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Introductory Mathematics & Statistics Chapter 1 Basic Mathematics Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-1 Learning Objectives Carry out calculations involving whole numbers Carry out calculations involving fractions Carry out calculations involving decimals Carry out calculations involving exponents Use and understand scientific notation Use and understand logarithms Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-2 1.1 Whole numbers • The decimal system consists of – Numerals Symbols, i.e. 0, 1, 2, 3 are numerals Represent natural numbers or whole numbers Used to count whole objects or fractions of them – Integers Another name for whole numbers A positive integer is a number greater than zero A negative integer is a number less than zero – Digits Numerals consist of one or more digits Example: a three-digit number (e.g. 841) lies between 100 and 999 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-3 Basic mathematical operations There are four basic mathematical operations that can be performed on numbers: • Multiplication: represented by • Division: represented by either • Addition: represented by or / • Subtraction: represented by Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-4 Rules for mathematical operations Order of operations: Multiplication and Division BEFORE Addition and Subtraction – However, to avoid any ambiguity, we can use parentheses (or brackets), which take precedence over all four basic operations – For example 5 4 9 can be written as 5 ( 4 9) to remove this ambiguity. – As another example, if we wish to add numerals before multiplying, we can use the parentheses as follows: (4 9) 3 13 3 39 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-5 Rules for mathematical operations (cont…) • Multiplication – There are several ways of indicating that two numbers are to be multiplied E.g. 4 multiplied by 6 can be expressed as 4 6 or 6 4 46 ( 4)( 6) 4(6) or ( 4)6 – Multiplying the same signs gives a positive result 5 6 30 – Multiplying different signs gives a negative result 5 4 20 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-6 Rules for mathematical operations (cont…) • Division – There are several ways of indicating that two numbers are to be divided. E.g. 6 3, 6 / 3, 6 3 The number to be divided (6) is called the numerator or dividend The number that is to be divided by (3) is called the denominator or divisor The answer to the division is called the quotient – Dividing the same signs gives a positive result 6 2 3 – Dividing different signs gives a negative result 3 1 6 2 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-7 Rules for mathematical operations (cont…) • Addition – Addition does have symmetry E.g. 56 65 – like signs—use the sign and add – unlike signs—use sign of greater and subtract • Subtraction Two signs next to each other – minus and a minus is a plus –(– 3) = 3 – minus and a plus is a minus –(+3) = –3 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-8 1.2 Fractions • A fraction can be either proper or improper: – Proper fraction—numerator less than denominator • E.g. 6 23 156 , , 9 52 238 – Improper fraction—numerator greater than denominator 3 56 856 , , 2 32 249 • • The number on top of the fraction is called the numerator and the bottom number is called the denominator The denominator cannot be zero, because if it is, the result is undefined Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-9 Addition & subtraction of fractions • Same denominators Step 1: Add or subtract the numerators to obtain the new numerator Step 2: The denominator remains the same • Different denominators Step 1: Change denominators to lowest common multiple (LCM) 1 2 5 6 4 15 25 7 1 3 9 6 18 18 18 LCM is the smallest number into which all denominators will divide Step 2: Add or subtract the numerators to obtain the new numerator. Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-10 Multiplication & division of fractions • Multiplication Step 1: Multiply numerators to get new numerator Step 2: Multiply denominators to get new denominator Step 3: Use any common factors to divide the numerator and denominator, to simplify the answer. • Division Step 1: Invert the second fraction Step 2: Multiply it by the first fraction Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-11 1.3 Decimals • Any fractions can be expressed as a decimal by dividing the numerator by the denominator. • A decimal consists of three components: – an integer – then a decimal point – then another integer • E.g. 0.3, 1.2, 5.69, 45.687 • Any zeros on the right-hand end after the decimal point and after the last digit do not change the number’s value. – E.g. 0.5, 0.50, 0.500 and 0.5000 all represent the same number. Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-12 Rules for decimals Addition and subtraction – Align the numbers so that the decimal points are directly underneath each other. Example of an addition Question : Add Step 1: align Step 2: add 2.3 0.34 1.672 2 .3 0.34 1.672 4.312 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-13 Rules for decimals (cont…) Multiplication • • • • Step 1: Count the number of digits to the right of each decimal point for each number Step 2: Add the number of digits in Step 1 to obtain a number, say x Step 3: Multiply the two original decimals, ignoring decimal points Step 4: Mark the decimal point in the answer to Step 3 so that there are x digits to the right of the decimal point Division • • • • Step 1: Count the number of digits that are in the divisor to the right of the decimal point. Call this number x Step 2: Move the decimal point in the dividend x places to the right (adding zeros as necessary). Do the same to the divisor Step 3: Divide the transformed dividend (Step 2) by the transformed divisor (which now has no decimal point) The quotient of this division is the answer Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-14 1.4 Exponents • An exponent or power of a number is written as a superscript to a number called the base • The base number is said to be in exponential form • This tells us how many times the based is multiplied by itself – E.g. • 23 2 2 2 8 Exponential form—an – where a is the base – where n is the exponent or power Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-15 Rules for exponents • Positive exponents n m – If numbers with same base, a and a , then product will have the same base. The exponent will be the sum of the two original exponents an am anm – For the quotient, if the two numbers have the same base, the exponent will be the difference between the original exponents am an amn Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-16 Rules for exponents (cont…) • Positive exponents (cont…) – A number in exponential form is raised to another exponent; the result is the original base raised to the product of the exponents a n m a nm • Negative exponents – A number expressed with a negative exponent is equal to the reciprocal of the same number with the negative sign removed. a n 1 n a Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-17 Rules for exponents (cont…) • Fractional exponents – Exponents can be expressed as a fraction 1 n is of the form k (where k is an integer) 1 k a is said to be the ‘kth root of a’. The kth root of a number is one such that when it is multiplied by itself k times, you get that number m n 1 k a a k a a a n m 1 m n • Zero exponent – Any base raised to the power of 0 equals 1 a0 1 – Except for 00 , which is undefined Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-18 1.5 Scientific notation – Scientific notation is a shorthand way of writing very large and very small numbers – It expresses the number as a numeral (less than 10) multiplied by the base number 10 raised to an exponent – The rule for writing a number N in scientific notation is: where N N' 10c N’ = the digit before the reference position, followed by the decimal point and the remaining digits in number N. c = the number of digits between the reference position and the decimal point Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-19 1.5 Scientific notation (cont…) • When c is positive – If the decimal point is to the right of the reference position, the value of c is positive e.g. 6325479.3 in scientific notation = 6.3254793 106 • When c is negative – If the decimal point is to the left of the reference position, the value of c is negative e.g. 0.0005849 in scientific notation = 5.849 104 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-20 1.6 Logarithms • Definition – The logarithm of a number N to a base b is the power to which b must be raised to obtain N logb N That is, if x logb N, then N bx E.g. log 4 64 3, so 43 64 • Characteristics and mantissa – Suppose that the logarithm is expressed as an integer plus a non-negative decimal fraction. Then: the integer is called the characteristic of the logarithm the decimal fraction is called the mantissa of the logarithm Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-21 1.6 Logarithms (cont…) • Antilogarithms – The antilogarithm is the value of the number that corresponds to a given logarithm E.g. Find the antilogarithm of 2.8756 From Table 5, the mantissa of 0.8756 corresponds to N = 7.51. The characteristic of 2 corresponds to a factor of 102. Hence, the required number is 7.51 × 102 = 751. Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-22 1.6 Logarithms (cont…) • Calculations involving logarithms – Using the following properties we can find solutions to problems containing logarithms log A B log A log B log A B log A log B log An n log B Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-23 Summary • A thorough knowledge of fractions, decimals and exponents is essential for an understanding of basic mathematical principles. • You should not be too reliant on modern technology to solve every problem. • You are far better prepared if you are also aware of the processes that the calculator is undertaking when performing calculations. Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 1-24