TYPES OF NUMBERS AND DEFINITION

What is natural number ?

Recall that the counting numbers 1,2,3, ... constitute the system of natural numbers. These are the numbers which we use in our day-to-day life.

There is no greatest natural number, for if 1 is added to any natural number, we get the next higher natural number, called its successor.

What is whole number ?

When a natural number is subtracted from itself we can not say what is the left out number. To remove this difficulty, the natural numbers were extended by the number zero (0), to get what is called the system of whole numbers Thus, the whole numbers are

0, 1, 2, 3, ................

What is Integers ?

While dealing with natural numbers and whole numbers we found that it is not always possible to subtract a number from another.

The integers are

..., – 7, – 6, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, 6, 7, ...

Consider the situation, when an integer a is divided by another non-zero integer b. The following cases arise:

What is rational number ?

A number which can be put in the form p/q, where p and q are integers and q ≠ 0, is called a rational number

What is irrational number ?

The number that which cannot be written as a proportion (or fraction). 

What is real numbers ?

The real numbers are the set of numbers containing all the irrational numbers and all the rational numbers. The real numbers consist of “all the numbers” on the number scale.

Complex numbers ?

The complex numbers are the set {a+bi | the real numbers are a and b}, where the imaginary unit is i, −1−√-.

Problem 1 :

True or False? If false, correct the statement.

1) If a number is in integer, then the number is also rational. _____

2) If a number is real, then it is also rational. _____

3) 3.456 is an irrational number. _____

4) 11 is a real number. _____

5) Zero is an natural number. _____

6) 9 is an integer. _____

7) If a number is natural, then it also whole. _____

Solution :

1) If a number is in integer, then the number is also rational.

True

For example, let us consider 6 it is a integer and it can be written as 6/1. Then it is rational number also.

2) If a number is real, then it is also rational. _____

Real number includes all positive, negative values and 0. So, they are real numbers as well. So, it is true.

3) 3.456 is an irrational number. _____

The given number is ending decimal. So, it can be converted into fraction. The it is rational number. Given that it is irrational. It is false.

4) 11 is a real number. _____

Yes it is real number, the given is true.

5) Zero is an natural number. _____

Yes, zero is a rational number. True.

6) 9 is an integer. _____

Yes, 9 is an integer. It is true.

7) If a number is natural, then it also whole. _____

Natural numbers will start with 1. But whole numbers start with 0. Then the given statement is false.

Problem 2 :

1. Write down the whole numbers less than 5?

2. How many whole numbers are there up to 100, including 100?

3. Write down the counting numbers greater than 7 but less than or equal to 10.

4. List the next 4 even numbers after 149.

5. Write down 5 consecutive odd numbers between 45 and 60.

6. List the 3 square numbers that end with the digit 9.

7. Write down all the whole numbers between 23/5 and 69/8.

8. Draw a number line and locate all the integers between -3 and 2.

9.The square of the sum of two odd numbers is __________ (even/odd).

10. Which even number is also a prime number?

Solution :

1) Whole numbers will start with 0. So, the required numbers are 0, 1, 2, 3, 4.

2) From 1 to 100 (including), we have 100 numbers. Since whole numbers start with 0, adding that also we have 101 whole numbers upto 100.

3)  8, 9, 10.

4) Even numbers after 149 are

150, 152, 154, 156.

5)  Odd numbers will end with 1, 3, 5, 7 or 9.

47, 49, 51, 53, 55, 57, 59

6) 9, 49, and 169 are the square numbers which end with 9.

7) Converitng the given numbers 23/5 and 69/8 as decimal, we get

23/5 = 4.6

69/8 = 8.625

8) Showing the number in between -3 and 2 on the number line, we get

9) Let us consider odd numbers 3 and 5

• square of 3 = 9
• square of 5 = 25
• 9 + 25 = 34
• Which is even 

10) 2 is the only one even prime number in all numbers.