Binary Systems
Denary Place Values
4326 reads as: Four thousands Three hundreds Two tens Six ones
In base ten, starting from right side you have columns or “places” for 10^{0} = 1, 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000, and so forth.
Binary Place Values
The Binary number can also be organized as its place values.
In base two, starting from right side you have columns or “places” for 2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, and so forth.
Let’s consider a binary number 1110
Each number has place value |
|||
Eights |
Fours |
Twos |
Ones |
8s |
4s |
2s |
1s |
2^{3} |
2^{2} |
2^{1} |
2^{0} |
1 |
1 |
1 |
0 |
You can calculate equivalent Denary number as:
(1 x 8) + (1 x 4) + (1 x 2) + (0 x 1) = 8 + 4 + 2 = 14
Converting Denary to Binary
Converting denary numbers to binaries is simple: just divide by 2.
Divide the starting number by 2.
- If it divides evenly, the binary digit is 0.
- If there is a remainder the binary digit is 1.
For example, let’s convert the denary number 18 to binary.
Put the remainders in reverse order to get the binary number: 10010.
Converting Binary to Denary
Converting from binary to denary number is simple, as long as you remember that each digit in the binary number represents a power of two.
For example, take a binary number 1110.
Each number has place value |
|||
Eights |
Fours |
Twos |
Ones |
8s |
4s |
2s |
1s |
2^{3} |
2^{2} |
2^{1} |
2^{0} |
1 |
1 |
1 |
0 |
By looking at the place values, we can calculate the equivalent denary number.
Step 1: Write each binary digit along with multiplication sign and number 2 (base 2).
(1 x 2) (1 x 2) (1 x 2) (0 x 2)
Step 2: Add ‘Plus’ sign between brackets
(1 x 2) + (1 x 2) + (1 x 2) + (0 x 2)
Step 3: Using place values convert each digit to the power of two that it represents
( 1 x 2^{3} ) + ( 1 x 2^{2} ) + ( 1 x 2^{1} ) + ( 0 x 2^{0 })
Step 4: Simplify
( 1 x 8 ) + ( 1 x 4 ) + ( 1 x 2 ) + ( 0 x 1 )
8 + 4 + 2 + 0 = 14_{10}
Example 2: Convert 10101010_{2 }_{ }to equivalent denary number
Following the above steps we can calculate the denary number as:
( 1 x 2^{7 }) + ( 0 x 2^{6 }) + ( 1 x 2^{5 }) + ( 0 x 2^{4 }) + ( 1 x 2^{3 }) + ( 0 x 2^{2 }) + ( 1 x 2^{1 }) + ( 0 x 2^{0 })
( 1 x 128 ) + ( 0 x 64 ) + ( 1 x 32 ) + ( 0 x 16 ) + ( 1 x 8 ) + ( 0 x 4 ) + ( 1 x 2 ) + ( 0 x 1 )
128 + 32 + 8 + 2 = 170_{10}
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