1. Introductory Mathematics & Statistics
Chapter 1
Basic Mathematics
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2. 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
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3. 1.1 Whole numbers The decimal system consists of Numerals Integers
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 and 999
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4. 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
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5. 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 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:
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6. 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
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7. Rules for mathematical operations (cont…)
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
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8. Rules for mathematical operations (cont…)
Addition
Addition does have symmetry
E.g.
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
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9. 1.2 Fractions A fraction can be either proper or improper:
Proper fraction—numerator less than denominator
E.g.
Improper 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
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10. 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)
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

11. Multiplication & division of fractions
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
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12. 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,
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, and all represent the same number.
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13. 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
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14. 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
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15. 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
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16. 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
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17. 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.
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18. 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 is
one 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
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19. 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
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20. 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 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 in scientific notation =
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21. 1.6 Logarithms Definition Characteristics and mantissa
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
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22. 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
From Table 5, the mantissa of corresponds to N = The characteristic of 2 corresponds to a factor of 102.
Hence, the required number is 7.51 × 102 = 751.
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23. 1.6 Logarithms (cont…) Calculations involving logarithms
Using the following properties we can find solutions to problems containing logarithms
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

24. 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