Hey!

I recently started a programming course and we have begun looking into binary numbers, how they are converted from decimals and back etc.

I have a hard time understanding why 1024 equals 1000 0000 0000 and not 0100 0000 0000 as 1024 fits in the 11th column.
I know there are two methods "Descending Powers of Two and Subtraction" & " Short Division by Two with Remainder." But I still can'tt understand why its 12th and not 11th column.

This is what I can find busing google "Increase the exponent by 1 until you reach the number close to the decimal you are starting with". But 1024 fits so why use 2048 and divide by 2?

Appreciate your help and sorry for the bad explanation

Solved it. I added a 0 too much

log_2 (1024) = 10
1 00 0000 0000

using log base 2 can make it faster

Binary is just the same as decimal, but instead of every position being 10 to a power (ones place = 10^0, tens place = 10^1, hundreds place = 10^2, etc.), it is 2 to a power (one = 2^0, two = 2^1, four = 2^2, etc.)

And with binary, all you can get is the power of two because the only number you can stick in that place is a 1. You can't multiply by a number like in decimal for that position. For instance the number 35 in decimal is 5*10^0 + 3*10^1.

If you want to find 1024, that is the same as 2^ 10. Since we start the farthest position as 2^0, there will be a total of 11 digits.

100 0000 0000 = 1*2^ 10 + 0*2^9 + 0*2^8 + 0*2^7+ 0*2^6 + 0*2^5 + 0*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 0*2^0 = 1024

Hopefully you passed your programming course!

This post was edited by Pelnied on Feb 19 2022 05:00pm

This.

Just remember that changing from a base A to a base B can be done using iterations of INT(log_A(x)) and exp_B(x) because a sum of n numbers in base A is EQUAL to the sum of the same numbers in base B

For example : transform 1030_10 in a number base 2 (X_2).
Step 1 ) INT(log2(1030)) = INT(10,something) = 10, exp_10(10) = 10 000 000 000, we have to transform the remaining part that is 1030 - exp_10(2) = 1030 - 1024 = 6
Step 2 ) INT(log2(6)) = INT(2,something) = 2, exp_10(2) = 100, remaining part is 6 - exp_2(2) = 6-4 = 2
Step 3 ) INT(log2(2)) = INT(1) = 1 (we will stop at step 3 because no more decimals to be transformed), exp_10(1) = 10

Summing steps will give that 1030_10 = 1024_10 + 4_10 + 2_10 = 10000000000_2 + 100_2 + 10_2 = 1 000 000 110

This is not the formal demonstration of course, the mathemathics behind it starts from the explanation given by @pelnied , I provided a different PoV because this is a programmer forum and the one I explained is pretty much one of the common algorithms to solve these problems.

This post was edited by NomadJe on Mar 26 2022 04:32am

floor(log_2(n)) + 1 gives you the "position" of the highest 1 in binary of form for number n. If n is a power of 2, only one 1 is needed, thus 1024 becomes 1 + 10 zeros

1024 does NOT equal 1000 0000 0000 in binary. IT IS 0100 0000 0000