books are the result of the mathematicians
efforts to solve mathematical problems with ease.
Methods of arriving at solutions may differ, but
the end results or answers are always the same.
These different approaches to mathematical
problems make the study of mathematics more
interesting, either by individual study or as a
You can supplement your study of mathe-
matics with the following training manuals:
1. Mathematics, Vol. 1, N A V E D T R A
2. Mathematics, Vol. 2-A, N A V E D T RA
Vol. 2-B, N A V E D T RA
4. Mathematics, Vol. 3, N A V E D T R A
TYPES OF NUMBERS
Positive and negative numbers belong to the
class called REAL NUMBERS. Real numbers and
imaginary numbers make up the number system
in algebra. However, in this training manual, we
will deal only with real numbers unless otherwise
A real number may be rational or irrational.
The word rational comes from the word ratio. A
number is rational if it can be expressed as the
quotient, or ratio, of two whole numbers.
Rational numbers include fractions like 2/7,
whole numbers (integers), and radicals if the
radical is removable. Any whole number is
rational because it could be expressed as a
quotient with 1 as its denominator. For instance,
8 equals 8/1, which is the quotient of two integers.
A number like vTZ is rational since it can be
expressed as the quotient of the two integers in
the form 4/1. An irrational number is a real
number that cannot be expressed as the ratio of
two integers. The numbers
and 3.1416 (n) are examples of irrational numbers.
An integer may be prime or composite. A
number that has factors other than itself and
1 is a composite number. For example, the
number 15 is composite. It has the factors 5
and 3. A number that has no factors except itself
and 1 is a prime number. Since it is advantageous
to separate a composite number into prime
factors, it is helpful to be able to recognize a few
prime numbers. The following are examples of
prime numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, and
A composite number may be a multiple of two
or more numbers other than itself and 1, and it
may contain two or more factors other than itself
and 1. Multiples and factors of numbers are as
follows: Any number that is exactly divisible by
a given number is a multiple of the given number.
For example, 24 is a multiple of 2, 3, 4, 6, 8, and
12 since it is divisible by each of these numbers.
Saying that 24 is a multiple of 3, for instance,
is equivalent to saying that 3 multiplied by
some whole number will give 24. Any number is
a multiple of itself and also of 1.
The most general definition of a fraction states
that a fraction is an indicated division. Any
division may be indicated by placing the dividend
over the divisor with a line between them. By the
above definition, any number, even a so-called
whole number, may be written as a common
fraction. The number 20, for example, may be
written as 20/1. This or any other fraction
that amounts to more than 1 is an IMPROPER
fraction. For example, 8/3 is an improper
fraction, The accepted practice is to reduce an
improper fraction to a mixed fraction (a whole
number plus a proper fraction). Perform the
indicated division and write the fractional part of
the quotient in its lowest term. In this case,
8/3 would be 2 2/3. A fraction that amounts to
less than 1 is a PROPER fraction, such as the
To refresh your memory, we are including the
following rules in the solution of fractions:
1. If you multiply or divide both the
numerator and denominator of a fraction by the
same number, the value does not change. The
resulting fraction is called an EQUIVALENT
2. You can add or subtract fractions only if
the denominators are alike.
3. To multiply fractions, simply find the prod-
ucts of the numerators and the products of the
denominators. The resulting fractional product
must be reduced to the lowest term possible.