© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1Slide 1

Introductory Mathematics & Statistics for Business

4th Edition

John S. Croucher

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 2Slide 2

Basic mathematics

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

Chapter M1

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 3Slide 3

Whole numbers

The decimal system– Numerals

• symbols i.e. 0, 1, 2, 3are numerals

• represent natural numbers or whole numbers

• used to count whole objects or fractions of them

– Integer • is another name for a whole number

– Digits• numerals consist of one or more digits

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 4Slide 4

Mathematical operations

Four basic mathematical operations performed on numbers– multiplication represented by: x– division represented by:

– addition represented by: +– subtraction represented by: -

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 5Slide 5

Rules for mathematical operations

Order of operations:

Multiplication and divisionMultiplication and division BEFORE

Addition and subtractionAddition and subtraction

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 6Slide 6

Rules for mathematical operations Multiplication and division

– same signs give positive result

– different signs give negative result

– perform calculations in brackets first

54

201165

&

2

1

6

3&2045

39763

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 7Slide 7

Rules for mathematical operations

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

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

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 8Slide 8

Fractions

A fraction appears as:

–Proper fractionProper fraction—numerator less than denominator

–Improper fractionImproper fraction—numerator greater than denominator

atormindeno

numerator

b

a

8

3

7

15

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 9Slide 9

Addition & subtraction of fractions

Different denominators

– change denominatorsdenominators to lowest common multiplelowest common multiple

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

18

71

18

25

18

1546

6

5

9

2

3

1

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 10Slide 10

Multiplication & division of fractions

– Multiply numeratorsnumerators to get new numerator

– Multiply denominatorsdenominators to get new denominator

– Cancel common factors of nominators and numerators by multiplying

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 11Slide 11

Decimals

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

A decimal consists of three components:

• an integer• a decimal point• another integer.

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 12Slide 12

Rules for decimals

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

directly underneath each other.

312.4

1.672

34.0

3.2

672.134.03.2 Add

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 13Slide 13

Rules for decimals

Multiplication and division1. Count the number of digits to the right of each

decimal point for each number.

2. Add the number of digits in Step 1 to obtain a number, say x.

3. Multiply the two original decimals, ignoring decimal points.

4. Mark the decimal point in the answer to Step 3 so that there are x digits to the right of the decimal point.

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 14Slide 14

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.

Exponential form—an

» where a is the base

» where n is the exponent or power

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 15Slide 15

Rules for exponents Positive exponents

• Two numbers with same base—an & am

• The product will have the same base; the exponent will be the sum of the two original exponents—an x am = an+m

• The quotient of the two numbers will have the same base; the exponent will be the difference between the original exponents—an am = an-m

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 16Slide 16

Rules for exponents

Positive exponents– A number in exponential form is raised to another exponent.

The result is the original base raised to the product of the

exponents. (an )m = anm

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

the reciprocal of the same number with the negative sign removed.

nn

aa

1

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 17Slide 17

Rules for exponents

Fractional exponents– Exponents can be expressed as a fraction

• where k is an integer and is said to be the kth root of a

• when k=2 it is the square root; k=3 is the cube root

ka1

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 18Slide 18

Rules for exponents

Scientific notation– Scientific notation is a shorthand way of writing very large

and very small numbers.– Scientific notation expresses the number as a numeral (less

than 10) multiplied by the base number 10 raised to an exponent.

– The reference position for the decimal point in a number is immediately to the right of the first non-zero digit.

© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 19Slide 19

Logarithms Logarithms are closely connected to the theory of

exponents. Calculations using logarithms have been replaced by

calculators since the 1970s. An understanding of logarithms can be useful in

statistics, physics, engineering etc. The logarithm of a number N to a base b is the power

to which b must be raised to obtain N.

logbN

That is, if x = logbN, then N = bx