Number Systems

//Denary → binary: take what fits

Write the place values down first, then go left to right. At each one ask: does it fit in what is left? If yes write 1 and subtract it; if no write 0. Here is 200.

128  64  32  16   8   4   2   1
  1   1   0   0   1   0   0   0

128 fits  -> 200 - 128 = 72
 64 fits  ->  72 -  64 =  8
 32 no,  16 no
  8 fits  ->   8 -   8 =  0
answer: 11001000

//Binary → denary: add up the 1s

Write the place values above the digits and add the ones with a 1 underneath.

  1   0   1   1   0   1
 32  16   8   4   2   1
 32 +      8 + 4 +     1  =  45

//Hexadecimal is binary in groups of four

Split the binary into nibbles from the RIGHT, then convert each nibble on its own. This is the whole trick — you never need to divide by 16.

1100 1000
  |    |
  C    8      ->  C8

A=10  B=11  C=12  D=13  E=14  F=15

//Binary addition, and spotting overflow

Add column by column, carrying just like denary. 1+1 = 10 (write 0, carry 1). If a carry falls off the left-hand end and you only have that many bits, that is overflow — the answer is wrong and you must say so.

  1 0 1 1   (11)
+ 0 1 1 0   (6)
---------
1 0 0 0 1   (17)

In 4 bits only 0001 could be stored -> OVERFLOW

//Logical shifts

Shifting left by 1 multiplies by 2; shifting right by 1 divides by 2 (whole number). Zeros come in at the empty end, and any bits pushed off the end are LOST — always mention the loss if the question asks about the effect.

0000 1010  = 10
shift LEFT 1
0001 0100  = 20

0000 1010  = 10
shift RIGHT 1
0000 0101  = 5

KEY TERMS

CHECK YOURSELF