#### Binary Digits

Recap:
• Decimal digits have 10 possible values ranging from 0 to 9
• Bits have only two possible values: 0 and 1.
• A binary number is composed of only 0s and 1s
The Value of a Binary Number:
To figure out what the value of the binary number we follow the method below:
Let;s say we wanted to find out wat the binary number 1001 was in Decimal Equivalent.
In other words  1001 - base two =  ???? (base 10)

The number 1001 has 4 digits located in 4 "positions"

 1 0 0 1 <===== Binary Digits 3 2 1 0 <===== Position 1x23 0x22 0x21 1x20 <==== Equivalent Power of 2

Step by step solution

Step 1: Write down the binary number: 1001 and note its digits in the 4 bits from right to left

Step 2: Multiply each digit of the binary number by the corresponding power of two:

1x23 + 0x22 + 0x2+ 1x20

Step 3: Solve the powers:

1x8 + 0x4 + 0x2 + 1x1 = 8 + 0 + 0 + 1

Step 4: Add up the numbers written above:

8 + 0 + 0 + 1 = 9.

Solution: The number 9, therefore, is the decimal equivalent of the binary number 1001.

You can see that in binary numbers, each bit holds the value of increasing powers of 2.

4 bits will accommodate the decimal numbers from 0 (Zero) to 15
The value of the number 15 in binary equivalent is 1111

Let's take a look:
 1 1 1 1 <===== Binary Digits 3 2 1 0 <===== Position

1x23 + 1x22 + 1x2+ 1x20 = 8 + 4 + 2 + 1 = 15
This is the largest number that 4 bits can accommodate.
To accommodate the number 16 we will need to add another bit:
 1 0 0 0 0 <===== Binary Digits 4 3 2 1 0 <===== Position

Therefore
1x24 0x23 + 0x22 + 0x2+ 0x20 = 16 + 0 + 0 + 0 = 16

Converting Decimal numbers to Binary Numbers

To convert decimal numbers to binary you kep dividing the number by 2, collect the remainders, until the final quotient is Zero.
Example to convert the number 85 to binary we will proceed as follows:
(85)10 = ??? ()2

Step 1: Keep dividing (85)10 successively by 2, collect all the remainders until the quotient is 0:

85/2 = 42, remainder is 1... note here that 85/2 = 42 + 0.5; the remainder of the division is actually 0.5 but in these cases it is always rounded off to 1
42/2 = 21, remainder is 0
21/2 = 10, remainder is 1
10/2 = 5, remainder is 0
5/2 = 2, remainder is 1  ... note here again that 5/2 = 2 + 0.5; the remainder of the division is actually 0.5 but it was rounded off to 1
2/2 = 1, remainder is 0
1/2 = 0, remainder is 1 ... again the remainder was 0.5 but was rounded off to 1

Step 2: Read and collect all the binary digits from the bottom to the top  as 1010101.

Therefore the binary number 1010101 is the binary equivalent of the decimal number 85