereas before there were two. We can now divide top and bottom by (x - 2): (2x - 1) (x + 2)
Number Sequences In the sequence 2, 4, 6, 8, 10... there is an obvious pattern. Such sequences can be expressed in terms of the nth term of the sequence. In this case, the nth term = 2n. To find the 1st term, put n = 1 into the formula, to find the 4th term, replace the n's by 4's: 4th term = 2 × 4 = 8. Example: What is the nth term of the sequence 2, 5, 10, 17, 26... ? n = 1 2 3 4 5 n² = 1 4 9 16 25 n² + 1 = 2 5 10 17 26 This is the required sequence, so the nth term is n² + 1. There is no easy way of working out the nth term of a sequence, other than to try different possibilities. Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably be a three in the formula, etc. In many cases, square numbers will come up, so try squaring n, as above. Also, the triangular numbers formula often comes up. This is n(n + 1)/2 . Example: Find the nth term of the sequence: 2, 6, 12, 20, 30... n = 1 2 3 4 5 n(n + 1)/2 = 1 3 6 10 15 Clearly the required sequence is double the one we have found the nth term for, therefore the nth term of the required sequence is 2n(n+1)/2 = n(n + 1). The Fibonacci sequence is an important sequence which is as follows: 1, 1, 2, 3, 5, 8, 13, 21, ... . The next term of this well-known sequence is found by adding together the two previous terms.
Numbers Revision Bookmark this page 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 (see 'surds'). 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.
Percentages A percentage is a fraction whose denominator is 100 (the numerator of a fraction is the top term, the denominator is the bottom term). So 30% = 30/100 = 3/10 = 0.3 To change a decimal into a percentage, multiply by 100. So 0.3 = 0.3 × 100 = 30% . Example: Find 25% of 10 (remember 'of' means 'times'). 25 × 10 (divide by 100 to convert the percentage to a decimal) 100 = 2.5 Percentage Change % change = new value - original value × 100 original value Example: The price of some apples is increased from 48p to 67p. By how much percent has the price increased by? % change = 67 - 48 × 100 = 39.58% 48 Percentage Error % error = error × 100 real value Example: Nicola measures the length of her textbook as 20cm. If the length is actually 17.6cm, what is the percentage error in Nicola's calculation? % error = 20 - 17.6 × 100 = 13.64% 17.6 Original value Original value = New value × 100 100 + %change Example: A dealer buys a stamp collection and sells it for £2700, making a 35% profit. Find the cost of the collection. It is the original value we wish to find, so the above formula is used. 2700 × 100 = £2000 100 + 35 Percentage Increases and Interest New value = 100 + percentage increase × original value 100 Example: £500 is put in a bank where there is 6% per annum interest. Work out the amount in the bank after 1 year. In other words, the old value is £500 and it has been increased by 6%. Therefore, new value = 106/100 × 500 = £530 . Compound Interest If in this example, the money was left in the bank for another year, the £530 would increase by 6%. The interest, therefore, will be higher than the previous year (6% of £530 is more than 6% of £500). Every year, if the money is left sitting in the bank account, the amount of interest paid would increase each year. This phenomenon is known as compound interest. The simple way to work out compound interest is to multiply the money that was put in the bank by nm, where n is (100 + percentage increase)/100 and m is the number of years the money is in the bank for, i.e: So if the £500 had been left in the bank for 9 years, the amount would have increased to: Percentage decreases: New value = 100 - percentage decrease × original value 100 Example: At the end of 1993 there were 5000 members of a certain rare breed of animal remaining in the world. It is predicted that their number will decrease by 12% each year. How many will be left at the end of 1995? At the end of 1994, there will be (100 - 12)/100 × 5000 = 4400 At the end of 1995, there will be 88/100 × 4400 = 3872 The compound interest formula above can also be used for percentage decreases. So after 4 years, the number of animals left would be: 5000 x [(100-12)/100]4 = 2998