GCSE Link: 3.03 (Binary Arithmetic)

You should already know how to add binary numbers from GCSE. You should also already know how to multiply a binary number by a power of two using shifts, but now we're going to learn how to multiply a binary number by any number.

The way we would do this is we would split the second number into powers of two, then multiply the first number by each power of two (using shifts), then sum the results of each multiplication.

Let's take an example. We'll use 19 × 13.

• Step 1: Write out both numbers in binary.
  
  This is 100112 × 11012.
• Step 2: Split the second number into powers of two from its binary representation.
  
  There are 1s in the 20, 22, and 23 columns. This means that 13 = 20 + 22 + 23.
• Step 3: Calculate the shifts. (We'll use the operator << to denote left-shift.)
  
  10011<<0 == 00010011
10011<<2 == 01001100
10011<<3 == 10011000
• Step 4: Sum the results.
  
    0 0 0 1 0 0 1 1
  0 1 0 0 1 1 0 0
+ 1 0 0 1 1 0 0 0
-----------------
  1 1 1 1 0 1 1 1
       1 1
• Step 5: Convert back to decimal and check the answer.
  
  This is 247 in decimal, which is the correct answer.

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Perform the calculation 110102 × 10012 in binary.

Step 1: 110102 × 10012
Step 2: 23 + 20
Step 3: 11010000, 00011010
Step 4: 11101010
Step 5: 26 × 9 = 234 ✓

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