Note for Problems on Numbers - PN by Placement Factory
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MODULE - 1 Number Systems Algebra 1 Notes 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 (rationals and irrational) numbers Mathematics Secondary Course 3
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MODULE - 1 Number Systems Algebra Notes • 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 a rational number on the number line; • cites examples of irrational numbers; • represent • 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. 2, 3, 5 on the number line; 1.1 EXPECTED BACKGROUND KNOWLEDGE Basic knowledge about counting numbers and their use in day-to-day life. 1.2 RECALL OF NATURAL NUMBERS, WHOLE NUMBERS AND INTEGERS 1.2.1 Natural Numbers 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 4 Mathematics Secondary Course
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MODULE - 1 Number Systems Algebra 12 6 = 6 is a natural number but is not defined in natural numbers. Thus, we can say that 2 4 i) a) addition and multiplication of natural numbers again yield a natural number but Notes 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. • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 ........... 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. 1.2.2 Whole Numbers (i) 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. (ii) 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 Division by zero (0) is not defined. (iii) Four fundamental operations can be performed on whole numbers also as in the case of natural numbers (with restrictions for subtraction and division). (iv) Whole numbers can also be represented on the number line as follows: • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 ........... 1.2.3 Integers While dealing with natural numbers and whole numbers we found that it is not always possible to subtract a number from another. Mathematics Secondary Course 5
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MODULE - 1 Number Systems Algebra Notes For example, (2 – 3), (3 – 7), (9 – 20) etc. are all not possible in the system of natural numbers and whole numbers. Thus, it needed another extension of numbers which allow such subtractions. Thus, we extend whole numbers by such numbers as –1 (called negative 1), – 2 (negative 2) and so on such that 1 + (–1) = 0, 2 + (–2) = 0, 3 + (–3) = 0..., 99 + (– 99) = 0, ... Thus, we have extended the whole numbers to another system of numbers, called integers. The integers therefore are ..., – 7, – 6, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, 6, 7, ... 1.2.4 Representing Integers on the Number Line We extend the number line used for representing whole numbers to the left of zero and mark points – 1, – 2, – 3, – 4, ... such that 1 and – 1, 2 and – 2, 3 and – 3 are equidistant from zero and are in opposite directions of zero. Thus, we have the integer number line as follows: • .......... –4 • –3 • –2 • –1 • 0 • 1 • 2 • 3 • 4....... We can now easily represent integers on the number line. For example, let us represent – 5, 7, – 2, – 3, 4 on the number line. In the figure, the points A, B, C, D and E respectively represent – 5, 7, – 2, – 3 and 4. • –7 • –6 A • –5 • –4 C D • • –3 –2 • –1 • 0 • 1 • 2 • 3 E • 4 • 5 • 6 B • 7 • 8 We note here that if an integer a > b, then ‘a’ will always be to the right of ‘b’, otherwise vise-versa. For example, in the above figure 7 > 4, therefore B lies to the right of E. Similarly, – 2 > – 5, therefore C (– 2) lies to the right of A (–5). Conversely, as 4 < 7, therefore 4 lies to the left of 7 which is shown in the figure as E is to the left of B ∴ For finding the greater (or smaller) of the two integers a and b, we follow the following rule: i) a > b, if a is to the right of b ii) a < b, if a is to the left of b Example 1.1: Identify natural numbers, whole numbers and integers from the following:Solution: 15, 22, – 6, 7, – 13, 0, 12, – 12, 13, – 31 Natural numbers are: 7, 12, 13, 15 and 22 whole numbers are: 0, 7, 12, 13, 15 and 22 Integers are: – 31, – 13, – 12, – 6, 0, 7, 12, 13, 15 and 22 6 Mathematics Secondary Course
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