Numeric Transformations


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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


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






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