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### number system

1. 1. NUMBER SYSTEM BY : IKA KOMALA SARI NILA PATMALA R. ULFAHTUL HASANAH VIERA VIRLIANI B.INGGRIS MATEMATIKA
2. 2. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 NUMBER SYSTEM Human beings have trying to have a count of their belonging, goods, ornaments, jewels, animals, trees, goats, etc. by using techniques. 1. putting scratches on the ground 2. by storing stones-one for each commodity kept taken out This was the way of having a count of their belongings without knowledge of counting
3. 3. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 NUMBER SYSTEM The questions of the type: HOW MUCH? HOW MANY? Need accounting knowledge
4. 4. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 The functions of learning number system Are 11 functions, that to: Illustrate the extension of system of number from natural number to real (rational and irrational) numbers Identify different types of numbers Express an integers as a rational number Express a rational number as a terminating or non-terminating repeating decimal and vice-versa
5. 5. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 The functions of learning number system Find rational numbers between any two rationals Represent a rational number on the number line Cites example of irrational numbers Represent 2, 3, 4 on the number line Find irrational numbers between any two given numbers 2 3 4
6. 6. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 The functions of learning number system Round off rational and irrational numbers to given number of decimal places Perform the four fundamental operation of addition, subtraction, multiplication, and division on real numbers
7. 7. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.1 EXPECTED BACKGROUND KNOWLEDGE It is about the accounting numbers in use on the day to day life Accounting numbers Day life
8. 8. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers Natural Numbers 1, 2, 3, … There is no greatest natural number, for if 1 added to any natural numbers. we get the next higher natural number, call its successor. Example : 4+2=6 12:2=6 22-6=16 12×3=36
9. 9. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers Addition and multiplication of natural numbers again yield a natural numbers But the subtraction and division of two natural number may or may not yield a natural numbers Example: Number line of natural numbers 1 2 3 4 5 6 7 8 9 … 2-6 = -4 6 : 4 = 3/2
10. 10. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers Whole Numbers The natural number were extended by zero (0) 0, 1, 2, 3, … There is no greatest whole numbers 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 0 is not defined
11. 11. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers The whole number in four fundamental operation is same The line number of whole number 0 1 2 3 4 5 6 7 …
12. 12. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers Integers Another extension of numbers which allow such subtractions. It is begin from negative numbers until the whole number. The number line of integers … -3 -2 -1 0 1 2 3 4 …
13. 13. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.2 Recall of Natural Numbers, Whole Numbers, and Integers Representing Integers on number line A B C D … -4 -3 -2 -1 0 1 2 3 4 5 … Then A = -3 C = 2 B = -1 D = 3 A < B, D > C, B < C, C > A The rule: 1. A > B, if A is to the right of B 2. A < B, if A is to the left of B
14. 14. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number Rational Numbers Consider the situation, when an integer a is divided by another non-zero integer b. The following case arise: 1. When A multiple of B A = MB, where M is natural number or integer. Then, A/B =M
15. 15. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number 2. Rational number is when A is not A multiple B. A/B is not an integer. Thus, a number which can be put in the form p/q, where p and q are integers and q ≠ 0. Example: All Rational Numbers -2 5 6 11 3 -8 2 7
16. 16. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number Positive and Negative Rational Number 1. p/q is said positive numbers if p and q are both positive or both negative integers 2. p/q is said negative if p and q are of different sign. Example: + - 3 4 -1 -5 -7 4 6 -5
17. 17. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number Standard Form of a Rational Number We can see that -p/q = -(p/q) -p/-q = -(-p)/-(-q)= p/q p/-q= (-p)/q -p p -p p q -q -q q
18. 18. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number Notes: A rational number is standard form is also referred to as “a rational lowest form” . There are two terms interchangeably Example: 18/27 can be written 2/3 in standard form (lowest form)
19. 19. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.3 Rational Number Some important result: 1. Every natural number is a rational number but vice-versa is not always true 2. Every whole number and integer is a rational number but vice-versa is not always true
20. 20. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS Addition of Rational Numbers 1. Consider the addition of rational numbers , + = for example : + = =
21. 21. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS 2. Consider the two rational numbers p/q and r/s p/q + r/s = ps/qs + rq/sq = for example : ¾ + 2/3 = = =
22. 22. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 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.
23. 23. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS b)The sum of two rational numbers with different denominator 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 product of the numerator of first rational with the denominator of second and the product of numerator of second rational number and the denominator of the first rational number.
24. 24. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS Examples: Add the following rational numbers : (i) 2/7 and 6/7 (ii) 4/17 and -3/17 Solution: (i) 2/7 + 6/7 = 8/7 (ii) 4/17 + (-3)/17 = 1/17
25. 25. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS Add each of the following rational numbers, examples: (i) 3/4 and 1/7 Solution : (i) we have 3/4 + 1/7 = 3x7/4x7 + 1x4/7x4 = 21/28 + 4/28 = 25/28 3/4 + 1/7 = 25/28 or 3x7+4x1 / 4x7 = 21+4/728 = 25/28
26. 26. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS Subtraction of Rational Numbers (a) p/q – r/q = p-r/q Example : 7/4 – ¼ = … 7/4 –1/4 = 7 – 1 4 = 6/4 = 2x3 = 3/2 2x2 3/5 – 2/15 = … 3x12/5x12 – 2x5/12x5 = 36/60 – 10/60 = 26/60 = 13x2/30x2 = 13/30
27. 27. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS  Multiplication and Division of Rational Numbers (i) Multiplication of two rational number (p/q) and (r/s) , q 0, s 0 is the rational number pr/ps where qs 0 = product of numerators/product of denominators (ii) Division of two rational numbers p/q and r/s , such that q 0, s 0, is the rational number ps/qr, where qr 0 In the other words (p/q) (r/s) = p/r x (s/r) Or (First rational number) x (Reciprocal of the second rational number) Let us consider some examples
28. 28. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS Examples : (i) 3/7 and 2/9 (ii) 5/6 and (-2/19) Solution : (i) 3/7 x 2/9 = 3x2/7x9 = 3x2/7x3x3 = 2/21 (3/7))x(2/9) = 2/21 (ii) 5/6 x (-2/19) = 5x(-2)/6x19 = - 2x5/2x3x19 = -5/57 5/6 x (-2/19) = -5/57
29. 29. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS (i) (3/4) (7/12) Solution: (i) (3/4) (7/12) = (3/4) x (12/7) [Reciplocal of 7/12 is 12/7] = 3x12/4x7 = 3x3x4/7x4 = 9/7 (3/4) (7/12) = 9/7
30. 30. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 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 rational number into decimal from is to carryout the process of long division using decimal notation. Example: Represent each one the following into a decimal number (i) (ii) : 5 12 25 27
31. 31. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER Solution: Using long division, we get (i) Hence , = 2,4 (ii) (-1, 08) hence, = -1, 08 4,2 0,2 0,2 10 ,12 5 x 5 12 x 200 200 25 27 25  25 27
32. 32. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER From the above example, 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 that in the above division, the denominators of the rational numbers had only 2 or 5 or both as the only prime factor
33. 33. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER Alternatively, we could have written as = = 2,4 Other examples: Here the remainder 1 repeats. The decimal is not a terminating decimal = 2,333… or 2,3 5 12 25 212 x x 10 24 33,2 00,1 9 0,1 6 00,7 3 3 7
34. 34. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER ` = 0,2 85714 Note: A bar over a digit or a group of digits implies that digit or group of digits starts repeating itself indefinitely. 28571428,0 4 56 60 14 20 28 30 7 10 49 50 35 40 56 60 14 000.2 7 7 2
35. 35. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER Expressing decimal expansion of rational number in p/q form Examples: Express in form p/q ! Express in in form p/q ! 0,48 =100 48 25 12 Let x = 0,666 (A) 10 x = 6,666 (B) (B)-(A) gives 9x = 6 or x = 2/3 0,666 The example above illustrates that: A terminating decimal or a non- terminating recurring decimal represents a rational number
36. 36. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER Note : The non-terminating recurring decimals like 0,374374374… are written as 0,374. The bar on the group of digits 374 indicate that group of digits repeats again and again.
37. 37. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.9 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 example. Example : Find rational number between and Let us try to find the number ( + ) ( ) = now, = = And = = 4 3 5 6 2 1 4 3 5 6 2 1 20 2415 40 39 4 3 104 103 x x 40 30 5 6 85 86 x x 40 48 abviously, < <40 30 40 39 40 48
38. 38. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.9 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS is a rational number between the rational numbers and Note : = 0,75. = 0, 975 and = 1,2 Than: 0,75 < 0, 975 < 1,2 This can be done by either way : (i) reducing each of the given rational number with a common base and then taking their average (ii) by finding the decimal expansions of the two given rational numbers and then taking their average 4 340 39 5 6 4 3 40 39 5 6
39. 39. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.4 Equivalent Forms of a Rational Number A rational number can be written in an equivalent form by multiplying or dividing the numerator and denominator of the given rational number by the same number Example : 2/3 = 2x2 = 4/6 and 2/3 = 2x4 = 8/12 3x2 3x8 It’s mean 4/6 and 8/12 are equivalent form of the rational number 2/3
40. 40. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.5 Rational Numbers on the Number Line We know how to represent intergers on the number line. Let us try to represent ½ on the number line. The rational number ½ is positive and will be represented to the right of zero. As 0<½<1, ½ lies between 0 and 1. divide the distance OA in two equal parts. This can be done by bisecting OA at P
41. 41. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.5 Rational Numbers on the Number Line Let P represet ½. Similarly R, the mid- point of OA’, represents the rational number -½. A R 0 P A … -2 -1 0 1 2 3 … Similarly , can be represented on the number line as below: B’ A’ O A P B C … -2 -1 0 1 2 3 … As 1 < 4/3 < 2 therefore, 4/3 between 1 and 2 3 4 -1/2 1/2 4/3
42. 42. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.6 COMPARISON OF RATION NUMBER In order to compare to rational number, we follow any of the following methods: (i)If two rational numbers, to be compare have the same denominator compare their numerators. The number having the greater numerator is the greater rational number. Thus for the two rational numbers and , with the same positive denominator. as 9>5. so, 17 5 17 9 17 5 17 9 ,17  17 5 17 9 
43. 43. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.6 COMPARISON OF RATION NUMBER (ii) If two rational number are having different denominator, make ther denominator equal by taking their equivalent form and then compare the numerator of the resulting rational numbers. The number having a greater numerator is greater rational number. For example, to compare two rational numbers and , we first make their denominator same in the following manner: 7 3 11 6
44. 44. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.6 COMPARISON OF RATION NUMBER (iii) By plotting two given rational numbers on the number line we see that rational number to the righ of the other rational number is greater. 77 33 117 113  x x 77 42 711 79  x x As 42>33, or 77 33 77 42  7 3 11 6 
45. 45. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 For example, take and , we plot these number on the number line as below: -2 -1 0 1 2 3 3 2 4 3 1.6 COMPARISON OF RATION NUMBER
46. 46. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 1.6 COMPARISON OF RATION NUMBER 0<⅔<1 and 0< ¾<1. it means ⅔ and ¾ both lie between 0 and 1. by the method of diving a line Into equal number of parts, A represent ⅔ and B represent ¾ As B is to the right of A, ¾>⅔ or ⅔<¾ So, out of ⅔ and ¾, ¾ is greter number.
47. 47. Number System 1.1 1.2 1.5 1.6 1.3 1.7 1.8 1.9 Exit 1.4 Thank’s for your attention