For any mathematical calculations, use of a number
system is important. In this chapter we present some fundamental
properties of number system, which you will require while solving problems in
all competitive exams
Important Definitions
Face
Value
Face Value of a digit in a number is the value of the
digit itself wherever it may be.
Example: in 720, the face value of 2 is 2 only.
Place
Value
As the name suggest, place
value of a digit in a number is the digit multiplied by the place of the digit.
Ex: in 576, the place value of 5 is 5x100=500, and the
place value of 6 is 6x1=6.
Various
Types of Numbers
Natural
Number:
All counting numbers are called Natural Numbers. It is denoted by ‘N’.
N = {1,2,3,4….}
Whole
Numbers:
All counting numbers including zero ‘0’ is called Whole Numbers. This is denoted by ‘Z’.
Z= {0,1,2,3….}
Integers:
Which are zero, the natural numbers,
and the negatives of the naturals:
..., –6, –5,
–4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...
Rational Numbers:
Which are in the form of p/q,
where p and q are integers and q not equal to 0.
It is denoted by ‘Q’.
Ex: 2/3, 15/26…..
Irrational Numbers:
The numbers which are in the form of
Non repeating and non terminating decimal is called Irrational Numbers.
Ex: 0.33333,3.141…..
Real Numbers:
All rational an Irrational numbers
are called Real Numbers. It is denoted by ‘R’.
R=2/3, 15/26, 0.33333,3.141......
True or False:
- True or False: An integer is a rational number.
Since any
integer can be formatted as a fraction by putting it over 1, then this is true.
- True or False: A rational is an integer.
Not
necessarily; 4/1 is an integer, but 2/3 is
not! So this is false.
- True or False: A number is either a rational or an irrational,
but not both.
True! In decimal form, a number is either non-terminating
and non-repeating (so it's an irrational) or not (so it's a rational); there is
no overlap between these two number types!
Classify according to number type;
some numbers may be of more than one type.
This is a
terminating decimal, so it can be written as a fraction: 45/100 = 9/20.
Since this fraction does not reduce to a whole number, then it's not an integer
or a natural. And everything is a real, so the answer is: rational,
real
- 3.14159265358979323846264338327950288419716939937510...
You probably
recognize this as being pi, though this may be more decimal places than you
customarily use. The point, however, is that the decimal does not repeat, so pi
is an irrational. And everything (that you know about so far) is a real, so the
answer is: irrational, real
Don't let this
fool you! Yes, you often use something like this as an approximation of
pi, but it isn't pi! This is a rounded decimal approximation, and, since this
approximation terminates, this is actually a rational, unlike pi
which is irrational! The answer is: rational, real
Obviously, this
is a counting number. That means it is also a whole number and an integer.
Depending on the text and teacher (there is some inconsistency), this may also
be counted as a rational, which technically-speaking it is. And of course it's
also a real. The answer is: natural, whole, integer, rational (possibly),
real
This is a
fraction, so it's a rational. It's also a real, so the answer is: rational,
real
This can also
be written as 5/3, which is the same as the previous
problem. The answer is:rational, real
Your first
impulse may be to say that this is irrational, because it's a square root, but
notice that this square root simplifies: –sqrt(81) = –9, which is
just an integer. The answer is: integer, rational, real
This is a
fraction, but notice that it reduces to –3, so this may also count as an
integer. The answer is: integer (possibly), rational, real
Even
Numbers:
The numbers which are divisible by 2 are called Even
numbers.
Ex: 4, 64, 56, 199998….
Odd
numbers
The numbers which are not divisible by 2 are called Odd
Numbers.
Ex: 3, 57, 444445….
Prime
Numbers
The numbers which is divisible by 1 and itself is called
Prime Numbers.
Ex: 3,5,7….
Some Important Points:
- Odd x Odd = Odd
- Even x Odd = Even
- Even x Even = Even
- Odd + Odd = Even
- Odd + Even = Odd
- Even + Even = Even
