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# Number System

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### Number System

1. 1. Number System T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
2. 2. Number system  In earlier days, people used to exchange their things for other things. The requirement for numbers primarily originated from the need to count.  They used the numbers 1,2,3,.that served the people for many years because all they needed to count was their crops, and animals.  Later on numbers such as zero, integers, rational numbers, irrational numbers were introduced.  There is evidence that as early as 30,000 BC our ancient ancestors were tallying or counting things. That is where the concept of number systems began. © iTutor. 2000-2013. All Rights Reserved
3. 3. Numbers  Natural Numbers: A natural number is a number that comes naturally, Natural numbers are greater than zero we can use this numbers as counting numbers: {1, 2, 3, 4, 5, 6 ….…, }.  Whole numbers: Whole numbers are just all the natural numbers plus a zero: {0, 1, 2, 3, 4, 5, ……………… , }.  If our system of numbers was limited to the Natural Numbers then a number such as –2 would have no meaning. The next number system is the Integers. © iTutor. 2000-2013. All Rights Reserved
4. 4. numbers  Integers: Integers include the Natural numbers, zero, and the negative Natural numbers. Numbers in the form of negative and positive numbers { ….-4, -3, -2, -1, 0, 1, 2, 3,4, …. }.  Rational number: Which can be written in the form of . Where p and q are integers and q ≠ 0 is called a rational number, so all the integers are rational number . q p © iTutor. 2000-2013. All Rights Reserved
5. 5. numbers  Irrational numbers : The number can not be written in the form of . Pythagorean in Greece were first to discover irrational number .  2, 3, are irrational number . q p © iTutor. 2000-2013. All Rights Reserved
6. 6. numbers  Real numbers:  All the numbers including rational and irrational numbers are called real number  The official symbol for real numbers is a bold R.  Prime numbers:  The real number which is divisible by 1 and itself is called prime number Ex- 1,2,3,5,7,11,13,17, ….. © iTutor. 2000-2013. All Rights Reserved
7. 7. The Real Number System Real Numbers (all numbers are real) Rational Numbers Irrational Numbers …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Integers Whole Numbers Natural Numbers …any number that is not rational Example: = 3.14159…… e= 2.71828….. Which can be written in the form of . q p © iTutor. 2000-2013. All Rights Reserved
8. 8. Number system  A number system defines how a number can be represented using distinct symbols.  A number can be represented differently in different systems.  For example, the two numbers (2A)16 and (52)8 both refer to the same quantity, (42)10, but their representations are different. © iTutor. 2000-2013. All Rights Reserved
9. 9. Common Number Systems Number system can be categorized as Decimal number system Binary number system Octal number system Hexadecimal Number System © iTutor. 2000-2013. All Rights Reserved
10. 10. Common Number Systems  Each number system is associated with a base or radix The decimal number system is said to be of base or radix 10  A number in base r contains r digits 0,1,2,...,r-1 Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9 © iTutor. 2000-2013. All Rights Reserved System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- decimal 16 0, 1, … 9, A, B, … F No No
11. 11. The decimal system (base 10)  The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols. S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. © iTutor. 2000-2013. All Rights Reserved Binary system (base 2)  The word binary is derived from the Latin root bini (or two by two).  In this system the base b = 2 and we use only two symbols, S = {0, 1}  The symbols in this system are often referred to as binary digits or bits.
12. 12. The hexadecimal system (base 16)  The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten).  In this system the base b = 16 and we use sixteen symbols to represent a number.  The set of symbols is S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}  The symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively.  The symbols in this system are often referred to as hexadecimal digits. © iTutor. 2000-2013. All Rights Reserved
13. 13. The octal system (base 8)  The word octal is derived from the Latin root octo (eight).  In this system the base b = 8 and we use eight symbols to represent a number.  The set of symbols is: S = {0, 1, 2, 3, 4, 5, 6, 7} © iTutor. 2000-2013. All Rights Reserved
14. 14. 14 Converting Decimal to Binary  To convert a decimal integer into binary, keep dividing by 2 until the quotient is 0. Collect the remainders in reverse order  To convert a fraction, keep multiplying the fractional part by 2 until it becomes 0. Collect the integer parts in forward order  Example: 162.375: So, (162.375)10 = (10100010.011)2 162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem 1 0.375 x 2 = 0.750 0.750 x 2 = 1.500 0.500 x 2 = 1.000 © iTutor. 2000-2013. All Rights Reserved
15. 15. 15 Octal and Hexadecimal Numbers The octal number system: Base-8 Eight digits: 0,1,2,3,4,5,6,7  The hexadecimal number system: Base-16  Sixteen digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F  For our purposes, base-8 and base-16 are most useful as a “shorthand” notation for binary numbers 10 1012 8 )5.87(84878281)4.127( 10 0123 16 )46687(16151651661611)65( FB © iTutor. 2000-2013. All Rights Reserved
16. 16. The end For more information call us 1-855-694-8886 Visit www.iTutor.com