Number Systems
We use numbers not only to show values but also for calculating purposes. So a number system which is easy to manipulate was needed when more complex calculations come across with the evolution of man kind.
Different number systems were used in the past with different notations. At the early stages they have used one character such as '1' to represent one unit. this is called as Unary numeral system. As example
And when they started counting more lager numbers, using the above number system was impossible so they started using a pattern to count.
In the above number system they have count things by grouping them into five.'1111 ' symbol is used to represent a group of five while '1' symbol is used to represent units in a one group. if we take twenty three, it can be divide into four set of fives and another three is remaining. this can be represented as below by using the above number system
1111 1111 1111 1111 111
This representation is called as sign-value system. the same concept is used in Roman. They have grouped things in to ones, fives, tens, fifties, hundreds, five hundreds and thousands.
Thousand seven hundred eighty six (1786) Can be group in to one(1) thousand, one(1) five hundred, two(2) hundreds, five(5) fifty, three(3) tens and four(1) ones.
MDCCLXXXVI
They write it from left to right in descending order (largest group to the smallest group)
Though this is the basic concept of Roman number system, it can get more complex.
[ If a smaller group symbol used before a large group symbol then as together they represent the value that comes after deducting the smaller group value from the larger group value. as example Thousand nine hundred (900) can be represent as MCM in Roman number system.]
When humans were evolving they need number systems not only to count but also for other purposes such as trading and architectural designs.. and so on... So they needed more logical number systems.
The Positional system or also called as place-value number system is used in different forms in different cultures. in positional system, it use set of symbols to represent numbers. the number of symbols we used in this number system is called as base. the value of that symbol is determined by it's position in the number.
As example in Hindu Arabic number system it use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 symbols represent numbers. This number system have ten different symbols. So the base is Ten. These symbols individually represent the first position of a number which represents the number of 'Ones' or the value. The positional values of the number systems are determined by its base's multiplicities.
The below example is in Hindu - Arabic number system which use base - 10 positional value system
Unlike other number systems this number system use the symbol 0 to represent that there are no value.
If we take Thousand six hundred and nine (1609) as example, this means that there are,
Nine 'Ones'
Zero 'Tens'
Six 'Hundreds'
One 'Thousands'
which is (1000x1)+(100x6)+(10x0)+(1x9).
following is a example for base-3 positional number system. a, b, c are the three symbols represent values zero, one, two respectively if we take them individually(at the first position).
the different combinations of these symbols can represent any value. Since it have three symbols the base of this number system is three(3) If we take two of those symbols we can represent different combinations.
If we take 'bca' as example, this means that there are,
Zero Ones
Two Threes
One Thrundreds (32 number of Nines) [ English words are used to represent base-10 number systems]
which is (9x1)+(3x2)+(1x0) =15 in base-10
The same concept goes with the number systems we used in computer systems.
In computer systems we use,
Base-2 number system which calls as Binary [0,1]
Base-8 number system which calls as Octal [0, 1, 2, 3, 4, 5, 6, 7]
Base-16 number system which calls as Hexa Decimal [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F]
Different number systems were used in the past with different notations. At the early stages they have used one character such as '1' to represent one unit. this is called as Unary numeral system. As example
| One | 1 |
| Two | 11 |
| Three | 111 |
| Four | 1111 |
| Five | 11111 |
| Six | 111111 |
| Seven | 1111111 |
| Eight | 11111111 |
| Nine | 111111111 |
| Ten | 1111111111 |
| Eleven | 11111111111 |
| One | 1 |
| Two | 11 |
| Three | 111 |
| Four | 1111 |
| Five | |
| Six | |
| Seven | |
| Eight | |
| Nine | |
| Ten | |
| Eleven |
In the above number system they have count things by grouping them into five.'
This representation is called as sign-value system. the same concept is used in Roman. They have grouped things in to ones, fives, tens, fifties, hundreds, five hundreds and thousands.
| Ones | I |
| Fives | V |
| Tens | X |
| Fifties | L |
| Hundreds | C |
| Five Hundreds | D |
| Thousands | M |
Thousand seven hundred eighty six (1786) Can be group in to one(1) thousand, one(1) five hundred, two(2) hundreds, five(5) fifty, three(3) tens and four(1) ones.
MDCCLXXXVI
They write it from left to right in descending order (largest group to the smallest group)
Though this is the basic concept of Roman number system, it can get more complex.
[ If a smaller group symbol used before a large group symbol then as together they represent the value that comes after deducting the smaller group value from the larger group value. as example Thousand nine hundred (900) can be represent as MCM in Roman number system.]
When humans were evolving they need number systems not only to count but also for other purposes such as trading and architectural designs.. and so on... So they needed more logical number systems.
The Positional system or also called as place-value number system is used in different forms in different cultures. in positional system, it use set of symbols to represent numbers. the number of symbols we used in this number system is called as base. the value of that symbol is determined by it's position in the number.
As example in Hindu Arabic number system it use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 symbols represent numbers. This number system have ten different symbols. So the base is Ten. These symbols individually represent the first position of a number which represents the number of 'Ones' or the value. The positional values of the number systems are determined by its base's multiplicities.
The below example is in Hindu - Arabic number system which use base - 10 positional value system
_____
|
_____
|
_____
|
_____
|
10x10x10
|
10x10
|
10
|
1
|
103
|
102
|
101
|
100
|
| Number of Thousands | Number of Hundreds | Number of Tens | Number of Ones |
Unlike other number systems this number system use the symbol 0 to represent that there are no value.
If we take Thousand six hundred and nine (1609) as example, this means that there are,
Nine 'Ones'
Zero 'Tens'
Six 'Hundreds'
One 'Thousands'
which is (1000x1)+(100x6)+(10x0)+(1x9).
following is a example for base-3 positional number system. a, b, c are the three symbols represent values zero, one, two respectively if we take them individually(at the first position).
| Zero | - a |
| One | - b |
| Two | - c |
| Position 2 | Position 1 | Calculation in decimal | Value | ||
|---|---|---|---|---|---|
| a | a | a | (30 x 0) | Zero | |
| b | a | b | (30 x 1) | One | |
| c | a | c | (30 x 2) | Two | |
| ba | b | a | (31 x 1)+(30 x 0) | Three | |
| bb | b | b | (31 x 1)+(30 x 1) | Four | |
| bc | b | c | (31 x 1)+(30 x 2) | Five | |
| ca | c | a | (31 x 2)+(30 x 0) | Six | |
| cb | c | b | (31 x 2)+(30 x 1) | Seven | |
| cc | c | c | (31 x 2)+(30 x 2) | Eight |
Zero Ones
Two Threes
One Thrundreds (32 number of Nines) [ English words are used to represent base-10 number systems]
which is (9x1)+(3x2)+(1x0) =15 in base-10
The same concept goes with the number systems we used in computer systems.
In computer systems we use,
Base-2 number system which calls as Binary [0,1]
Base-8 number system which calls as Octal [0, 1, 2, 3, 4, 5, 6, 7]
Base-16 number system which calls as Hexa Decimal [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F]
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