Binary Numbers

Humans use a base 10 number system for counting numbers, most likely because have 10 fingers. For example, using your fingers to do 8 + 4, since we don’t have 11 fingers we close our hands and have to hold a 10 in our mind, reset our fingers and add two: + 2 = 12.



This number system is natural for us but imagine a dolphin counting with it’s 2 fins. 1 + 2 = 3, but they don’t have 3 fins so they have to hold in their mind increments of 2. Computers are just like this in that they use a base 2 system(aka binary) of couting in contrast to our base 10(decimal).

Recall(in the language chapter) that 4 bits gives us a maximum of 16 unique combinations because (2⁴ = 16).

Every 1 toggles corresponding powers of two:

If we need more to represent numbers over 16, we need to expand to 5 bits, 2⁵ = 32

#How to convert a binary number to a decimal number?

1. Look at where there is a 1

   8	4	2	1
   0	1	1	1

   Notice there are 1’s underneeth the 4,2, and 1 headers

2. Sum up the corresponding power of 2 number

   you can think of it as an on/off switch

   8	4	2	1
   0	1	1	1

   0 + 4 + 2 + 1 = 7; thus our answer is 0111 represents 7

Check your understanding!

#How to convert a decimal number to a binary number?

Subtraction method

1. Find the biggest power of 2 that is less than the decimal number and place a 1 in that position

   
   Let’s try converting the number 13

   8	4	2	1
   1	0	0	0

   8 is the biggest power of 2 that is less than 13, hence we put a 1 there.

2. Subtract it from the original number and repeat step 1 until you subtract to 0

   13-8 = 5

   8	4	2	1
   1	1	0	0

   Now we need to look for the next biggest power of 2 that is less than 5, which is 4, hence we put a 1 there.
   

   5 - 4 = 1

   Lastly 1 fits with 1, so 1 - 1 = 0, and we are done.

   8	4	2	1
   1	1	0	1

   8+4+0+1 = 13; thus 1101 represents 7

Check your understanding!

Division method

1. Modulo the number by two

   Let’s try for the number 233
   233/2 = 116 remainder 1

2. Recursively modulo the quotient while keeping track of the remainder

   11101001

   Division	Quotient	Remainder
   233/2	116	1
   116/2	58	0
   58/2	29	0
   29/2	14	1
   14/2	7	0
   7/2	3	1
   3/2	1	1
   1/2	0	1

   
   Note: The remainder will always be 0 or 1

3. Gather all the remainders from bottom to top

   From bottom to top in the remainders column, 11101001, which is the binary representation of 233

Check your understanding!

Use the division method