Why does the decimal representation of (10^x * 10^y) always suffix the same representation in binary?

Question

10d * 10d = 100d ~ (100 in decimal) - last three digits of result are 100
100d =  1100100b ~ (100 in binary)  - last three digits of result are 100

If you multiply any (10^x * 10^y) together, you always get the decimal result at the end of the binary result ...

1000d * 100d =       100000d - last 6 digits are 100000
100000d = 11000011010100000b - last 6 digits are 100000

...

1000000d * 10000000d =                          10000000000000d
10000000000000d = 10010001100001001110011100101010000000000000b

... i think you get the idea by now!!

Answer 1

Just look at powers of ten:

$10^n = 5^n \times 2^n$

so $10^n$ is $1$ followed by $n$ $0$s in decimal

and $2^n$ is $1$ followed by $n$ $0$s in binary

while $5^n$ (being odd) ends with a $1$ in binary

so $10^n$ ends with exactly $n$ $0$s in binary

Answer 2

Just worked 1t 0ut ...

it's because 10d is 1010b

so every time you multiply 1010b by a multiple of itself, you can forget the first two digits, and just pretend you're multiplying 10d * 10d.