1. Essential Concepts of AlgebraEssential Concepts of Algebra
Business Mathematics
Lecture : 1
By: Lamya Bint-al Islam
Eastern University
Faculty of Business Administration
2. Numbers & Integers
• Numbers: A number is a digit or a collection
of digits. Numbers can be positive, negative,
odd, even, fractions, decimals and even weird
numbers such as √2.
• Integers: All whole numbers are integers, they
can be positive, negative and zero, thus, the
set of integers is {……-3,-2,-1,0,1,2,3,…....}.
3. Numbers & Integers
• The difference between ‘number’ and ‘integer’ is
that number can mean fractions or whole
number, 3 is not the only number between 2 & 4,
there are many numbers in between such as 2.5,
2.9, and 3.9. While integer only means whole
number, so 3 is the only integer between 2 & 4.
• Only integers can be even or odd. Fractions,
decimals and other non-integers can never be
even or odd.
5. Real Number
• The set of all rational and irrational numbers is
called the set of real numbers.
6. Rational Numbers
• The integers combined with the fractions form
the set of rational numbers. Thus a rational
number is a number that can be expressed in the
form of a fraction that has integers as numerator
and denominator, such as p/q where p & q are
integers and q ≠ 0. Example: 5/4, 9/10, 6/1. Here
5/4= 1.25, 1/3 = 0.33333, 1/22 = 0.045454545,
15/14 = 1.0714285714285
• So every rational number can be expressed as a
terminating or repeating decimal.
7. Irrational Number
• Irrational numbers cannot be expressed as a
simple fraction, because the decimals do not
terminate or repeat, such as √2, Π, e, and √15.
• √2= 1.414213…. Π= 3.14159265….
√7= 2.645751….
8. Complex Numbers
• Square root of a negative number is called an
imaginary number such as √-1=i, numbers
with an imaginary component are called
complex numbers such as a+ib.
9. Properties of Zero
Zero is a special number with some unique properties:
• O is even
• It is an integer but it is neither positive nor negative.
• O + any other number is equal to that number.
• O multiplied by any other number is equal to 0.
• Any number divided by 0 will be infinite or undefined.
Any number/ 0 = undefined or ∞
• 0 divided by any number equals to 0.
0/any number = 0.
• 00
is undefined.
11. Order of Operations
• 4 (1-3) + 5x 6/2 = 4 (-2) + 5 x 6/2
= - 8 + 5 x 3
= - 8 + 15
= 7
12. Properties of Algebra
Property Addition Multiplication
Commutitative If a and b are real, then
a + b = b + a
If a and b are real, then
a.b = b. a
Associative If a, b and c are real, then
(a+b) + c = a + ( b+c)
If a, b and c are real, then
(ab) c = a (bc)
Distributive If a, b and c are real, then
a (b+c) = a.b + a.c
14. Fractions
• A fraction is a number of the form a/b where a
and b are both integers and b ≠ 0.
• The integer a is called the numerator and b is
called the denominator of the fraction. For
example, -7/ 5 is a fraction where -7 is the
numerator and 5 is the denominator.
• If both the numerator a and denominator b are
multiplied by the same nonzero integer then the
resulting fraction will be equal to a/b. For
example, (-7)4 / (5)(4) = -28/ 20 = -7/5
15. Rules of Fractions
• A fraction with a negative sign in either the
numerator or denominator can be written
with the negative sign in front of the fraction,
for example, -7 /5 = 7/ -5 = - (7/5)
• If both the numerator and denominator have
a common factor, then the numerator and
denominator can be factored and reduced to
an equivalent fraction, for example, 40/ 72 =
(8) (5) / (8) (9) = 5/9
16. Addition & Subtraction of Fractions
• To add two fractions with the same denominator,
we add the numerator and keep the same
denominator, for example,
-8/11 + 5/ 11 = -8 + 5/11 = -3/11
• To add two fractions with different
denominators, we first find the LCM of the
denominators, then add the numerators .
For example, 1/3 + -2/5 = 5+ (-6) / 15 = 1/15
• The same method applies to subtraction of
fractions.
17. Multiplication & Division of Fractions
• To multiply two fractions, multiply the two
numerators and multiply the two
denominators. For example,
(10/7) ( -1/ 3) = -10/ 21
• To divide one fraction by another, first invert
the second fraction then multiply the first
fraction by the inverted fraction. For example,
• 17/8 ÷ 3/4 = (17/8) (4/3) = (17/2) (1/3) = 17/6
18. Mixed Number
• An expression such as 4⅜ is called a mixed
number. It consists of an integer part and a
fraction part, the mixed number means
4⅜ = 4 + 3/8 = 35/8
