1.3.2 Number Systems

• Decimal

• Binary

• Hexadecimal

• Learning Exercise

The world of computing uses several number systems to represent data. While the decimal system, also known as base10, will be familiar to people, as it is the numbering system used in everyday life, binary (base2) and hexadecimal (base16) are common number systems used in computing today.

Decimal

We will start our discussion on number systems by examining the decimal system as an example of a number system. The decimal number system contains ten values- 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each number in the decimal number system can be broken into digits by their "place" in the number. Using the number 43,872 as an example, 2 is in its 0th place, 7 is in its first place, 8 in its second, 3 in its third, and 4 in its fourth. Each place has a value that can be represented either exponentially or by its decimal values. The following table shows the exponential and decimal representation for each place in the number 43,872.

Place	4th	3rd	2nd	1st	0th
Digit	4	3	8	7	2
Exponential value of the place	104	103	102	101	100
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:

Place	1st	0th
Digit	2	1
Exponential value of the place	41	40
Decimal value of the place	4	1

Table 2 Exponential and decimal values corresponding to a digit's place in a number

2(4)¹ + 1(4)⁰ = 2× 4 + 1 × 1 = 8 + 1 = 9

So far, we have only discussed converting numbers to decimal. It is also important to be able to convert numbers from decimal to other numbering systems. Continuing with the base₄ system, let us convert 89 from decimal to base₄.

First, find the value p, where 4^p < = 89 < 4^p+1. In this case p = 3. p is the value of the highest place.

Now we can proceed by filling out the following chart:

Place	3	2	1	0
Exponential value of the place	43	42	41	40
Decimal value of the place	64	16	4	1
Calculation	89 ÷ 64	25 ÷ 16	9 ÷ 4	1 ÷ 1
Result	1	1	2	1
Remainder	25	9	1	0

Table 3 Converting numbers from base 10 to base 4

Therefore, 89₁₀ = 1121₄.

We are now going to review binary and hexadecimal more closely.

Binary

Since all numbering-systems are treated the same, you already have all the tools necessary to convert to and from binary. Let's review converting from binary to decimal the number 10110110₂.

The highest place, p, is obtained by counting the number of places in the binary number, starting from zero. In this case, p = 7.

1 × 2⁷ + 0 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 0 × 2⁰ = 1 × 128 + 0 × 64 + 1 × 32 + 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 0 × 1 = 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0 = 182

For example, in 100110₂ the largest place is 2^p, where p = 5. Because binary is the easiest numbering system to convert into decimal, it will help us later when we are convert hexadecimal numbers.