Transcript PPT Chapter 01 - McGraw Hill Higher Education

Introductory Mathematics
& Statistics
Chapter 1
Basic Mathematics
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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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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
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Basic mathematical operations
There are four basic mathematical operations that can be
performed on numbers:

• Multiplication: represented by
• Division: represented by either
• Addition: represented by
 or
/

• Subtraction: represented by

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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 5  4  9 can be written as 5  ( 4  9) to remove this
ambiguity.
– As another example, if we wish to add numerals before
multiplying, we can use the parentheses as follows:
(4  9)  3  13  3
 39
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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
4  6 or 6  4
46
( 4)( 6)
4(6) or ( 4)6
– Multiplying the same signs gives a positive result
 5   6   30
– Multiplying different signs gives a negative result
5   4   20
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Rules for mathematical operations
(cont…)
• Division
– There are several ways of indicating that two numbers are to be divided.
 E.g.
6  3, 6 / 3,
6
3
 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
6
 2
3
– Dividing different signs gives a negative result
3
1

6
2
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Rules for mathematical operations
(cont…)
• Addition
– Addition does have symmetry
 E.g.
56  65
– 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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1.2 Fractions
• A fraction can be either proper or improper:
– Proper fraction—numerator less than denominator
• E.g.
6 23 156
,
,
9 52 238
– Improper fraction—numerator greater than denominator
3 56 856
,
,
2 32 249
•
•
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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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)
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
Step 2: Add or subtract the numerators to obtain the new
numerator.
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Multiplication & division of fractions
• Multiplication
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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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.
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Rules for decimals
Addition and subtraction
– Align the numbers so that the decimal points are directly
underneath each other.
 Example of an addition
Question : Add
 Step 1: align
 Step 2: add


2.3  0.34  1.672
2 .3
0.34
1.672
4.312
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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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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.
•
23  2  2  2  8
Exponential form—an
– where a is the base
– where n is the exponent or power
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Rules for exponents
• Positive exponents
n
m
– If numbers with same base, a and a , then product
will have the same base. The exponent will be the sum
of the two original exponents
an  am  anm
– For the quotient, if the two numbers have the same
base, the exponent will be the difference between the
original exponents
am  an  amn
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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
a 
n m
a
nm
• 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
 n
a
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Rules for exponents (cont…)
• Fractional exponents
– Exponents can be expressed as a fraction
1
 n is of the form k (where k is an integer)
1
k
 a 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
m
n
1
k
a 
a k a
 a   a 
n
m
1
m n
• Zero exponent
– Any base raised to the power of 0 equals 1
a0  1
– Except for
00 , which is undefined
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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  N'  10c
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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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 =
6.3254793  106
• 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 =
5.849  104
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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
logb N
That is, if x  logb N, then N  bx
 E.g.
log 4 64  3, so 43  64
• 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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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.
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1.6 Logarithms (cont…)
• Calculations involving logarithms
– Using the following properties we can find solutions to
problems containing logarithms
log A  B  log A  log B
log A  B  log A  log B
log An  n  log B
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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.
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