You’re just scaling the sequence. 25 * [... 16 8 4 2 1] = [... 400 200 100 50 25]. This allows us to translate the problem easily into simple base 10 arithmetic, as per your example: 25 * 17 = 25 * (16 + 1) = 25 + 400 = 425.
Really, the beauty of binary is that we can only use each number of the sequence at most once. However, the same works with bases > 2, as long as you realize you can use each number of the sequence more than once.
For example, base 3 has the place sequences [... 27 9 3 1], and the base ten number 17 is given by 122 in base 3 (we’ve used the bottom 2 places twice each: 1*9 + 2*3 + 2*1). So, just as before: 25 * [... 9 3 1] = [... 225 75 25]. Also, just as before: 25 * 17 = 25 * (1*9 + 2*3 + 2*1) = 1*225 + 2*75 + 2*25 = 425
Clearly, the benefit of binary is ease of doubling and the fact that each place of the sequence is used at most once.
Snappy, he’s not writing a proof. There is no need to explain how he got the multiples of 25. It’s just assumed that most people can easily produce those numbers without much thought.
Interesting.
Thank you, that was awesome.
You’re just scaling the sequence. 25 * [... 16 8 4 2 1] = [... 400 200 100 50 25]. This allows us to translate the problem easily into simple base 10 arithmetic, as per your example: 25 * 17 = 25 * (16 + 1) = 25 + 400 = 425.
Really, the beauty of binary is that we can only use each number of the sequence at most once. However, the same works with bases > 2, as long as you realize you can use each number of the sequence more than once.
For example, base 3 has the place sequences [... 27 9 3 1], and the base ten number 17 is given by 122 in base 3 (we’ve used the bottom 2 places twice each: 1*9 + 2*3 + 2*1). So, just as before: 25 * [... 9 3 1] = [... 225 75 25]. Also, just as before: 25 * 17 = 25 * (1*9 + 2*3 + 2*1) = 1*225 + 2*75 + 2*25 = 425
Clearly, the benefit of binary is ease of doubling and the fact that each place of the sequence is used at most once.
Question:
How would one get the multiples of 25 in each case without the initial multiplication tables?
He conveniently writes down 25, 50, 75, 100, 200, 400 … without basis for arriving at their values.
Snappy, he’s not writing a proof. There is no need to explain how he got the multiples of 25. It’s just assumed that most people can easily produce those numbers without much thought.
Well, if it was on a computer, you can double a number by shifting the bits left by one and putting a 0 in the ones column.
Thank you Mr. Schneider! What a beautiful, elegant method of math. Where can I learn more?
but…how do u divide 10 by 7…or…8 by 6 … etc. ?????
u take whats left and divide by what you divided, i get 10by7=1,3/7 and 8by6=1,1/3
I am amazed I never learned this in school (and I am a master’s degree electrical engineer!!!
Well, that was worth digging up.