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Symbols and Notation .....................................................................................................................................1
Addition (+) .................................................................................................................................................1
Subtraction (-) ..............................................................................................................................................1
The Plus Or Minus Sign (±).........................................................................................................................1
Multiplication (×) ........................................................................................................................................1
Key Points When Multiplying Numbers: ................................................................................................2
Examples: ............................................................................................................................................2
Division (÷) .................................................................................................................................................2
Key Points When Dividing Numbers: .....................................................................................................3
Examples: ............................................................................................................................................3
Twiddle (~) ..................................................................................................................................................3
The Reciprocal Of A Number......................................................................................................................3
Greater Than (>) ..........................................................................................................................................4
Less Than (<) ...............................................................................................................................................4
Not Equal to (≠) ...........................................................................................................................................4
Order of Operations Rules ...........................................................................................................................4
Example ...................................................................................................................................................4
Sequences of Operations .............................................................................................................................5
Proportions: Fractions, Decimals, and Percentages .....................................................................................5
Example ...................................................................................................................................................5
The Denominator .........................................................................................................................................5
The Numerator.............................................................................................................................................6
Converting a Decimal to a Percentage .........................................................................................................6
Percentages as a Special Fraction ................................................................................................................6
Math With Percentages ................................................................................................................................6
Adding Negative Numbers ..........................................................................................................................6
Subtracting Negative Numbers ....................................................................................................................7
Multiplying and Dividing Negative Numbers - Summary...........................................................................7
Multiplication and Division Example ......................................................................................................7
Basic Algebra: Solving Equations ...............................................................................................................7
Equations with Variables .............................................................................................................................7
Equations as a Balance ................................................................................................................................8
Finding the Value of X ................................................................................................................................8
Exponents ....................................................................................................................................................8
Special Exponent Rules ...............................................................................................................................8
Square Roots
  ......................................................................................................................................9
General Introduction to Absolute Value ......................................................................................................9
Case 1: ...................................................................................................................................................10
Case 2: ...................................................................................................................................................10
Example 1: .........................................................................................................................................10
Example 2: .........................................................................................................................................10
Summary ...............................................................................................................................................11
The Factorial Symbol (!) ...........................................................................................................................11
Key Point Factorial notation: .................................................................................................................11
Example .............................................................................................................................................11
Powers: Definition and Simple Powers with Numbers ..............................................................................11
Powers with Variables ............................................................................................................................... 12
Algebraic Rules for Working with Powers ................................................................................................ 12
Summary of Algebraic Rules for Powers ..................................................................................................13
Arithmetical Expressions ...........................................................................................................................14
Two Signs Rule .........................................................................................................................................14
The BODMAS rule ....................................................................................................................................15
Key Points .............................................................................................................................................15
The Greek Alphabet............................................................................................................................... 16
More exercises for you to try .....................................................................................................................16
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Aids to Calculation ....................................................................................................................................17
Logarithm ..............................................................................................................................................17
Bases ......................................................................................................................................................17
Other Notations .....................................................................................................................................18
Exponential Functions ...........................................................................................................................18
Easier Computations .............................................................................................................................. 18
Example .................................................................................................................................................19
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Symbols and Notation
Addition (+)
We say that 4 + 5 is the sum of 4 and 5. Note that 4 + 5 is equal to 5 + 4 so that the order in which we write
down the numbers does not matter when we are adding them. Because the order does not matter, addition is
said to be commutative. When more than two numbers are to be added, as in 4+8+9, it makes no difference
whether we add the 4 and 8 first to get 12 + 9, or whether we add the 8 and 9 first to get 4 + 17. Whichever
way we work we will obtain the same result, 21. This second property of addition is called associativity.
Symbol: +
Example: 5 + 4 = 9
Subtraction (-)
We say that 8 - 3 is the difference of 8 and 3. Note that 8 - 3 is not the same as 3 - 8 and so the order in
which we write down the numbers is important when we are subtracting them i.e. subtraction is not
commutative. Subtracting a negative number is equivalent to adding a positive number, thus
7 - (-3) = 7 + 3 = 10.
Symbol: Example: 8 – 4 = 4
The Plus Or Minus Sign (±).
We often use the notation plus or minus, ±. For example, we write 12 ± 8 as shorthand for the two numbers
12 + 8 and 12 - 8, that is 20 and 4. If we say a number lies in the range 12 ± 8 we mean that the number can
lie between 4 and 20 inclusive.
Multiplication (×)
The instruction to multiply, or obtain the product of, the numbers 6 and 7 is written 6 × 7. Sometimes the
multiplication sign is missed out altogether and we write (6)(7). Note that (6)(7) is the same as (7)(6) so
multiplication of numbers is commutative. If we are multiplying three numbers, as in 2 × 3 × 4, we obtain
the same result whether we multiply the 2 and 3 first to obtain 6 × 4, or whether we multiply the 3 and 4
first to obtain 2 × 12. Either way the result is 24. This property of multiplication is known as associativity.
Recall that when multiplying positive and negative numbers the sign of the result is given by the following
rules:
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Key Points When Multiplying Numbers:
positive × positive = positive
positive × negative = negative
negative × negative = positive
negative × positive = negative
For example (-4) × 5 = -20, and (-3) × (-6) = 18. When dealing with fractions we sometimes use the word
“of” as in “find ½ of 36”. In this context “of” means multiply, that is ½ of 36 means ½ × 36 = 18.
Multiplication has several symbols:
×
()
·
Examples:
3 × 9 = 27
3(9) = 27
3·3 = 27
Division (÷)
The quantity 8 ÷ 4 means 8 divided by 4. This is also written as 8/4 or
8
and is known as the quotient of 8
4
and 4. In the fraction, the top line is called the numerator and the bottom line is called the denominator.
Note that 8/4 is not the same as 4/8 and so the order in which we write down the numbers is important.
Division is not commutative. When dividing positive and negative numbers recall the following rules for
determining the sign of the result:
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Key Points When Dividing Numbers:
positive
 positive
positive
positive
 negative
negative
negative
 negative
positive
negative
 positive
negative
Division has several symbols:
÷
/
Examples:
15 ÷ 3 = 5
15 / 3 = 5
Remember that a fraction is also division
15
5
3
Twiddle (~)
In mathematics, the tilde, often pronounced "twiddle," is often used to denote an equivalence relation
between two objects. Thus x ~ y means x "is equivalent to" y. (Note that this is quite different from stating
that x equals y.) Especially, it can be used to denote the asymptotical equality of two functions.
The Reciprocal Of A Number
The reciprocal of a number is found by inverting it. If the number
reciprocal of
3
2
is inverted we get
. So the
2
3
3
4
2
is . Because we can write 4 as , the reciprocal of 4 is ¼.
2
1
3
Now do this exercise
State the reciprocal of a)
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, b) , c) 7.
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Answer a)
1
11
, b) 5, c) .
7
6
Greater Than (>)
Example: 20 > 10
Less Than (<)
Example: 7 < 11
Not Equal to (≠)
Example: 7≠11
Order of Operations Rules
1.
Any calculation contained within parentheses is done first.
2.
Squaring (or raising to another exponent) is done second
3.
Multiplying or Dividing is done third – sets of these are done left to right
4.
Adding or Subtracting is done fourth
Example
Evaluate the expression:
(3 + 1)2 – 4 × 7/2
Do the stuff in parentheses first to get:
(4)2 – 4 × 7/2
Do the exponents next to get:
16 – 4 × 7/2
Perform the multiplication and division to get:
16 – 14
Do the addition and subtraction to get:
16 – 14 = 2
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Sequences of Operations
When you have several multiplication and divisions within an expression—work left to right.
If you don't work left to right, you will end up with the wrong answer!
Show your answer for each stage in the four steps in the Order of Operation Rules—like the preceding
example.
Proportions: Fractions, Decimals, and Percentages
A proportion is a part of a whole and can be expressed as

Fraction

Decimal

Percentage
Example
Say that out of a class of 40, 3 people fail an exam.
The proportion of the class that failed can be expressed as a fraction:
3/40
Or as a decimal value:
0.075
Or as a percentage:
7.5%
The Denominator
The denominator is the bottom number in a fraction. For example, in the fraction ¾
Four is the denominator because it is the bottom number.
The denominator tells how many equal sized pieces into which the whole is split.
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The Numerator
The value of the top number in a fraction tells us how many of the equal pieces specified in the bottom
number we are using. The figure to the right illustrates the fraction ¾.
To convert a fraction to a decimal, divide the numerator by the denominator.
¾ = 3 ÷ 4 = .75
Converting a Decimal to a Percentage
To convert a decimal to a percentage, multiply the decimal by 100 and place a percent sign after the result.
0.75 × 100 = 75%
Percentages as a Special Fraction
Remember that all a percentage is a special fraction where the bottom number is always 100.
That is why percentages are so universally used—we know that the percentage represents how many pieces
of “pie” we are considering out of 100.
67% = 67/100
Math With Percentages
Some calculators let you deal with fractions, but most don't.
The easiest way is often to convert the percentage to a decimal and plug it into a calculator.
32% × 50 = ?
.32 × 50 = 16
Adding Negative Numbers
When adding numbers that have negative signs, look at the negative sign as subtraction.
3 + (-2) + 5 = ?
3–2+5=6
Hint: When adding a long string of numbers, it is easier to add up all the positive values add up all of the
negative values subtract the negative sum from the positive sum
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Subtracting Negative Numbers
Change the negative number to a positive number and add.
4 – (-3) = ?
4+3=7
Multiplying and Dividing Negative Numbers - Summary
When two numbers being multiplied or divided have the same sign, then the result is always positive.
When two numbers have different signs, the result is always negative.
Multiplication and Division Example
3 × (-2) = -6
-2 × -2 = 4
-6 ÷ 2 = -3
-6 / -2 = 3
Basic Algebra: Solving Equations
An equation is a math statement that indicates that two things are identical—they have the same value.
We are all familiar with the general form:
12 = 4 + 8
1+1=2
Equations with Variables
Some equations will have an unknown quantity that is identified with a letter or symbol that stands in—like
a question mark—for what we do not know.
Algebra teachers love to use the letter X, but any letter or symbol will work.
12 = 8 + x
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Equations as a Balance
The equal sign is like the centre point of a balance. The idea is to make sure each side stays the same.
The equation is still “true” if you make sure to do the exact same thing to both sides—it stays balanced.
Finding the Value of X
Finding the value of the unknown quantity is called solving the equation.
When solving for x, keep the following 2 things in mind:
Your goal is to have the unknown value by itself on one side of the equal sign. You do this by removing all
the other symbols on that side of the equal sign.
The equation must be made to stay in balance by treating both sides exactly the same. For example, you
could add 10 points to both sides of the equation and not change the value of X.
Exponents
Whenever a number is multiplied by itself 2 or more times, a simplified notation can be used: A number—
called an exponent—is raised above the number that is multiplied by itself.
The number to be multiplied by itself is called the base.
The exponent tells us how many times the base is to be multiplied times itself.
52 = 5 × 5 = 25 the second power is said to be “squared”
23 = 2 × 2 × 2 = 8 the third power is said to be “cubed”
24 = 2 × 2 × 2 × 2 = 16 read “2 to the fourth power”
Special Exponent Rules
Any number raised to the first power equals itself
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The exponent applies only to the base that is just in front of it
If a negative number is raised to an exponent, the result will be positive for exponents that are even and
negative for exponents that are odd
If an exponent appears outside of parentheses, then any operation inside the parentheses is done first then
the result is raised to the power of the exponent. This is further expanded upon below.
Square Roots
 
The square root of a value is the number that when multiplied by itself gives the original value.
The square root is the reverse of an exponent.
If we square a number, we can “undo” it by taking the square root.
General Introduction to Absolute Value
The absolute value of a number tells you how far away that number is from 0 on the number line, so
it is a measure of distance, it is always a positive quantity.
Put simply the modulus of a number is the size of that number regardless of its sign. For example |4| is
equal to 4, and |- 3| is equal to 3. The modulus of a number is thus never negative.
The absolute value of a number is denoted by | |, with the number being referred to inside the bars. Say, for
example, we wanted to know how far away from 0 the number negative three is. The notation for that, then,
would be |-3|. And since -3 is three units away from 0, the absolute value of -3 is 3. Formally, |-3|=3.
Similarly, we know that the number 3 is also three units away from 0, so |3| = 3 also. From this we can see
that additive opposites have the same absolute value, because they are the same distance away from 0 on
the number line.
The formal definition of absolute value is given by:
 x if x  0
x 
 x if x  0
When working with absolute value we often see equations such as:
|x| = a.
These equations will have two solutions provided a > 0; If a = 0 it will have only one solution; and if a < 0
it will have no solutions. We can solve these equations simply using the definition of absolute value:
|x| = a
Giving us:
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x = a or –x = a.
This approach works, but we usually use a modified form of this:
|x| = a,
means :
x = ±a
When an absolute value occurs with an inequality, there are two basic situations which can occur:
| x | < a and | x | > a.
For both cases, we can derive the formula using the definition of absolute value:
Case 1:
|x|<a
- x < a if x < 0 , or x < a if x > 0.
This gives us:
x > -a if x < 0 , or x < a if x > 0.
-a < x <0 , or 0 <x < a.
This means that :
-a < x < a.
Case 2:
|x|>a
-x > a if x < 0 , or x > a if x > 0.
Similarly
x < -a if x < 0 , or x > a if x > 0.
This gives us:
x < -a or x > a.
Example 1:
|x-5|<3
Using the result from Case 1,
-3 < x - 5 < 3
Adding 5 to both sides, we get our answer:
2 < x < 8.
Example 2:
|x-5|>3
Using the result from Case 2,
x - 5 < -3 or x - 5 > 3
and,
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x < 2 or x > 8
Now do this exercise State the modulus of a) -17, b)
Answer a) 17, b)
1
1
, c) 
7
5
1
1
, c)
7
5
Summary
| x | < a gives -a < x < a.
| x | > a gives x < -a or x > a.
The Factorial Symbol (!)
Another commonly used notation is the factorial, denoted by the exclamation mark “!”. The number 5!,
read “five factorial”, or “factorial five”, is a shorthand notation for the expression 5 × 4 × 3 × 2 × 1, and the
number 7! is shorthand for 7 × 6 × 5 × 4 × 3 × 2 × 1. Note that 1! equals 1, and by convention 0! is defined
as 1 also. Your scientific calculator is probably able to evaluate factorials of small integers. It is important
to note that factorials only apply to positive integers.
Key Point Factorial notation:
If n is a positive whole number then n! = n × (n - 1) × (n - 2) ... 5 × 4 × 3 × 2 × 1
Example
(a) Evaluate 4! and 5! without using a calculator. (b) Use your calculator to find 10!.
Solution (a) 4! = 4 × 3 × 2 × 1 = 24. Similarly, 5! = 5 × 4 × 3 × 2 × 1 = 120. Note that, for example,
5! = 5 × 4!. (b) Using your calculator check that 10! = 3, 628, 800.
Now do this exercise Find the factorial button on your calculator and hence compute 11!. The button may
be marked !. Check that 11! = 11 × 10! Answer 39,916,800.
Powers: Definition and Simple Powers with Numbers
Even though you may not be aware of it, you have already encountered the concept of “raising a number to
a power”. Consider the following simple example: 3 × 3 = 9. We can re-write this as: 3(2)=9. This notation
simply means that if we take the number 3 and multiply it by itself we get 9. Let us consider:
3 × 3 × 3 = 3(3) = 27. This means that three threes multiplied together gives us 27.
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In the above examples we say that the number 3 is raised to a power. If 3 is raised to the 2nd. power, we get
9 and if 3 is raised to the 3rd. power we get 27. In fact, writing the number of times that we multiply
something into itself as a power - with the notation of the power as being a superscript to the right of the
number - can save us quite a lot of space! Suppose we were multiplying 3 by itself 6 times, and compare
the following:
3 × 3 × 3 × 3 × 3 × 3 = 729
36 = 729
Clearly, the 2nd. way of writing has the advantages of: (i) being more efficient and (ii) being easier to see
how many times the number is being multiplied into itself. Furthermore, as you will see below, this way of
indicating numbers and powers makes the algebraic manipulation of numbers and multiplication of large
values much easier.
Powers with Variables
To generalize the notion of powers further, we can imagine any number being multiplied by itself any
number of times. We call a general - under specified - number a variable, and the canonical notation for an
algebraic variable is x. Thus, we say that x2 is “x raised to the power of two” or equivalently “x squared”
(which is x multiplied by itself).
In fact, if we multiply the variable x by itself a number of times (say n - times) we get: xn. The number
represented here as x is called the base, and n is called the power or exponent. The basic definition of a
number written in exponential notation states that the base should be multiplied by itself the number of
times indicated by the exponent.
For instance:
52 = 5 × 5
b4 = b × b × b × b.
Algebraic Rules for Working with Powers
From the above definition of powers some basic algebraic rules follow. Consider the following; suppose we
multiply the following:
23 × 22 = (2 × 2 × 2) × (2 × 2) = 2 5 = 32
Notice that in multiplying the 2 to the 3rd. power by 2 to the 2nd. power we get number that is 2 to the fifth
power. Thus, in multiplying numbers in exponential notation - following the rules of number multiplication
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- the power must add: 23 × 22 = 2 3 + 2 = 25. This, in fact, generalizes to any base and any power, and we can
write it in terms of our variable notation introduced above.
From this, the division rule for powers is easy to see as well:
23 / 22 = (2 × 2 × 2) / (2 × 2) = 2 1 = 2
Thus, instead of adding we subtract powers when carrying out a division: 23 / 22 = 2 3 - 2 = 21.
Another rule that emerges out of the algebra of exponential powers is that if a number raised to a power and
is in the denominator of a fraction, then it can be written as “raised to a negative power”. For instance,
consider the following example:
1/22 = 2-2 = 1/4.
Another important observation about exponential powers rests in the fact that a number raised to some
power can itself be raised to a power again! An example can best illustrate this point. Consider:
(23) × (23) = 23+3 = 26
The number on the left side of the equations string is the quantity “2 cubed” multiplied by itself. According
to our definition above - of algebraic powers -we can write this as: (23)2 ... that is, 2 cubed raised to the
power of two. Thus, when a number to some power is raised to a power again, we multiply the two powers
to get the resultant value:
(23)2 = 2(3) × (2) = 26.
Finally, one more important piece of knowledge about exponents: a number raised to the power of zero is
always equal to 1. Thus: 20 = 130= -40 = 1.
These are the basic rules for working with and manipulating exponents and powers. You need to keep these
in mind and they will show up frequently. Here is a summary of what has been mentioned above, in more
general notation.
Summary of Algebraic Rules for Powers

Rule for Multiplication: xn × xm = xm+n

Rule for Division: xn ÷ xm = xn-m

Rule for Raising a Power to a Power: (xn)m = xnxm

Negative Exponents: A negative exponent indicates that the power is in the denominator: x-n=1/xn

Identity Rule: Any nonzero number raised to the power of zero is equal to 1, x0=1 (x not zero).
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Arithmetical Expressions
A quantity made up of numbers and one or more of the operations +, -, × and / is called an arithmetical
expression. Frequent use is also made of brackets, or parentheses, ( ), to separate different parts of an
expression. When evaluating an expression it is conventional to evaluate quantities within brackets first.
Often a division line implies bracketed quantities. For example in the expression
bracketing of the numerator and denominator i.e. the expression is
would be evaluated first resulting in the number
3  4
7  9
3 4
there is implied
79
and the bracketed quantities
7
.
16
Two Signs Rule
"Two like signs make a positive sign,
two unlike signs make a negative sign"
When We Multiply:
Example
×
two positives make a positive:
3×2=6
×
two negatives make a positive:
(−3) × (−2) = 6
×
a negative and a positive make a negative:
(−3) × 2 = −6
×
a positive and a negative make a negative:
3 × (−2) = −6
Similarly for division x ÷ y = x × (1/y)
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The BODMAS rule
When several arithmetical operations are combined in one expression we need to know in which order to
perform the calculation. This order is found by applying rules known as precedence rules which specify
which operation has priority. The convention is that bracketed expressions are evaluated first. Then orders
(or operators), that is powers, square roots etc. Any division and multiplication are then performed, and
finally any addition and subtraction. For short this is called the BODMAS rule.
Key Points
Brackets ( )
First priority: evaluate terms within brackets
Operators (Orders) n
Division ÷
Multiplication ×
Addition +
n/m
Second priority: evaluate all powers including roots
Third priority: carry out all multiplications and divisions
Fourth priority: carry out all additions and subtractions
Subtraction -
Alternately
Paranthesis ( )
Exponents
n n/m
Multiplication ×
Division ÷
Addition +
Subtraction -
First priority: evaluate terms within brackets
Second priority: evaluate all powers including roots
Third priority: carry out all multiplications and divisions
Fourth priority: carry out all additions and subtractions
Even the media are aware of the problem! 9 – 3 ÷ ⅓ + 1 = ?
If an expression contains only multiplication and division we evaluate by working from left to right.
Similarly, if an expression contains only addition and subtraction we evaluate by working from left to right.
In exponentiation, or raising to a power. We shall see that, in the simplest case, this operation is repeated
multiplication and it is usually carried out once any brackets have been evaluated.
Example Evaluate a) 4 - 3+7 × 2, b) 8 ÷ 2 - (4 - 5)
Solution (a) The BODMAS rule tells us to perform the multiplication before the addition and subtraction.
Thus 4 - 3+7 × 2 = 4 - 3+14 Finally, because the resulting expression contains just addition and subtraction
we work from the left to the right, that is 4 - 3 + 14 = 1 + 14 = 15
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(b) The bracketed expression is evaluated first: 8 ÷ 2 - (4 - 5) = 8 ÷ 2 - (-1) Division has higher priority than
subtraction and so this is carried out next giving 8 ÷ 2 - (-1) = 4 - (-1) Subtracting a negative number is
equivalent to adding a positive number. Thus 4 - (-1) = 4 + 1 = 5
Try each part of this exercise Evaluate a) 4 + 3 × 7, b) (4 - 2) × 5 Part (a) Use the BODMAS rule to decide
which operation to carry out first. Because multiplication has a higher priority than addition we find
Answer 25
Part (b) In the expression (4 - 2) × 5 the bracketed quantity must be evaluated first. Answer 2 × 5 = 10
Now do this exercise Evaluate
94
. - Remember that the dividing line implies that brackets are present
25  5
9  4
94

25  5 25  5
around the numerator and around the denominator. Thus
Answer
5 1

20 4
The Greek Alphabet
alpha
А
α
iota
Ι
ι
rho
Ρ
ρ
beta
В
β
kappa
Κ
κ
sigma
Σ
σ
gamma
Г
γ
lambda
Λ
λ
tau
Τ
τ
delta
Δ
δ
mu
Μ
μ
upsilon
Υ
υ
epsilon
Ε
ε
nu
Ν
ν
phi
Φ
φ
zeta
Ζ
ζ
xi
Ξ
ξ
chi
Χ
χ
eta
Η
η
omicron
Ο
ο
psi
Ψ
ψ
theta
Θ
θ
pi
Π
π
omega
Ω
ω
More exercises for you to try
1.
Simplify without using a calculator a) -5 ×-3, b) -5 × 3, c) 5 ×-3, d) 15 ×-4, e) -14 ×-3, f)
g)
18
,
3
36
21
, h)
 12
7
Answer (a) 15, (b) -15, (c) -15, (d) -60, (e) 42, (f ) -6, (g) -3, (h) 3
2.
Evaluate (a) 3 + 2 × 6, (b) 3 - 2 - 6, (c) 3 + 2 - 6, (d) 15 - 3 × 2, (e) 15 × 3 - 2, (f ) (15 ÷ 3) + 2,
(g) 15 ÷ 3+2, (h) 7+4 - 11 - 2, (i) 7 × 4+11 × 2, (j) -(-9), (k) 7 - (-9), (l) -19 - (-7),
(m) -19 + (-7).
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Answer (a) 15, (b) -5, (c) -1, (d) 9, (e) 43, (f) 7, (g) 7, (h) -2, (i) 50, (j) 9, (k) 16, (l) -12,
(m) -26
3.
Evaluate (a) | - 18|, (b) |4|, (c) | - 0.001|, (d) |0.25|, (e) |0.01 - 0.001, (f) 2!, (g) 8! - 3!, (h)
9!
8!
Answer (a) 18, (b) 4, (c) 0.001, (d) 0.25, (e) 0.009, (f) 2, (g) 40314, (h) 9
4.
Evaluate (a) 8 + (-9), (b) 18 – (-8), (c) -18 + (-2), (d) -11 - (-3)
Answer (a) -1, (b) 26, (c) -20, (d) -8
5.
State the reciprocal of (a) 8, (b)
Answer (a)
6.
9
13
1
13
, (b)
8
9
Evaluate (a) 7 ± 3, (b) 16 ± 7, (c) -15 ± ½ (d) -16 ± 0.05, (e) |- 8|± 13, (f) |- 2|± 8.
Answer (a) 4,10, (b) 9,23, (c) -15 ½, -14 ½, (d) -16.05, -15.95, (e) -5, 21, (f) –6, 10
7.
Which of the following statements are true? (a) -8 ≤ 8, (b) -8 ≤ -8! (c) -8 ≤ |8|, (d) |- 8| < 8, (e)
|- 8| ≤ -8, (f) 9! ≤ 8!, (g) 8! ≤ 10
Answer (a), (b), (c), (g) are true
Aids to Calculation
Logarithm
The logarithm is the mathematical operation that is the inverse of exponentiation (raising a constant, the
base, to a power).
The logarithm of a number x in base b is the number n such that x = bn. If n is a positive integer, b n means
the product of n factors equal to b.
Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to
multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical
operations easier to perform. They have important mathematical properties and are still used in many ways.
Bases
The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When
"log" is written without a base (b missing from logb), the intent can usually be determined from context:
natural logarithm (loge) in mathematical analysis
common logarithm (log10) in engineering and when logarithm tables are used to simplify hand
calculations
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To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
Other Notations
The notation "ln(x)" invariably means loge(x), i.e., the natural logarithm of x, but the implied base for
"log(x)" varies by discipline:
Mathematicians generally understand both "ln(x)" and "log(x)" to mean loge(x) and write "log10(x)" when
the base-10 logarithm of x is intended.
Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean
the natural logarithm of x, and take "log(x)" to mean log10(x).
On most calculators, the LOG button is log10(x) and LN is loge(x).
Exponential Functions
The natural exponential function exp(x), also written ex is defined as the inverse of the natural logarithm. It
is positive for every real argument x.
The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by
bp=exp(pln((b))
Easier Computations
Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this
makes certain operations easier:
Operation with numbers
Operation with exponents
Logarithmic identity
a×b
A+B
Log(ab)=log(a)+log(b)
a/b
A-B
Log(a/b)=log(a)-log(b)
ab
Ab
Log(ab)=b;og(a)
A/b
1
ln b a  lna 
b
b
a
 
These relations made such operations on two numbers much faster and the proper use of logarithms was an
essential skill before multiplying calculators became available.
To multiply two numbers, one found the logarithms of both numbers on a table of common logarithms,
added them, and then looked up the result in the table to find the product. This is faster than multiplying
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them by hand, provided that more than two decimal figures are needed in the result. The table needed to get
an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few
shelves.
To compute powers or roots of a number, the common logarithm of that number was looked up and
multiplied or divided by the radix. Interpolation could be used for still higher precision.
Example
For example
direct
calculation
0.99896900
0.3920025
using ln and
exp functions
ln(0.99896)
-0.0010405
900×ln(0.99896) -0.9364871
exp(900×ln(0.99896)) 0.3920025
using log and
powers of 10
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log(0.99896)
-0.0004519
900×log(0.99896)
10900×ln(0.99896)
-0.4067112
0.3920025
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