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# The function that converts a decimal integer to binary.

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Let's see how humans convert numbers to binary.

1) We divide the number x by 2 to the power of n and take its remainder from dividing by 2.

$$\Bigl\lfloor\dfrac{x}{2^{n}}\Bigr\rfloor\operatorname{mod}2=\lfloor2^{-n}x\rfloor \operatorname{mod} 2$$

We may also need the length of the binary x, so it can be obtained by the formula:

$$\lceil log_2x\rceil$$

2) When we have all remainders we concatenate them with each other. In math, we can replace concatenation with $$a*10^n+b*10^{n+1}+c*10^{n+2}...$$

3) Write all remainders in reversed order

So here's the formula that converts decimal to binary.

$$bin(x)=\sum^{\lceil\operatorname{log}_2x\rceil}_{n=0}\lfloor2^{-n}x\rfloor\operatorname{mod}2*10^n$$

Look at this in Desmos: https://www.desmos.com/calculator/2jabzvxn4t​

off-topic
Oct 3, 2019
edited by JoshuaGreen  Oct 22, 2019
edited by JoshuaGreen  Oct 23, 2019

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JG: That is very nice. Since you have a good understanding of deriving mathematical formulas, it would serve you very well to learn a programming language such as Python, C, C++.......etc. You could, for instance, program the above formula using very short one-line code to convert any number into binary, almost instantly, such as this line: n=123456789;m=2; a=n%m;printa,", ",;b=int(n/m);n=b;if(n>m, goto2, discard=0;printb,;

It will instantly convert 123456789 into binary like this:

1 , 0 , 1 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , 0 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 1 [read from right to left].

Oct 3, 2019
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Thanks for your reply. I want to say that I already know Python and Javascript, and I know how to convert decimal numbers to binary using programming, but I'm interested in doing this mathematically.

JoshuaGreen  Oct 3, 2019
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That's really cool!!

If you want to try it, here's another (similar, maybe easier) challenge:

Make a function that will return the sum of the digits. So for examples:

f(123) = 6

f(100) = 1

f(4532)  =  14

f(3) = 3

f(0) = 0

And let's also assume here that  n ≤ 32

Oct 3, 2019
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$$f(x)=\sum^{\lfloor\log_{10}x\rfloor}_{n=0}\Bigl\lfloor\dfrac{x}{10^n}\Bigr\rfloor\operatorname{mod}10$$

look at this in Desmos: https://www.desmos.com/calculator/6tktu9vizl

JoshuaGreen  Oct 3, 2019
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Ah, I did not think about using floor(log(x)) as the upper limit!! That's even better!

hectictar  Oct 4, 2019
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Hi hectictar: Just for my own curiosity, I took up you suggestion and hastily wrote this very short code to sum up the digits of any number. It can easily add up the first 1,000 digits of Pi, for example. In this short code, it sums up:123456789 and counts them as well:

n=123456789;s=0;p=0;cycle: s=s+(n%10);p=p+1;n=int(n/10);if(n!=0, goto cycle,0);print"Total Sum =",s;print"Total Num =",p

Total Sum = 45
Total Num = 9

Oct 3, 2019