chapter 01

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

Chapter 1

Basic Mathematics

Introductory Mathematics & Statistics


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


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


Basic mathematical operations

There 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:

945 )94(5

39

3133)94(


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

– Multiplying the same signs gives a positive result

– Multiplying different signs gives a negative result

3065

2045

6)4(or)6(4

)6)(4(

64

46or64



• Division– There 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 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

– Dividing different signs gives a negative result

3

6,3/6,36

23

6

2

1

6

3



• Addition– Addition does have symmetry

E.g.

– like signs—use the sign and add– unlike signs—use sign of greater and subtract

• SubtractionTwo signs next to each other– minus and a minus is a plus –(– 3) = 3– minus and a plus is a minus –(+3) = –3

5665


1.2 Fractions• A fraction can be either proper or improper:

– Proper fractionProper fraction—numerator less than denominator

• E.g.

– Improper fractionImproper fraction—numerator greater than denominator

• 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

238

156,

52

23,

9

6

249

856,

32

56,

2

3


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 denominatorsdenominators to lowest common multiple (LCM)lowest common multiple (LCM)

LCM LCM is the smallest number into which all denominators will divide

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

18

71

18

25

18

1546

6

5

9

2

3

1


Multiplication & division of fractions

• MultiplicationStep 1: Multiply numeratorsnumerators to get new numerator

Step 2: Multiply denominatorsdenominators 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 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.


Rules for decimals

Addition and subtraction– Align the numbers so that the decimal points are directly

underneath each other. Example of an addition

Step 1: align

Step 2: add

672.134.03.2: AddQuestion

312.4

672.1

34.0

3.2


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


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.

• Exponential form—an

– where a is the base– where n is the exponent or power

822223


Rules for exponents

• Positive exponents– If 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

mn aanda

mnmn aaa

nmnm aaa


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 exponents– A number expressed with a negative exponent is equal to the

reciprocal of the same number with the negative sign removed.

nn

a

1a

nmmn aa


Rules for exponents (cont…)• Fractional exponents

– Exponents 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 exponent– Any base raised to the power of 0 equals 1

– Except for , which is undefined

k

1

k1

a

kk1

aa n1

mmnn

m

aaa

1a0 00


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’ = 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

c' 10NN


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 =

• 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 =

6103254793.6

410849.5


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

E.g.

• 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

Nlogb

xb bNthen,Nlogxif,isThat

644,364log 34 so


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.


1.6 Logarithms (cont…)

• Calculations involving logarithms– Using the following properties we can find solutions to

problems containing logarithms

BlogAlogBAlog

BlogAlogBAlog

BlognAlog n


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.

chapter 01

Documents

mathematical

original exponents

5e rules

decimal point

reference

kth root

accompany

introductory