Back

An introduction to binary - Answers

Q1. Can you fill in the following partially completed table showing what each position in binary is worth? 

27  26  25  24 23  22 21 20 
 128 64 32  16  8 4  2 1 

Q2. Consider the denary number 6. Let's convert it into binary. The biggest binary number we can fit into 6 is 4. That leaves us with 2 left over. The biggest binary number we can fit into 2 is 2. that leaves us with 0. We can't fit any 1s in 0. The denary number 6 is therfore the same as the binary number 110.

Consider the denary number 23. The biggest binary number we can fit into 23 is 16. That leaves us with 7. We can't fit any 8s into 7. We can fit a 4, and that leaves us with 3. We can fit a 2 into 3 and that leaves us with a 1. We can fit a 1 into 1. The denary number 23 is therefore the same as the binary number 10111.

Consider the denary number 34. The biggest binary number we can fit into 34 is 32. That leaves 2. We can't fit any 16s into 2. We can't fit any 8s into 2. We can't fit any 4s into 2. We can fit a 2 into 2 and that leaves 0. We can't fit a 1 into 0. The denary number 34 is the same as the binary number 100010.

Whenever you are going to convert numbers, always write down first what each bit position is worth. It will really help you avoid making mistakes. For the moment, 8 positions is more than enough. 

128  64  32 16  8  4 2  1
               
    1. The binary equivalent of the denary number 9 is 1001
    2. The binary equivalent of the denary number 18 is 10010
    3. The binary equivalent of the denary number 73 is 1001001

Q3. We only have 2 digits in binary. They are 0 and 1. We can use our table showing what each position is worth to help us count in binary. Study this table carefully, to see how we worked out the binary version of each denary number as we counted up from zero. Complete the table up to 25.

Denary 128  64  32  16 8 4 2 1
0               0
1               1
2             1 0
3             1 1
4           1 0 0
5           1 0 1
6           1 1 0
7           1 1 1
8         1 0 0 0
9         1 0 0 1
10         1 0 1 0
11         1 0 1 1
12         1 1 0 0
13         1 1 0 1
14         1 1 1 0
15         1 1 1 1
16       1 0 0 0 0
17       1 0 0 0 1
18       1 0 0 1 0
19       1 0 0 1 1
20       1 0 1 0 0
21       1 0 1 0 1
22       1 0 1 1 0
23       1 0 1 1 1
24       1 1 0 0 0
25       1 1 0 0 1

Q4. Convert each of the following numbers into a binary byte:

    1. 7 is 0000 0111
    2. 46 is 0010 1110
    3. 127 is 0111 1111

Q5. The smallest number that you can represent in binary using a byte is 0000 0000. This is 0 in denary.
Q6. The biggest number that you can represent in binary using a byte is 1111 1111. This is 255 in denary.
Q7. The smallest number that you can represent in binary using a nibble is 0000. This is 0 in denary.
Q8. The biggest number that you can represent in binary using a nibble is 1111. This is 15 in denary.
Q9. The bit positions are:

128  64  32 16  8  4 2  1
 Bit 7  Bit 6  Bit 5  Bit 4 Bit 3   Bit 2  Bit 1  Bit 0

Q10. a) Bit comes from Binary digit.
b) Use subscripts if there is any doubt which numbering system is being used or discussed. e.g. 10110 is 101 in base 10 or denary whereas 1012 is 101 in base 2 or binary. More often than not, they can be left out because it is clear to everyone what numbering system is being used. for example, in everyday life, we don't use subscripts for prices because we all know prices use the denary system!

Back