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Numbers

(i) Naturalnumber system

1, 2, 3, … are called natural numbers (or counting numbers). The set of all natural numbers is denoted by N.

N = {1, 2, 3, …}

(ii) Whole number system

0, 1, 2, 3, … are called whole numbers. The set of all whole numbers is denoted by W.

W = {0, 1, 2, 3, 4, …}

All natural numbers are whole numbers, but the whole number 0 is not a natural number. Addition of two (or more) natural numbers or two (or more) whole numbers is also a natural number of a whole number respectively.

(e.g.,) 23 + 17 = 40. 40 is a natural number

0 + 20 = 20. 20 is a whole number

What applies to addition, also applies to multiplication.

Subtraction of one natural or whole number from another natural or whole number need not always be a natural number or a whole number.

For example, 20 – 25 = -5, -5 is neither a natural number nor a whole number.

0 – 7 = -7. Again -7 is neither a whole number nor a natural number.

What applies to subtraction also applies to division.

For example, $$\frac{28}{13} = 2\frac{2}{13}$$

is neither a natural number nor a whole number. What about division by zero? It is not permitted.

(iii) Integers:

Since zero is not an element in the set of natural numbers N, the set of whole numbers W was invented. 0 is called the additive identity for 0 + any whole number = the same whole number. (0 + 8) = 8 (0 + 0 = 0). Both the natural number system and the whole number system do not have negative numbers. To answer this need came the set of integers denoted by Z.

Z = {… -3, -2, -1, 0, 1, 2, 3, …}

(Zero is neither positive nor negative).

All whole numbers and natural numbers are integers but all integers are not whole numbers or natural numbers.

In Z, addition, subtraction and multiplication have got solutions, but not division.

For example, (i) 12 – 20 = -8. -8 belongs to Z

(ii) (-10) + (-17) = -27, -27 belongs to Z

(iii) 40 x 0 = 0, 0 belongs to Z

(iv) -8 x -7 = 56, 56 belongs to Z

But $$\frac{-10}{3} = -3\frac{1}{3}$$

does not belong to Z. $$ -3\frac{1}{3}$$ is not an integer

Hence the necessity of another system to include such non integer numbers arose.

(iv)Rational numbers

A number of the form $$\frac{p}{q}$$ where p and q are integers and q ? 0 is called a rational number. The set of all rational numbers is denoted by Q. Every integer (every whole number and every natural number) is a rational number. As -17 is an integer. It can be expressed as $$\frac{-17}{1}$$ which is a rational number. But the converse is not true (i.e., Every rational number need not be an integer). For $$\frac{12}{13}$$ is not an integer, but only a rational number.

(v) Decimal representation of rational numbers

A rational number can be expressed as a decimal. For example $$\frac{2}{5}=0.4$$ which is a terminating decimal. Again $$\frac{3}{8}=0.375$$ which is also a terminating decimal.

Now consider $$\frac{3}{7}$$ when 3 is divided by 7,

$$\frac{3}{7}=0.428571428571\ldots$$

It is a non terminating repeating (or recurring) decimal.

are examples of non terminating recurring decimals. From this it is observed that all terminating decimals and repeating (recurring) decimals are rational numbers.

A terminating decimal can be given as a rational number (of the form p/q).

For example (i) -0.35 (a terminating decimal)

$$=\frac{-35}{100}=\frac{-7}{20}$$

(ii) $$1.075=\frac{1075}{1000}=\frac{43}{40}$$

A recurring decimal can also be rewritten as a rational number.

For example

examples

Thus every terminating decimal as well as every recurring decimal is a rational number.

(vi) Non-terminating and non-recurring decimals ? Irrational numbers

0.10100100100001 … is an example for non-terminating non-recurring decimal.

$$\sqrt{2}=1.4142135 \ldots$$

$$\sqrt{3}=1.7320508 \ldots$$

are also examples for non terminating, non-recurring decimals.

Such non-terminating, non recurring decimals are called irrational numbers.

A rational number is either a terminating or non-terminating but recurring decimal.

An irrational number is a non-terminating and non-recurring decimal.

It cannot be given in the form

$$\frac{p}{q}$$

where p and q are integers (q ? 0)

chart

Real numbers:

Rational numbers and irrational numbers taken together form the real number system. The set ofreal numbersis denoted by the letter R.

Q is the set of rational numbers and Q’ is the set of irrational numbers

R = Q U Q’

The set of real numbers includes all natural numbers, whole numbers, integers, rational numbers and irrational numbers.