Introduction to Binary Numbers
Binary numerals
Binary numerals are a way of expressing any whole numbers (1,2,3,...)
using only the two binary digits, also called bits: 0 and 1.
For those of you who have studied
arithemtic in different bases, binary numerals are base 2
numbers. Binary numerals are a positional number system, where
the powers of 10: 1's 10's 100's 1000's etc
for decimal numerals are replaced by the powers of 2:
1's 2's 4's 8's 16's 32's 64's for binary numbers.
Thus, the binary numeral 1011 represents 11 because it has a
1 in the 8's, 2's, and 1's places and so represents the number
with one 8, one 2, and one 1, that is, 8+2+1 = 11.
place 128 64 32 16 8 4 2 1
1 0 1 1
================
8 + 2+ 1 = 11
Exercise 1: You should try to convert the following binary numerals
to decimal: 111, 10001, 101010, 1111111. The answers are
here
For example, the binary numerals
for 0 and the first several integers are as follows:
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
16 10000
17 10001
18 10010
19 10011
20 10100
21 10101
22 10110
23 10111
24 11000
25 11001
26 11010
27 11011
28 11100
29 11101
30 11110
31 11111
32 100000

Observe that with five bits we can represent every number between 0 and 31, inclusive.
This gives us 32 different numbers. In general with k bits you can represent 2^k different
numbers, where 2^k is 2 to the power k, which is the product of 2 by itself k times. Thus,
with 10 bits we can represent all numbers from 0 to 2^101 = 10241 = 1023.
This
fact can be used to count to 1000 on your fingers by resting
your palm on a table (or your knee) and then representing a zero
by lifting a finger off the table and a one by placing the finger
on the table. Letting your little finger represent the 1's, your
ring finger represent 2's, your middle finger 4's, pointer 8's and
thumb 16's, you can count to 31 with one hand. It turns out that
the counting motions are fairly easy to learn and if you then add
your other hand in, you can count to (2^10)1 = 1,023 with two hands.
See if you can find the pattern that allows you to count to 1023 with
your ten fingers.
Powers of two
The first 20 powers of two are listed in the following table:
1 2^0
2 2^1
4 2^2
8 2^3
16 2^4
32 2^5
64 2^6
128 2^7
256 2^8
512 2^9
1024 2^10
2048 2^11
4096 2^12
8192 2^13
16384 2^14
32768 2^15
65536 2^16
131072 2^17
262144 2^18
524288 2^19
1048576 2^20
Observe in particular that 2^10 is about 1000 and 2^ 20 is about
one million. One convenient fact about binary numbers is that the
largest binary number you can represent with k bits is 2^k1,
as shown in the following table:
k largest k bit its value
binary number in decimal
1 1 1= 1
2 11 3= 2+1
3 111 7= 4+2+1
4 1111 15= 8+4+2+1
5 11111 31= 16+8+4+2+1
6 111111 63= 32+16+8+4+2+1
7 1111111 127= 64+32+16+8+4+2+1
8 11111111 255= 128+64+32+16+8+4+2+1
9 111111111 511= 256+128+64+32+16+8+4+2+1
10 1111111111 1023= 512+256+128+64+32+16+8+4+2+1
Thus, using 10 bits we can express about 1000 numbers.
Similarly,
20 bits can represent about 1 million numbers,
30 bits represent about 1 billion, etc.
IP addresses and sizes of subnets
Recall that an IP address is a number that is 4 bytes long.
Thus, there are exactly 256*256*256*256 = 4,294,967,296
IP addresses. This is about 4 billion. The number of
IP addresses in subnets of levels 1,2,3 can be calculated
in the same way:
level IP address number of addresses
0 xxx.yyy.zzz.www 256*256*256*256 = 4,294,967,296
1 AAA.xxx.yyy.zzz 256*256*256 = 16,777,216
2 AAA.BBB.xxx.yyy 256*256 = 65,536
3 AAA.BBB.CCC.xxx 256 = 256
Converting Decimal to Binary
You can convert from decimal to binary by expressing your
decimal number as a sum of powers of two (with no repeats).
The easiest way to do this for small numbers is by
repeatedly subtracting the biggest possible
powers of two from your number. Thus, to convert 43 to binary,
we would
 first find the highest power of two that isn't
bigger than 43 and subtract it from 43. This would be 32
and 43  32 = 11. In other words,
43 = 32 + 11.
 find the highest power of two that isn't bigger than 11
and subtract from 11.
This would be 8, and 118 = 3, so 43 = 32 + 8 + 3.
 Finally, 3 = 2 + 1, so 43 = 32 + 8 + 2 + 1.
If we then put a 1 in the 32,8,2, and 1 places and 0 in
the other places we get the binary representation for 43:
101011.
place 128 64 32 16 8 4 2 1
1 0 1 0 1 1
==========================================
32 + 8+ 2+ 1 = 43
Binary numbers that consist of 8 bits (i.e. 8 place binary numbers)
are called bytes and they play a prominent role in today's computer
technology. Observe that one byte can represent any number from 0
to 128+64+32+16+8+4+2+1 = 255.
place 128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
==========================================
128+ 64+ 32+ 16+ 8+ 4+ 2+ 1
You should try converting the following decimals to binary:
Exercise 2: You should try to convert
the following decimals
to binary: 10, 99, 100, 111.
here
Answers to exercises

Exercise 1. Converting Binary Numerals to Decimal:
 111 base 2 = 4+2+1 = 7
place 128 64 32 16 8 4 2 1
1 1 1
==========================================
4+ 2+ 1 = 7
 10001 base 2 = 16 + 1 = 17
place 128 64 32 16 8 4 2 1
1 0 0 0 1
==========================================
16+ 1 = 17
 101010
place 128 64 32 16 8 4 2 1
1 0 1 0 1 0
==========================================
32+ 8+ 2 = 42
 1111111
place 128 64 32 16 8 4 2 1
1 1 1 1 1 1 1
==========================================
64+ 32+ 16+ 8+ 4+ 2+ 1 = 127

Exercise 2. Converting Decimal to Binary:
 10 = 8 + 2 = 1010 base 2
place 128 64 32 16 8 4 2 1
1 0 1 0
==========================================
8+ 2 = 10
 99 = 1100011 (base 2)
99 = 64 + 35
= 64 + 32 + 3 =
= 64 + 32 + 2 + 1
place 128 64 32 16 8 4 2 1
1 1 0 0 0 1 1
==========================================
64+ 32+ 2+ 1 = 99
 100 = 1100100 (base 2)
100 = 64 + 36
= 64 + 32 + 4
place 128 64 32 16 8 4 2 1
1 1 0 0 1 0 0
==========================================
64+ 32+ 4 = 100
 111
111 = 64 + 47
= 64 + 32 + 15
= 64 + 32 + 8 + 7
= 64 + 32 + 8 + 4 + 3
= 64 + 32 + 8 + 4 + 2 + 1
place 128 64 32 16 8 4 2 1
1 1 0 1 1 1 1
==========================================
64+ 32+ 8+ 4+ 2+ 1 = 111