Binary Systems

Description: 

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 100 = 1, 101 = 10, 102 = 100, 103 = 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 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and so forth.

Let’s consider a binary number 1110

Each number has place value

Eights

Fours

Twos

Ones

8s

4s

2s

1s

23

22

21

20

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

23

22

21

20

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 23 ) + ( 1 x 22 ) + ( 1 x 21 ) + ( 0 x 2)

Step 4: Simplify

( 1 x 8 ) + ( 1 x 4 ) + ( 1 x 2 ) + ( 0 x 1 )

8 + 4 + 2 + 0 = 1410

Example 2: Convert 10101010 to equivalent denary number

Following the above steps we can calculate the denary number as:

1 x 2) + ( 0 x 2) + ( 1 x 2) + ( 0 x 2) + ( 1 x 2) + ( 0 x 2) + ( 1 x 2) + ( 0 x 2)

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 = 17010

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