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Multiplying binary numbers

When we want to multiply a denary number by 10, we just add a 0. For example:

5 * 10 = 50
36 * 10 = 360
178 * 10 = 1780

If we want to multiply a binary number by 2 (or 10 in binary), we can also just add a 0. For example:

100 * 2 = 1000      ( 4 * 2 = 8)
1111 * 2 = 11110      (15 * 2 = 30) 
01001 * 2 = 010010      (9 * 2 = 18)

This is a simple quick rule to remember that we can apply for multiplying by 2. If we want to multiply by other numbers, we need to remember the following:

0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1

Example 1
Multiply  4 by 3

Always start on the right hand side of the number you are using to multiply by, which is 3 in this case. 

     We go to 3 and always start with the units (the 1s column). This digit is a 1.
     With the digits in 4 and starting on the right hand side as always, we multiply each digit in turn by 1.
     1 * 0 = 0
     1 * 0 = 0
     1 * 1 = 1
     This is in green in the table.
     
     Now we go to the next digit in 3. This is in the 2s column and also happens to be a 1.
     We now need to remember, however, to start writing our answers shifted along one place.
     1 * 0 = 0
     1 * 0 = 0
     1 * 1 = 1
     This is in red in the table. 

     Next, we add the rows in green and red.
     This gives us our answer in purple: 1100, or 12 in denary.

It's a good idea to check the answer: (1 * 8) + ( 1 * 4) + (0 * 2) + (0 * 1) = 12

Example 2
Multiply  13 by 5

Always start on the right hand side of the number you are using to multiply by, which is 5 in this case. 

     We go to 5 and always start with the units (the 1s column). This digit is a 1.
     With the digits in 13 and starting on the right hand side as always, we multiply each digit in turn by 1.
     1 * 1 = 1
     1 * 0 = 0
     1 * 1 = 1
     1 * 1 = 1
     This is in green in the table.
     
     Now we go to the next digit in 5. This is in the 2s column and is a 0.
     We now need to remember, however, to start writing our answers shifted along one place.
     0 * 1 = 0
     0 * 0 = 0
     0 * 1 = 0
     0 * 1 = 0
     This is in red in the table.
     
     Now we go again to the next digit in 5. This is in the 4s column and is a 1.
     We now need to remember again to start writing our answers shifted along one place.
     1 * 1 = 1
     1 * 0 = 0
     1 * 1 = 1
     1 * 1 = 1
     This is in light blue in the table.

     Next, we add the rows in green, red and light blue.
     This gives us our answer  in purple: 1000001, or 65 in denary. 

It's a good idea to check the answer: (1 * 64) + ( 0 * 32) + (0 * 16) + ( 0 * 8) + (0 * 4)  + ( 0 * 2) + (1 * 1) = 65

As long as you can remember:

0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1

and that you must shift along one place every time you multiply by the next digit, you wll be able to multiply in binary after a few goes.

Multiply the following questions in binary without a calculator. Lay out your answers exactly like above. 

Q1. 6 * 3
Q2. 7 * 2
Q3. 10 * 3
Q4. 12 * 2
Q5. 11 * 3
Q6. 6 * 4
Q7. 9 * 5
Q8. 14 * 5
Q9. 15 * 5
Q10. 16 * 8
Q11. 20 * 8
Q12. 21 * 10
Q13. 22 * 2
Q14. 63 * 2
Q15. 50 * 5

Extension work
Write binary multiplication problems and test your neighbour.

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