And now back to our regularly scheduled program, already in progress…

Getting back into the groove regarding binary, the key is associating it with our more-familiar decimal system of math. I will frankly admit that it took me from second through eighth grade to master my times tables completely, but the 10’s were easy…you just added a zero to get the result! The whole powers of ten (1, 10, 100, 1000) was pretty simple, 10 to the first power was 10, 10 to the second power was 100, and so on. If you want to represent the decimal number 1,200,20,000, you can map it out to the powers of ten like this:

10^{7} 10^{6} 10^{5} 10^{4} 10^{3} 10^{2} 10^{1} 10^{0}

10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

0 1 2 1 0 0 0 0

Simple, right? You multiply the 1 by 1,000,000 (106) place, multiply the 2 by 100,000 (giving you 200,000) and the 1 by 10,000 (104 place), and it all adds up. Binary works the same way, but instead of powers of 10, you use powers of two, 20, 21, etc. The math is simple, the first place is always 1, then 2 is next, followed by 2×2 (4), 2x2x2 (8), 2x2x2x2 (16), 2x2x2x2x2 (32), 2x2x2x2x2 (64), and finally 2x2x2x2x2x2x2 (128). Another way to think of is is that the numbers double in every “place.” Now, let’s map the number 192 in binary, using the same format as before:

**2 ^{7} 2^{6} 2^{5} 2^{4} 2^{3} 2^{2} 2^{1} 2^{0}**

**128 64 32 16 8 4 2 1**

**1 1 0 0 0 0 0 0**

By looking at the numbers and doing a little math you can see that 128+64=192, so by placing a one in the 128 column (107) and the 64 column (106) and zeroes everywhere else, you get the binary number 1100000. You can get decimal from binary by just doing the reverse, and multiplying 1 by 128 and 1 by 64 and adding them together.

This may seem a little strange when you first try to wrap you head around it, but it truly does explain how everything else in networking operates. My own favorite personal example is my “aha moment” regarding Routing Information Protocol. Ever wonder why RIP has a 15 hop limit? Simple–it has a FOUR BIT METRIC (binary 0000 through 1111); when you map the 1111 into the power of two diagram you realize that 1111 (8+4+2+1) is 15. The protocol simply cannot count any higher! Hopefully that gives you a small sample of why binary really is the secret decoder ring to certification and computing understanding.

– Joe