Numbers
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A number is an abstract entity used originally to describe quantity.
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CGF Lecture 2 Numbers
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i.e. 80 Students etc
The most familiar numbers are the natural numbers {0, 1, 2, ...} or {1, 2, 3, ...},
used for counting, and denoted by N or N
This set is infinite but countable by definition.
To be unambiguous about whether zero is included or not, sometimes an
index "0" is added in the former case, and a superscript "*" is added in the
latter case:
No = {0, 1, 2, ...}N ∗ = {1, 2, ....}
Integers
Number Lines
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The integers consist of the positive natural numbers (1, 2, 3, -), their
negatives (-1, -2, -3, ...) and the number zero.
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The number line is a diagram that helps visualise numbers and their
relationships to each other.
The set of all integers is usually denoted in mathematics by Z or , Z
which stands for Zahlen (German for "numbers").
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The numbers corresponding to the points on the line are called the
co-ordinates of the points.
They are also known as the whole numbers, although that term is
also used to refer only to the positive integers (with or without
zero).
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The distance between to consecutive integers is called a unit and is
the same for any two consecutive integers.
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The point with co-ordinate 0 is called the origin.
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Real Numbers
Rational Numbers
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In mathematics, a rational number (or informally fraction) is a ratio
or quotient of two integers
a
It is usually written as the vulgar fraction b or a/b, where b is not
zero.
Each rational number can be written in infinitely many forms, for
example
3
2
1
= =
6
4
2
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Real numbers may be rational or irrational; algebraic or
transcendental; and positive, negative, or zero.
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Real numbers measure continuous quantities.
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Measurements in the physical sciences are almost always conceived
as approximations to real numbers.
They may in theory be expressed by decimal fractions that have an
infinite sequence of digits to the right of the decimal point;
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For every rational number there is a point on the number line
1
For example the number 2 corresponds to a point halfway between 0 and 1 on
the number line.
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and − 4 corresponds to a point one and one quarter units to the left of 0.
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The set of points that corresponds to all points on a number line are called the
set of Real Numbers.
5
Since there is a correspondence between the numbers and points on the
number line the points are often referred to as numbers
Real Numbers
Real Numbers
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Computers can only approximate most real numbers with rational
numbers; these approximations are known as floating point
numbers or fixed-point numbers.
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Computer algebra systems are able to treat some real numbers
exactly by storing an algebraic description (such as "sqrt(2)") rather
than their decimal approximation.
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Mathematicians use the symbol R or alternatively R to represent
these numbers
Irrational Numbers
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All rational numbers are real numbers, but there are points on the
number line that do not correspond to rational numbers.
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Those real numbers that are not rational are called irrational
numbers
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An irrational numbers cannot be written as a ratio of integers.
√
It can be show that numbers such as 2 and π are irrational
Irrational numbers are not as easy to represent as rational numbers
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The absolute value of a number is the numbers distance from 0
(origin) on the number line.
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For example the numbers 5 and -5 are both five units away from 0
on the number line.
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So the absolute value of both these number is 5
We write |a| for the absolute value so
|5| = 5 and |−5| = 5
The notation |a| represents distance, and distance is never negative
so |a| is greater than or equal to zero for any real number
Usually they are represented using symbols such as
√ √
2, 3 and
π
When we perform computations with irrational numbers we use
rational approximations for them
• For example :
√
• 2 ≈ 1.414 and π ≈ 3.141
√
• Not that not all square roots are Irrational for example 9 = 3
because 3 × 3 = 9
Absolute Value
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Irrational Numbers II
Opposites
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Two numbers that are located on opposite sides of zero and have
the same absolute value are called opposites
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The number 5 and -5 are opposite to each other.
The symbol "-" is used to indicate opposite as well as negative.
When the negative sign is used before a number it should be read
as "negative"
Opposites
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When it is used in front of parentheses or a variable it should be read as
"opposite"
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-(5) = -5 means the "opposite" of 5 is "negative" 5
In general -a means "the opposite of a"
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If a is positive, -a is negative.
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The result of writing numbers in a meaningful combination with the
arithmetic operations is usually called and arithmetic expression or
expression
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For example
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The parentheses (or brackets) are used as grouping symbols and
indicate which operation to perform first
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Because of the parentheses these expressions have different values
If a is negative, -a is positive.
Using this we can give a symbolic definition of absolute value
|a| =



a
−a
if a is positive or zero
if a is negative
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For example
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In general an expression of the form an is called an exponential
expression and is defined as follows
2 × 2 × 2 = 23 and 5 × 5 = 52
For any counting number n,
an = a × a × a × . . . × a
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n factors
we call a the base and n the exponent
(3 + 2) × 5 and 3 + (2 × 5)
(3 + 2) × 5 = 5 × 5 = 25
3 + (2 × 5) = 3 + 10 = 13
Exponential Expressions
An arithmetic expression with repeated multiplication can be
written by using exponents
Arithmetic Expressions
Exponential Expressions
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The expression an is read "a to the nth power"
If the exponent is 1, it is usually omitted
e.g. 91 = 9
Order of Operations
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When we evaluate expressions, operations within grouping symbols are always
performed first
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For example
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To make expressions look simpler we sometimes omit some or all of the
parentheses
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To do this we must agree in which order the operation are always carried out
Order of Operations
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(3 + 2) × 5 = (5) × 5 = 25 and (2 × 3)2 = 62 = 36
We do multiplication before addition and exponential expressions before
multiplication.
If no grouping symbols are present, evaluate expressions in the
following order:
1. Evaluate each exponential expression (in order from left to right).
2. Perform multiplication and division (in order from left to right).
3. Perform addition and subtraction (in order from left to right).
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For operations within grouping symbols, use the above order within
the grouping symbols.
3 + 2 × 5 = 3 + 10 = 13 and 2 × 32 = 2 × 9 = 18
Algebraic Expressions
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Since variables (or letters) are used to represent numbers, we can use variables
in arithmetic expressions.
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The result of combining numbers and variables with the ordinary operations of
arithmetic is called an algebraic expression
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For example
a−b
x + 2, πr2 , b2 − 4ac, and
c−d
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Expressions are often named by the last operation to be performed
in the expression.
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For example the expression x + 2 is a sum because the only
operation in the expression is addition
Algebraic Expressions II
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The expression a − bc is referred to as a difference because
subtraction is the last operation to be performed
• The expression 3(x − 4) is a product
is a quotient
• The expression
• The expression (a + b) is a square because the addition is performed
3
x−4
2
before the square is found
Translating Algebraic Expressions
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Algebra is useful because it can be used to solve problems.
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Since problems are often communicated verbally, we must be able to translate
verbal expressions into algebraic expressions and translate algebraic expression
into verbal expressions
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Consider the following examples of verbal expressions and their corresponding
algebraic expressions
Verbal Expression
Algebraic Expression
The sum of 5x and 3
The product of 5 and x + 3
5x + 3
5(x + 3)
The
The
The
The
sum of 8 and x3
quotient of 8 + x and 3
difference of 3 and x2
square of 3 − x
8+
8+x
3
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The value of an algebraic expression depends on the values given to
the variables.
For example, the value of x − 2y where x = −2 and y = −3
is found by replacing x and y by − 2 and − 3 respectively
x − 2y = −2 − 2(−3) = −2 − (−6) = 4
If x = 1 and y = 2, the value of x − 2y is found by replacing x by 1 and y by 2
x − 2y = 1 − 2(2) = 1 − 4 = −3
x
3
or (8 + 3)/3 or (8 + x) ÷ 3
3 − x2
(3 − x)2
Properties of the Real Numbers
Equations
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Evaluating Algebraic Expressions
An equation is a statement of equality of two expressions
For example
11 − 5 = 6 , x + 3 = 9 , 2x + 5 = 13 and x
are all equations.
x
−4=1
2
In an equation involving a variable any number that gives a true statement when
we replace the variable with a number is said to satisfy the equation and is
called a solution or root to the equation
For example in the equation 2x + 5 = 13 the solution for x is x = 4 as (2 × 4) + 5 = 13
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There are a number of different properties associated with numbers
They are used in arithmetic.
In Algebra we must understand these to allow us to manipulate
equations
The Commutative Properties
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The Associative Properties
We get the same results whether we evaluate 3 + 5 or 5 + 3
This is an example of the commutative property of addition
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Consider the computation 2+3+6 using the order of operations we
add 2 and 3 to get 5 and then add 5 and 6 to get 11
The fact that 4 × 6 and 6 × 4 show that multiplication is
commutative.
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If we reverse the order we get the same result so
In General :
For any real numbers a and b
a + b = b + a and ab = ba
(2 + 3) + 6 = 2 + (3 + 6)
•We also have an associative property of multiplication.
(2 × 3) × 4 = 6 × 4 = 24 or 2 × (3 × 4) = 2 × 12 = 24
So For any Real Numbers a, b and c
(a + b) + c = a + (b + c) and (ab)c = a(bc)
The Distributive Property
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Example :
3(4 + 5) = 3 × 9 = 27 or 3 × 4 + 3 × 5 = 12 + 15 = 27
• We say the multiplication by 3 is distributed over the addition.
Distributive Property
So For any Real Numbers a, b and c
a(b + c) = ab + ac and a(b − c) = ab − ac
The identity properties
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The numbers 0 and 1 have special properties.
Multiplication of a number by 1 does not change the number
Addition of 0 to a number does not change the number.
That is why 1 is called the multiplicative identity and 0 is called the additive
identity
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Identity Properties
For any real number a
a × 1 = 1 × a = a and a + 0 = 0 + a = a
The Inverse Properties
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Every non-zero real number a also has a multiplicative inverse or reciprocal
This is written
1
1
such that a × = 1
a
a
Inverse Properties
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Properties of Integers
The following table lists some of the basic properties of addition
and multiplication for any integers a, b and c.
closure
For any real number a there is a number -a such that a + (−a) = 0
1
For any nonzero real number a there is a number such that
a
a×
1
=1
a
References
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Elementary and Intermediate Algebra, Mark Dugopolski McGraw Hill 1st Ed
2002
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Engineering Mathematics, K. A. Stroud, Macmillan 3rd Edition 1987
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http://en.wikipedia.org/wiki/Number
Collins Dictionary of Mathematics E. Borowski & J. M. Borwein Harper Collins
1989
http://en.wikipedia.org/wiki/Rational_number
http://en.wikipedia.org/wiki/Real_numbers
http://en.wikipedia.org/wiki/Floating-point_number
associativity
commutativity
addition
Multiplication
a + b is an integer
a × b is an integer
a + (b + c) = (a + b) + c
a+b=b+a
identity element
a+0=a
inverse element
a + (−a) = 0
distributivity
a × (b + c) = (a × b) + (a × c)
a × (b × c) = (a × b) × c
a×b=b×a
a×1=a