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Number System
- 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.
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- 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.
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- 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
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- 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
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- 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, …..
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- 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
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- 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.
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- 9. Common Number Systems
Number system can be categorized as
Decimal number system
Binary number system
Octal number system
Hexadecimal Number System
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- 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
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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. 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}.
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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. 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.
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- 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}
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- 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
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- 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
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