Binary and decimal for networking

Why computers use binary

Inside a computer, a router or a switch, information is stored and sent as signals that are either on or off: a voltage is high or low, a light pulse in a fibre is bright or dark. A single on or off value is called a bit (short for binary digit). We write on as 1 and off as 0. Binary is the number system that uses only these two digits.

We humans count in decimal, base 10, because we have ten fingers. Each place in a decimal number is worth ten times the place to its right: ones, tens, hundreds. Binary works the same way, but each place is worth two times the place to its right.

An octet: 8 bits, 0 to 255

Network devices group bits in sets of 8. A group of 8 bits is called a byte, and in networking we usually call it an octet (octo means eight). An IPv4 address is 32 bits long, which is 4 octets. Each octet is written as a normal decimal number, and the four numbers are separated by dots: 192.168.10.1. This is called dotted decimal notation.

With all eight bits off you get 00000000, which is 0. With all eight on you get 11111111, which is 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. So one octet can hold 256 different values, from 0 to 255. That is why you will never see 256 or 300 in a valid IPv4 address.

Decimal to binary: the subtraction method

Go through the place values from left to right. If the number is at least as big as the place value, write 1 and subtract that value. If not, write 0 and move on. Keep going until all eight places are filled.

1. 192: 192 is at least 128, write 1, left 64. 64 is at least 64, write 1, left 0. The other six places are 0. Answer 11000000.
2. 168: take 128 (left 40), skip 64, take 32 (left 8), skip 16, take 8 (left 0), then 0, 0, 0. Answer 10101000.
3. 10: skip 128, 64, 32 and 16, take 8 (left 2), skip 4, take 2 (left 0), skip 1. Answer 00001010.
4. 255: every place value is taken. Answer 11111111.
5. 172: take 128 (left 44), skip 64, take 32 (left 12), skip 16, take 8 (left 4), take 4 (left 0), then 0, 0. Answer 10101100.

Binary to decimal: add the place values

Write the place values above the bits and add up the values that sit above a 1. For 11000000, only 128 and 64 are on, so it is 192. For 10101100, the ones sit under 128, 32, 8 and 4, so it is 172. For 00000001, only the 1 place is on, so it is 1.

So the address 192.168.10.1 in binary is 11000000.10101000.00001010.00000001. Network engineers write the dots between octets so the bits are easier to read.

print(bin(192))              # 0b11000000  (the 0b prefix means binary)
print(format(10, "08b"))     # 00001010    (8 digits, padded with zeros)
print(int("10101100", 2))    # 172         (read a string as base 2)

Powers of two you should know

Subnetting is mostly about powers of two: 2 multiplied by itself a number of times. Learn this short list by heart, it will save you time in every later topic.

Common mistakes

• Starting from the wrong end: the leftmost bit is worth 128, the rightmost is worth 1.
• Thinking an octet goes up to 256. It holds 256 values, but they run from 0 to 255.
• Reading the binary string as a decimal number. 11000000 is "one one zero zero zero zero zero zero", not eleven million.