
Binary Numbers
How would we count if we could only have two digits: 0 and 1? The beginning is easy. We could suggest the we would denote 0 by 0 and 1 by 1. But what should we do next?
Let's think carefully about our usual base10 (decimal) notation.
When we write 2095, we mean 2 * 1000 + 0 * 100 + 9 * 10 + 5 * 1.
Or, if we like using exponents, 2095 = 2 * 10^{3} + 0 * 10^{2} + 9 * 10^{1} + 5 * 10^{0}.
Note that each digit in the sum is a number between zero and nine; there is no need for symbols greater than 9, because ten times 10^{k} is reckoned for by adding one to the 10^{k+1} place.
Binary representation works on the same principle, except that each place now represents a power of 2, i.e., 2^{k}.
So the binary number 1101 represents 1*2^{3}+1*2^{2}+0*2^{1}+1*2^{0}=8+4+0+1=13.
Note that we only need digits 0 and 1 for binary representation (just as we only needed digits 0 to 9 in decimal representation), because two times 2^{k} is reckoned for by adding one to the 2^{k+1} place.
Practice
Conversion from Binary to Decimals
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Last Modified: August 2008

