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How decimal numbers differ from binary numbers?

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2011-09-12 20:19:40
2011-09-12 20:19:40

Decimal numbers are in base 10 and binary numbers are in base 2. Let's take the number 135. This is equal to 100 + 30 + 5. Or 1 * 10^2 + 3 * 10^1 + 5 * 10^0. (the caret being the sign for exponent and any number greater than 0 to the exponent 0 equal 1 hence 10^0 = 1). In binary format, the base is two so the first basic number is 2^0 = 1, then 2^1 = 2, then 2^2 = 4 and so on, 8, 16, 32, 64...

to build a number like 135 in binary you may ask how many 2^8 = 256 can I fit in it? The answer is 0, then how many 2^7=128, the answer is 1. Having already allocated 128 in the number, there is only a value of 7 left to insert. That is to say that there won't be any 2^6, 2^5, 2^4, or 2^3 in it. Since 2^2 = 4 and that fits into the remaining 7 we put 1 of those. We do the same for 2^1=2 and 2^0=1 and we have the final value (0*2^8) + (1*2^7) + (0*2^6) + (0*2^5) + (0*2^4) + (0*2^3) + (1*2^2) + (1*2^1) + (1*2^0) or using only the coefficients: 010000111. Since the first zero is irrelevant, 135 in binary is 10000111!

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