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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
Slide 1
1
Chapter M1
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 2
2
Whole numbers

The decimal system
– 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
– 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
Slide 3
3
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

+
-
Slide 4
4
Rules for mathematical operations
Order
of operations:
Multiplication and division
BEFORE
Addition and subtraction
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 5
5
Rules for mathematical operations

Multiplication and division
– same signs give positive result
 5   6   11 &
 20
5
4
– different signs give negative result
5   4   20
&
3
1

6
2
– perform calculations in brackets first
3 6  7  39
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 6
6
Rules for mathematical operations

Addition
– like signs—use the sign and add
– unlike signs—use sign of greater and subtract

Subtraction
Two 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
Slide 7
7
Fractions

A fraction appears as:
a
numerator

b deno minator
–Proper fraction—numerator less than denominator
3
8
–Improper fraction—numerator greater than denominator
15
7
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 8
8
Addition & subtraction of fractions

Different denominators
– change denominators to lowest common multiple
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 9
9
Multiplication & division of fractions
– Multiply numerators to get new numerator
– Multiply denominators 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
Slide 10
10
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
Slide 11
11
Rules for decimals

Addition and subtraction
– Align the numbers so that the decimal points are
directly underneath each other.
Add
2.3  0.34  1.672
2 .3
0.34
1.672
4.312
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 12
12
Rules for decimals

Multiplication and division
1.
2.
3.
4.
Count the number of digits to the right of each decimal
point for each number.
Add the number of digits in Step 1 to obtain a number,
say x.
Multiply the two original decimals, ignoring decimal
points.
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
Slide 13
13
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
Slide 14
14
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
Slide 15
15
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.
a
n

1
an
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 16
16
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
1
ak
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 17
17
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
Slide 18
18
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 19
19