Let’s look at a simple example. Let’s determine the binary equivalent of 23.

23/2 = 11 --- Remainder 1
11/2 = 5 --- Remainder 1
5/2 = 2 --- Remainder 1
2/2 = 1 --- Remainder 0
1/2 = 0 --- Remainder 1

The sequence of remainders going up gives the answer. So, the binary equivalent of the decimal 23 is 10111.

Just as the place values in the decimal system increase by powers of 10, the place values in the binary system increase by powers of 2 from right to left (…32, 16, 8, 4, 2,…). That is, from right to left a binary system has a ones place, a two’s place, a four’s place, an eight’s place, a sixteen’s place etc.

For example, in the binary number 11,

The 1 on the left is in the two’s place and its value is 1 × 2 = 2.
The 1 on the right is in the ones place and its value is 1 × 2^0 = 1 × 1 = 1.

Observe that the value of the 1 on the left is twice as large as that of the 1 on the right.

Let’s now talk about binary fractions.

A binary fraction is the same as a decimal fraction, but with the base of 2 instead of 10.

In a binary fraction, a binary point separates the integer part of a binary number from its fractional part. The binary point indicates the place where values change from positive to negative powers of 2. Each place to the left of the binary point represents a positive power of 2 (2, 4, 8, 16, 32, and so on) and each place to the right of the binary point represents a negative power of 2 [2^(-1), 2^(-2), 2^(-3), 2^(-4), and so on]. When a base is raised to a negative power, it just means the reciprocal of the base raised to the positive power. So:

2^ (-1) = 1/ (2^1) = 1/2
2^ (-2) = 1/ (2^2) = 1/4
2^ (-3) = 1/ (2^3) = 1/8 etc.

The place values to the right of the binary point are one-half, one-fourth, one-eighth and so on. Negative powers of 2 are small numbers, less than 1 and greater than 0.

For example,

The binary number 11.011 represents 1 x 2^ (1) + 1 x 2^ (0) + 0 x 2^ (-1) + 1 x 2^ (-2) + 1 x 2^ (-3)

= 1 x 2 + 1 x 1 + 0 x 1/ (2^1) + 1 x 1/ (2^2) + 1 x 1/ (2^3)
= 1 x 2 + 1 x 1 + 0 x 1/2 + 1 x 1/4 + 1 x 1/8
= 2 + 1 + 0 + 1/4 + 1/8
= 3 + 3/8
= 3 + 0.375
= 3.375

Converting a decimal fraction to a binary fraction requires more steps.

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