Chapter 01 (1)

1-*Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

Chapter 1

Basic MathematicsIntroductory Mathematics & Statistics

Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e


Learning Objectives

Carry out calculations involving whole numbersCarry out calculations involving fractionsCarry out calculations involving decimalsCarry out calculations involving exponentsUse and understand scientific notationUse and understand logarithms



1.1 Whole numbersThe decimal system consists ofNumeralsSymbols, i.e. 0, 1, 2, 3 are numeralsRepresent natural numbers or whole numbers Used to count whole objects or fractions of them

Integers Another name for whole numbersA positive integer is a number greater than zeroA negative integer is a number less than zero

DigitsNumerals consist of one or more digitsExample: a three-digit number (e.g. 841) lies between 100 and 999



Basic mathematical operationsThere are four basic mathematical operations that can be performed on numbers:

Multiplication: represented by

Division: represented by either or

Addition: represented by

Subtraction: represented by



Rules for mathematical operationsOrder 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 can be written as to remove this ambiguity. As another example, if we wish to add numerals before multiplying, we can use the parentheses as follows:



Rules for mathematical operations (cont)MultiplicationThere are several ways of indicating that two numbers are to be multipliedE.g. 4 multiplied by 6 can be expressed as

Multiplying the same signs gives a positive result

Multiplying different signs gives a negative result



Rules for mathematical operations (cont)DivisionThere are several ways of indicating that two numbers are to be divided.

E.g.

The number to be divided (6) is called the numerator or dividendThe number that is to be divided by (3) is called the denominator or divisorThe answer to the division is called the quotient

Dividing the same signs gives a positive result

Dividing different signs gives a negative result



Rules for mathematical operations (cont)AdditionAddition does have symmetry

E.g.

like signsuse the sign and addunlike signsuse sign of greater and subtract

SubtractionTwo signs next to each otherminus and a minus is a plus ( 3) = 3minus and a plus is a minus (+3) = 3



1.2 FractionsA fraction can be either proper or improper:

Proper fractionnumerator less than denominator

E.g.Improper fractionnumerator greater than denominator

The number on top of the fraction is called the numerator and the bottom number is called the denominatorThe denominator cannot be zero, because if it is, the result is undefined



Addition & subtraction of fractionsSame denominators

Step 1: Add or subtract the numerators to obtain the new numeratorStep 2: The denominator remains the same

Different denominators

Step 1: Change denominators to lowest common multiple (LCM)

LCM is the smallest number into which all denominators will divide

Step 2: Add or subtract the numerators to obtain the new numerator.



Multiplication & division of fractionsMultiplicationStep 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.DivisionStep 1: Invert the second fraction

Step 2: Multiply it by the first fraction



1.3 DecimalsAny fractions can be expressed as a decimal by dividing the numerator by the denominator.

A decimal consists of three components:an integerthen a decimal pointthen another integerE.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 numbers value.E.g. 0.5, 0.50, 0.500 and 0.5000 all represent the same number.



Rules for decimalsAddition and subtractionAlign the numbers so that the decimal points are directly underneath each other.Example of an addition

Step 1: align

Step 2: add



Rules for decimals (cont)Multiplication Step 1: Count the number of digits to the right of each decimal point for each numberStep 2: Add the number of digits in Step 1 to obtain a number, say xStep 3: Multiply the two original decimals, ignoring decimal pointsStep 4: Mark the decimal point in the answer to Step 3 so that there are x digits to the right of the decimal point

DivisionStep 1: Count the number of digits that are in the divisor to the right of the decimal point. Call this number xStep 2: Move the decimal point in the dividend x places to the right (adding zeros as necessary). Do the same to the divisorStep 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



1.4 ExponentsAn 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.

Exponential formanwhere a is the basewhere n is the exponent or power



Rules for exponentsPositive exponentsIf numbers with same base, , then product will have the same base. The exponent will be the sum of the two original exponents

For the quotient, if the two numbers have the same base, the exponent will be the difference between the original exponents



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

Negative exponentsA number expressed with a negative exponent is equal to the reciprocal of the same number with the negative sign removed.



Rules for exponents (cont)Fractional exponentsExponents can be expressed as a fraction

n is of the form (where k is an integer)

is said to be the kth root of a. The kth root of a number isone such that when it is multiplied by itself k times, you get that number

Zero exponentAny base raised to the power of 0 equals 1

Except for , which is undefined



1.5 Scientific notationScientific notation is a shorthand way of writing very large and very small numbersIt expresses the number as a numeral (less than 10) multiplied by the base number 10 raised to an exponentThe rule for writing a number N in scientific notation is:

where 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



1.5 Scientific notation (cont)When c is positiveIf the decimal point is to the right of the reference position, the value of c is positivee.g. 6325479.3 in scientific notation =

When c is negativeIf the decimal point is to the left of the reference position, the value of c is negativee.g. 0.0005849 in scientific notation =



1.6 LogarithmsDefinitionThe logarithm of a number N to a base b is the power to which b must be raised to obtain N

E.g.

Characteristics and mantissaSuppose that the logarithm is expressed as an integer plus a non-negative decimal fraction. Then:the integer is called the characteristic of the logarithmthe decimal fraction is called the mantissa of the logarithm



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 2.8756From 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.



1.6 Logarithms (cont)Calculations involving logarithmsUsing the following properties we can find solutions to problems containing logarithms



SummaryA 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.


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