This course is the first course at the undergraduate level, designed primarily for students in mathematics, pure and applied sciences. However, it also meets the need of students in other fields. The course’s focus is to impart useful skills on the students in order to enhance their knowledge in ...
Number Systems
Indices and Logarithms
Surds
Mathematical Induction
Elementary Set Theory
Equations
Series and Sequences
Partial Fractions
Binomial Theorem and Binomial Series
Complex Numbers
,NUMBER SYSTEM
Natural number are the counting numbers 1, 2, 3, 4,…
{1, 2,3, }
Whole numbers include the natural numbers plus zero. Natural numbers are closed under the
operations of addition and multiplication only.
Integer: This is a set of numbers consisting of positive numbers, zero and negative numbers
(additive inverses of each natural numbers). It is denoted by . The operations of addition,
multiplication and subtraction are closed in the set of integers.
Rational numbers: These are set of number that consist of numbers that can be expressed in ratio
a
i.e where b 0 e.g 1/3, 11/7. Obviously, all integers are included in the set of of rational
b
number e.g 3.33=10/3. The set of rational number is denoted by . All the four basic
operations are closed in a set of rational numbers except the case case where we divide by
0 which is undefined.
Irrational numbers: These are numbers that cannot be expressed in ratios. In decimal form,
irrational numbers do not repeat or never end. Examples include , e, 2 etc. Irrational
c
numbers are denoted by . Irrational numbers are divided into two parts Algebraic and
transcendental numbers. Algebraic numbers are solutions of some polynomial equation
while transcendental numbers are not.
Real numbers: This is a set of numbers containing rational numbers and irrational numbers. It
complete the number line. i.e c
. They are denoted by .
Complex numbers: This a set of numbers containing the real part and the imaginary part i.e a bi
Hence,
0
, INDICES AND LOGARITHMS
Indices is the power or exponent which is raised to a number or a variable.
Laws of indices
Addition law: x a x b x a b
xa
Subtraction law: x a x b x a b
xb
Multiplication law: ( x a ) b x ab
1
Negative law: x a
xa
Zero law: x 0 1
a
Law of fractional Index: x b x a or ( b x ) a
b
Exercise
1) Simplify the following
a. 52 54
1 1
b. 49 2 7 0 7 2
c. 0.00013
1
d. 40 x 2 8 x
e. 10 x 0.00001
1
f. 3x
81
4
g. 64 x
2x
2) Solve the exponential equation
a. 52 x 26(5) x 25 0
b. 32 x 4(3x1 ) 27 0
c. 2 2 x 2 x1 8 0
d. 2 2 x 1 9(2 x ) 4 0
e. 2 x 1 13(2 x ) 4 0
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