Explained: "How to multiply 2 Binary Numbers?a=aN2aN2−1…a1,b=bN2bN2−1…b1"

Henry Winters

Answered question

2022-04-06

How to multiply 2 Binary Numbers?

a = a_{\frac{N}{2}} a_{\frac{N}{2} - 1} \dots a_{1}, b = b_{\frac{N}{2}} b_{\frac{N}{2} - 1} \dots b_{1}

granfury90210birm

Beginner2022-04-07Added 10 answers

Step 1
In base 10, the digit

a_{1}

can take values from 0 to 9. Any numbers higher than that can only displayed if you also use

a_{2}

. In your setting,

a_{1}

can take values from 0 to 65535 (which is 1111111111111111 in binary). Any numbers higher than that can only displayed if you also use

a_{2}

. So, in summary, we should consider base 65536 (instead of base 10).
The formula becomes

a \cdot b = a_{\frac{N}{2}} a_{\frac{N}{2} - 1} \dots a \cdot b_{\frac{N}{2}} b_{\frac{N}{2} - 1} \dots b_{1}

= \sum_{i = 1}^{\frac{N}{2}} \sum_{j = 1}^{\frac{N}{2}} (a_{i} b_{j}) 65536^{i + j - 2},

where instead of

65536^{i + j - 2}

we can also write

{(2^{16})}^{i + j - 2}

, which is the same as

2^{16 (i + j - 2)}

. The number

16 (i + j - 2)

in the exponent of the last expressions is the number of bits that you need to shift by. to do (because if you multiply by

2^{k}

this corresponds to a shift by k bits).

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