Data Representation Using Binary Digits

Exponential value of the place	10^4	10^3	10^2	10^1
10^0
Decimal value of the place	10,000	1,000	100	10	1

Table 1 Exponential and decimal values corresponding to a digit's place in a number
Note that the exponential values are raised to a power corresponding to the place of the digit. For example, the exponential value of the 4th place is 10⁴.
To determine the value of the number, multiply the digit contained in a column by the value that column represents. The following is a sample calculation for the previous example.
4 × 10⁴ + 3 × 10³ + 8 × 10² + 7 × 10¹ + 2 × 10⁰ =
4 × 10,000 + 3 × 1000 + 8 × 100 + 7 × 10 + 2 × 1 =
40,000 + 3000 + 800 + 70 + 2 =
43,872
While performing these calculations on a decimal number seems trivial, it demonstrates a pattern, or formula can be used to convert a number in any numbering system to decimal.
d_p(b)^p + d_p-1(b)^p-1 + . . . + d₀(b)⁰
Where p is the place, b is the base, d_p is the digit in the highest place in the number, and d_p-1 is the next highest place in the number, and so on.
Using the number example above, d_p = 4, d_p-1 = 3, b = 10, and p = 4.
4 × 10⁴ + 3 × 10³ + 8 × 10² + 7 × 10¹ + 2 × 10⁰ = 43,872
The formula above can be used to compute the decimal value of any number in a given base. Below is the calculation for converting 21₄ to its decimal value: