General Method of Transforming Numbers
Number we want to transform uses base X: we would like to transform this number into new one, that uses Y base. In this case we will use method of continuous dividing and multiplying (dividing numbers left of comma (,) with Y and multiplying numbers on the right side of comma, by Y - rational numbers). This method is best understood looking at these examples:
Example
Transform 5324 (decimal) into decimal number (?!) using method of continuous dividing – result 5324
- end of procedure
Example
Transform 0,8125 (decimal) into decimal number (?!) using method of continuous multiplying – result 0,8125
Decimal 2 Binary
Example
Transform 29 (decimal) into binary number - reading upwards, result is 111012
- end of procedure
Example
Transform 0,8125 (decimal) into binary number – reading downwards, result is 0,11012
Example
Transform 0,3 (decimal) into binary number – result is 0,01001 1001 1001… 2 shortly rounded = 0,010012
Example – Quick Method
Transform 53(decimal) into binary number using quick method
Decimal 2 Hex
Example
Transform 2540,34 (decimal) into hex number-result ~ 9EC, 570A316
a) left from”,”
b) right from “,”
Final result ~ 9EC , 570A316
Binary 2 Decimal
Example
Transform 110101 (binary) into decimal number
Example
Transform -11,101 (binary) into decimal number
Hex 2 Decimal
Example
Transform 9EC,570A3 (hex) into decimal number
Quick Method of Transforming between Binary, Hex and Octal System
Example
Transform -6F,A 16 into binary number
Example
Transform 11,000011001 2 into hex number
Example
Transform 37,24 8 into binary number
Example
Transform 1111011,10011101 2 into octal number
Technorati Tags: Numeric, Numerical, Transform, Transformation, Binary, Decimal, Octal, Hexadecimal, Hex, Base, Complement, Method
Number we want to transform uses base X: we would like to transform this number into new one, that uses Y base. In this case we will use method of continuous dividing and multiplying (dividing numbers left of comma (,) with Y and multiplying numbers on the right side of comma, by Y - rational numbers). This method is best understood looking at these examples:
Example
Transform 5324 (decimal) into decimal number (?!) using method of continuous dividing – result 5324
5324 : 10 = 532 , remaining 4 100 last digit
532 : 10 = 53 , remaining 2 101
53 : 10 = 5 , remaining 3 102
5 : 10 = 0 , remaining 5 103 first digit
- end of procedure
Example
Transform 0,8125 (decimal) into decimal number (?!) using method of continuous multiplying – result 0,8125
0,8125 * 10 = 8, 125 10-1 first digit after zero
0,125 * 10 = 1, 25 10-2
0,25 * 10 = 2, 5 10-3
0,5 * 10 = 5, 0 10-4 last digit
0,0 end of procedure
Decimal 2 Binary
Example
Transform 29 (decimal) into binary number - reading upwards, result is 111012
29 : 2 = 14 , remaining 1 20 last (smallest) digit
14 : 2 = 7 , remaining 0 21
7 : 2 = 3 , remaining 1 22
3 : 2 = 1 , remaining 1 23
1 : 2 = 0 , remaining 1 24 first digit
- end of procedure
Example
Transform 0,8125 (decimal) into binary number – reading downwards, result is 0,11012
0,8125 * 2 = 1, 625 2-1 first digit after zero
0,625 * 2 = 1, 25 2-2
0,25 * 2 = 0, 5 2-3
0,5 * 2 = 1, 0 2-4 last digit
0,0 end of procedure
Example
Transform 0,3 (decimal) into binary number – result is 0,01001 1001 1001… 2 shortly rounded = 0,010012
0,3 * 2 = 0, 6 2-1 first digit after zero
0,6 * 2 = 1, 2 2-2
--------
0,2 * 2 = 0, 4 2-3
0,4 * 2 = 0, 8 2-4
0,8 * 2 = 1, 6 2-5
0,6 * 2 = 1, 2 2-6
--------
…
…
procedure never ends
Example – Quick Method
Transform 53(decimal) into binary number using quick method
53 25 + 24 + 22 + 20 =
- 32 -> 25 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 =
-------- 110101 2
21
- 16 -> 24
--------
5
- 4 -> 22
--------
1
- 1 -> 20
--------
0 -> end of procedure
Decimal 2 Hex
Example
Transform 2540,34 (decimal) into hex number-result ~ 9EC, 570A316
a) left from”,”
2540 : 16 = 158 , remaining 12 => C 160 last digit
158 : 16 = 9 , remaining 14 => E 161
9 : 16 = 0 , remaining 9 => 9 162 first digit
end of procedure => 9EC 16
b) right from “,”
0,34 * 16 = 5,44 5 => 5 16-1 first digit after zero
0,44 * 16 = 7,04 7 => 7 16-2
0,04 * 16 = 0,64 0 => 0 16-3
0,64 * 16 = 10,24 10 => A 16-4
0,24 * 16 = 3,84 3 => 3 16-5
…
…
procedure could be continued => 0, 570A3… 16
Final result ~ 9EC , 570A316
Binary 2 Decimal
Example
Transform 110101 (binary) into decimal number
110101 2 = 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20
= 1*25 + 1*24 + 1*22 + 1*20
= 1*32 + 1*16 + 1*4 + 1*1
= 53 10
Example
Transform -11,101 (binary) into decimal number
-11,101 2 = - (1*21 + 1*20 + 1*2-1 + 0*2-2 + 1*2-3)
= - (1*21 + 1*20 + 1*2-1 + 1*2-3)
= - (1*2 + 1*1 + 1*0,5 + 1*0,125)
= - 3 , 625
Hex 2 Decimal
Example
Transform 9EC,570A3 (hex) into decimal number
9EC, 570A3 16 = 9*162 + 14*161 + 12*160 + 5*16-1 + 7*16-2 + 0*16-3 + 10*16-4 + 3*16-5
= 2304 + 224 +12 + 0,3125 + 0,02734375 + 0,0001525878... + 0,00000286102...
= 2540 , 33999919891357421875
Quick Method of Transforming between Binary, Hex and Octal System
Binary Hex Binary Octal
0000 0 000 0
0001 1 001 1
0010 2 010 2
0011 3 011 3
0100 4 100 4
0101 5 101 5
0110 6 110 6
0111 7 111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
Example
Transform -6F,A 16 into binary number
-6F,A 16 = - 0110 1111 , 1010 2 = - 1101111 , 101 2
Example
Transform 11,000011001 2 into hex number
11,000011001 2 = 11 , 0000 1100 1 2
= 0011 , 0000 1100 1000 2
= 3 , 0 C 8 16
= 3 , 0C8 16
Example
Transform 37,24 8 into binary number
37,24 8 = 011 111 , 010 100 2 = 11111 , 0101 2
Example
Transform 1111011,10011101 2 into octal number
1111011,10011101 2 = 1 111 011 , 100 111 01 2
= 001 111 011 , 100 111 010 2
= 1 7 3 , 4 7 2 16
= 173 , 472 16
Technorati Tags: Numeric, Numerical, Transform, Transformation, Binary, Decimal, Octal, Hexadecimal, Hex, Base, Complement, Method
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