Day 1 Chapter 1 NUMBER SYSTEM-1

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.

• 0.45

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

• 3.14159

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

• 10

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

• ⁵/₃

This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real

• 1 ²/₃

This can also be written as ⁵/₃, which is the same as the previous problem. The answer is:rational, real

• –sqrt(81)

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

• ^{– 9}/₃

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