GCSE Mathematics Numbers Md Marufur Rahman Msc Sustainable Energy Systems Beng (Hons) Mechanical Engineering Bsc (Hons) Computer science & engineering GCSE AQA Mathematics 215/16
Table of Contents Introduction:... 3 Addition (+):... 3 Subtraction (-):... 3 Multiplication ( ):... 3 Division ( ):... 4 Exercise Questions:... 4 Order of operations:... 5 Exercise Questions:... 5 Types of numbers:... 5 The Natural and Whole Numbers... 5 The Integers... 6 The Rational Numbers... 6 Prime Numbers... 7 The Real Numbers... 7 Squares, cubes and their roots... 8 Squaring a number... 8 Square roots... 8 Cubing a number... 9 Cube roots... 9 The Imaginary Numbers and the Complex Numbers ( higher level)... 9 2 P a g e
Introduction: Whole numbers are the numbers..-3, -2, -1,, 1, 2, 3 Whole numbers are also referred to as integers. The positive integers are 1, 2, 3, 4 The negative integers are..,-4, -3, -2, -1. In fact, given two or more whole numbers it is possible to perform an operation on them. The four arithmetic operations are addition (+), subtraction (-), multiplication ( ) and division ( ). Addition (+): We say that 4+5 is the sum of 4 and 5. Note that 4+5 is equal to 5+4 so that the order in which we write down the numbers does not matter when we are adding them. Because the order does not matter, addition is said to be commutative. Subtraction (-): We say that 8-3 is the difference of 8 and 3. Note that 8-3 is not same as 3-8 and so the order in which we write down the number is important when we are subtracting them. Subtraction is not commutative. Subtracting a negative number is equivalent to adding a positive number: thus 7- (-3) = 7+3 = 1. Multiplication ( ): The instruction to multiply the numbers 6 and 7 is written 6 7. This is known as product of 6 and 7. We can write different way multiplication operation such as (6)(7) or 6 7. Multiplication of number is commutative means that we can write (6)(7) is the same as (7)(6). Key points: (positive) (positive) = positive (positive) (negative) = negative (negative) (positive) = negative (negative) (negative) = positive Example: 4 6 = 24 5 (-3) = -15 (-4) 5 = -2 (-3) (-6) = 18 3 P a g e
Division ( ): The quantity 8 4 means 8 divided by 4. This is also written as 8/4 or and is known as the quotient of 8 and 4. In the fraction the top line is called the numerator and the bottom line is called denominator. Note that 8/4 is not same as 4/8. Division is not commutative. Division by is never allowed, it is called undefined and is division by number is zero or (). Key points: Example: 8 4 2 8 4 2 8 4 2 8 4 2 Key points: Example: 7 7 Exercise Questions: Sum of 9 and 4 Sum of 9 and -4 The difference of 6 and 3 The difference of 6 and -3 The product of 9 and 3 The product of -9 and 3 The product of -9 and -3 The quotient of 1 and 2 The quotient 1 and -2 The quotient -1 and -2 The quotient and 999 The quotient 999 and 4 P a g e
Order of operations: The order of four operations are carried out is very important but may not be obvious when looking at expression. Consider the following case. Suppose we wish to evaluate 2 3 + 4. If we carry out the multiplication first the result is 2 3 + 4 = 6 + 4 = 1. However, if we carry out addition first the result will be 2 3 + 4 = 2 7 = 14. Clearly we need some rules that specify the order in which the various operations are performed. To remind us of the order in which to carry out these operations we can make use of the BODMAS rule. Brackets Others Division Multiplication Addition Subtraction Exercise Questions: Evaluate: 2 3 + 4 [Answer: 1] Evaluate: 6 2 1 [Answer: 2] Evaluate: 2 (3 + 4) [Answer: 14] Evaluate: 6 (2 1) [Answer: 6] Evaluate: 12 2 4 3 [Answer: 18] Evaluate: 9 + 3-6 + 2-4 [Answer: 4] Evaluate: 27 (7-4) + 3 4 [Answer: 21] Evaluate: [(6 2) (1+2)] [ 3 + 5 7 + 3] [Answer: 1] Evaluate: (12 2) 6 [Answer: 36] Evaluate: 12 2 6 [Answer: 15] Evaluate: Evaluate:! # Types of numbers: [Answer: 2] [Answer: 1] There are a lot of numbers, infinitely many as it happens. Because of this mathematicians need a way to classify numbers, a way to group then and tell them apart from each other. We are going to take look at the different groups of numbers that exist. The Natural and Whole Numbers We start with the natural numbers. These are the numbers 1, 2, 3, (The symbol means that the sequence goes on forever.) They are used for counting. If we include zero then we get 5 P a g e
the whole numbers,, 1, 2, 3,. The natural and whole numbers are usually considered to be exact (e.g. there are 4 tires on a car, 8 legs on a spider). But sometimes they are approximate (e.g. there were 1 people in the crowd). Here they are, shown on the number line: The Integers Now for each of the numbers 1, 2, 3, let s create its opposite or negative and put it on the opposite side of the number line, like this: We say that 1 and 1 are opposites, 2 and 2 are opposites, etc. and we also say that 1 is the opposite of 1 and that 1 is the opposite of 1. Negative numbers are used to describe debts as opposed to assets, temperatures below zero as opposed to temperatures above zero, heights below sea level as opposed to heights above sea level, and so on. The set of numbers, 3, 2, 1,, 1, 2, 3, (the whole numbers and their opposites) is called the integers. The Rational Numbers Next are the rational numbers. A rational number is any number that can be expressed as the quotient of two integers. We can use any of these fraction notations to express rational numbers: a is called the numerator and b is called the denominator. The denominator cannot be zero. The notation means that we break something into b equal pieces and we have a of those pieces. For example if we break a pie into 4 pieces and take 1 piece then we have 1/4 of the pie: 6 P a g e
Rational numbers can also be written in decimal notation instead of fraction notation. For example: 1/4 =.25 The decimal notation.25 means literally 25/1 and 25/1 and 1/4 are equivalent fractions. Notice that some rational numbers have no exact decimal equivalent. For example 1/3 is approximately equal to 33/1 but not exactly: 1/3.33 (The symbol means is approximately equal to.) Rational numbers are usually considered to be exact. For this reason the Algebra Coach program will not convert fractions to decimals when it is running in exact mode. Notice that since 3/1 = 3 and 5/1 = 5, the rational numbers include all the integers. The Irrational Numbers The irrational numbers are those that cannot be expressed as a ratio of two integers. Examples are as well as the square roots of many other numbers, and special numbers like e and π. It turns out that there are as many irrational numbers as rational. Irrational numbers have no exact decimal equivalents. To write any irrational number in decimal notation would require an infinite number of decimal digits. Thus these are only approximations: 1.732, e 2.718 and π 3.14, For this reason the Algebra Coach program will not convert irrational numbers to decimals when it is operating in exact mode. Prime Numbers A prime number is a number which does not divide by anything apart from itself and 1. Only exceptional is 1, which is not a prime number. Apart from 2 and 5, all primes end in 1, 3, 7, or 9. So possible primes would be 71, 73, 77, 79, 11, 13, 17, 19 etc. However, not all of these are prime- we need to check carefully that they do not divide by anything. First order primes are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 The Real Numbers The rational numbers and the irrational numbers together make up the real numbers. The real numbers are said to be dense. They include every single number that is on the number 7 P a g e
line. The number line is useful for understanding the order of numbers. Smaller numbers are farther to the left and larger numbers are farther to the right. We use the symbol < to mean is less than and the symbol > to mean is greater than. Here are some examples of the use of these symbols: 5 < 8 because 5 is to the left of 8. 5 < 2 because 5 is to the left of 2. In fact any negative number is less than any positive number. 5 < 8 and 8 > 5 are two ways of stating the same fact. 3.14 < π < 3.15 I call this a less than sandwich. It means that 3.14 < π and also π < 3.15. In other words the number π is somewhere between 3.14 and 3.15. Real numbers often result from making measurements and measurements are always approximate. Squares, cubes and their roots Squaring a number 3 2 means '3 squared', or 3 x 3. The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself. Similarly 7 2 means '7 squared', or 7 x 7. And 1 2 means '1 squared', or 1 x 1. So, 1 2 = 1 x 1 = 1 2 2 = 2 x 2 = 4 3 2 = 3 x 3 = 9 4 2 = 4 x 4 = 16 5 2 = 5 x 5 = 25 etc 1, 4, 9, 16, 25 are known as square numbers. Square roots The opposite of a square number is a square root. We use the symbol to mean square root. So we can say that = 2 and = 5. 8 P a g e
However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25. So, in fact, = 2 or -2. And = 5 or -5. Remember that every positive number has two square roots. Cubing a number 2 x 2 x 2 means '2 cubed', and is written as 2 3. 1 3 = 1 x 1 x 1 = 1 2 3 = 2 x 2 x 2 = 8 3 3 = 3 x 3 x 3 = 27 4 3 = 4 x 4 x 4 = 64 5 3 = 5 x 5 x 5 = 125 etc 1, 8, 27, 64, 125 are known as cube numbers. Cube roots The opposite of a cube number is a cube root. We use the symbol to mean cube root. So is 2 and is 3. Each number only has one cube root. The Imaginary Numbers and the Complex Numbers ( higher level) If real numbers include every single number on the number line, then what other numbers could there be? To answer that question, consider how we built up the number system so far: We started with the whole numbers (numbers like 3) We wanted opposites for these numbers so we created the integers (numbers like 3) We wanted to divide these numbers but needed the rationals to describe some of the results (numbers like 3/4) We wanted to take the square root of these numbers but needed the irrationals to describe some of the results (e.g. ) 9 P a g e