 NUMBER SYSTEMS

From time immemorial human beings have been trying to have a count of their belongings- goods, ornaments, jewels, animals, trees, sheeps/goats, etc. by using various techniques – putting scratches on the ground/stones – by storing stones – one for each commodity kept/taken out. This was the way of having a count of their belongings without having any knowledge of counting. One of the greatest inventions in the history of civilization is the creation of numbers. You can imagine the confusion when there were no answers to questions of the type “How many?”, “How much?” and the like in the absence of the knowledge of numbers. The invention of number system including zero and the rules for combining them helped people to reply questions of the type:

(i) How many apples are there in the basket?

(ii) How many speakers have been invited for addressing the meeting?

(iii) What is the number of toys on the table?

(iv) How many bags of wheat have been the yield from the field? The answers to all these situations and many more involve the knowledge of numbers and operations on them. This points out to the need of study of number system and its extensions in the curriculum. In this lesson, we will present a brief review of natural numbers, whole numbers and integers. We shall then introduce you about rational and irrational numbers in detail. We shall end the lesson after discussing about real numbers.

OBJECTIVES

After studying this lesson, you will be able to

• illustrate the extension of system of numbers from natural numbers to real (rational and irrational) numbers Number Systems Notes Mathematics.

• identify different types

identify different types of numbers;

• express an integer as a rational number;

• express a rational number as a terminating or non-terminating repeating decimal, and vice-versa;

• find rational numbers between any two rationals;

• represent√2, √3, √5 on the number line;

• cites examples of irrational numbers;

• represent on the number line;

• find irrational numbers betwen any two given numbers;

• round off rational and irrational numbers to a given number of decimal places;  • perform the four fundamental operations of addition, subtraction, multiplication and division on real numbers

OBJECTIVES

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. Recall that 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. We have also studied about four-fundamental operations on natural numbers. For, example,

4 + 2 = 6, again a natural number;

6 + 21 = 27, again a natural number;

22 – 6 = 16, again a natural number, but

2 – 6 is not defined in natural numbers.

Similarly, 4 × 3 = 12, again a natural number

12 × 3 = 36, again a natural number  12/2= 6 is a natural number but is not defined in natural numbers. Thus, we can say that

1. (a) addition and multiplication of natural numbers again yield a natural number but

( b) subtraction and division of two natural numbers may or may not yield a natural number

(ii) The natural numbers can be represented on a number line as shown below

(iii) Two natural numbers can be added and multiplied in any order and the result obtained is always same. This does not hold for subtraction and division of natural numbers.

Whole Numbers

• 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, ………..

Again, like before, there is no greatest whole number.

• The number 0 has the following properties:

a + 0 = a = 0 + a

a – 0 = a but (0 – a) is not defined in whole numbers

a × 0 = 0 = 0 × a

Number System Nios Class 10th part 1st

Division by zero (0) is not defined.

• Four fundamental operations can be performed on whole numbers also as in the case of natural numbers (with restrictions for subtraction and division).
• Whole numbers can also be represented on the number line as follows:

Integers

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

. RATIONAL NUMBERS

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

(i) When ‘a’ is a multiple of ‘b’

Suppose a = mb, where m is a natural number or integer, then a/b = m

• When a is not a multiple of b

In this case a/b is not an integer, and hence is a new type of number. Such a number is called a rational number.

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

Thus, , 2/(3,) 5/(-8) 6/(2 ) 11/7 are all rational numbers.

Positive and Negative Rational Numbers

(i) A rational number q/p is said to be a positive rational number if p and q are both positive or both negative integers

(ii) If the integers p and q are of different signs, then q /p is said to be a negative rational number.

EQUIVALENT FORMS OF A RATIONAL NUMBER

A rational number can be written in an equivalent form by multiplying/dividing the numerator and denominator of the given rational number by the same number.

For example

the point P represents 3/ 4 on the number line.

COMPARISON OF RATIONAL NUMBERS

In order to compare two rational numbers, we follow any of the following methods:

(i) If two rational numbers, to be compared, have the same denominator, compare their numerators. The number having the greater numerator is the greater rational number. Thus for the two rational numbers 17/ 9 and 17/ 5 , with the same positive denominator

If two rational numbers are having different denominators, make their denominators equal by taking their equivalent form and then compare the numerators of the resultingrational numbers. The number having a greater numerator is greater rational number.

For example, to compare two rational numbers 6/11 and3/7we first make their denominators same in the following manner:

FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS

From the above two cases, we generalise the following rule:

(a) The addition of two rational numbers with common denominator is the rational number with common denominator and numerator as the sum of the numerators of the two rational numbers.

(b) The sum of two rational numbers with different denominators is a rational number with the denominator equal to the product of the denominators of two rational numbers and the numerator equal to sum of the product of the numerator of first rational number with the denominator of second and the product of numerator of second rational number and the denominator of the first rational number.

Let us take sone examples:

DECIMAL REPRESENTATION OF A RATIONAL NUMBER

You are familiar with the division of an integer by another integer and expressing the result as a decimal number. The process of expressing a rational number into decimal form is to carryout the process of long division using decimal notation.

Let us consider some examples.

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From the above examples, it can be seen that the division process stops after a finite number of steps, when the remainder becomes zero and the resulting decimal number has a finite number of decimal places. Such decimals are known as terminating decimals.

Note: Note that in the above division, the denominators of the rational numbers had only 2 or 5 or both as the only prime factors.

RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

Is it possible to find a rational number between two given rational numbers. To explore this, consider the following examples.

IRRATIONAL NUMBERS