Computer Number Systems

All digital computers, from supercomputers to your smartphone, are electronic devices and ultimately can do one thing: detect whether an electrical signal is on or off. That basic information, called a bit (binary digit), has two values: a 1 (or true) when the signal is on, and a 0 (or false) when the signal is off.

Larger values can be stored by a group of bits. For example, there are 4 different values stored by 2 bits (00, 01, 10, and 11), 8 values for 3 bits (000, 001, 010, 011, 100, 101, 110, and 111), 16 values for 4 bits (0000, 0001, …, 1111), and so on. However, large numbers, using 0s and 1s only, are quite unwieldy for humans. For example, a computer would need 19 bits to store the numbers up to 500,000! We use different number systems to work with large binary strings that represent numbers within computers.

Different Number Systems

A number system is the way we name and represent numbers. The number system we use in our daily life is known as the decimal number system or base 10 because it is based on 10 different digits: 0, 1, 2, …, 8, and 9. The base of a number is written as a subscript. For example, the decimal number “twelve thousand three hundred and forty-five” is written as $12345_{10}$. Without a base or explicit context, it’s assumed that a number is in base 10.

In computer science, apart from the decimal system, three additional number systems are commonly used: binary (base-2), octal (base-8), and hexadecimal or just hex (base-16). Binary numbers are important because that is how numbers are stored in the computer. Octal and hexadecimal are used to represent binary numbers in a user-friendly way. Each octal symbol represents 3 binary bits and each hex digit represents 4 binary bits. For example, the decimal number 53201, stored in the computer as the binary number 10000001111000101100, is represented by the octal number 2017054 and the hex number 81E2C. The table below compares the number systems:

Number System	Base	Digits Used	Examples
Binary	2	0, 1	$10110_2$, $10110011_2$
Octal	8	0, 1, 2, 3, 4, 5, 6, 7	$75021_8$, $231_8$, $60012_8$
Decimal	10	0, 1, 2, 3, 4, 5, 6, 7, 8, 9	$97425_{10}$ or simply 97425
Hexadecimal	16	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A–F	$54A2DD0F_{16}$

In general, $N$ bits have $2^N$ different values. The computers aboard the Apollo spacecraft had 8-bit words of memory. Each word of memory could store 256 different values, and the contents were displayed using 2 hex characters.

The following table shows the first 20 numbers in decimal, binary, octal, and hexadecimal:

Decimal	Binary	Octal	Hexadecimal
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10
17	10001	21	11
18	10010	22	12
19	10011	23	13
20	10100	24	14

Converting to Decimal

As we learned in the first or second grade, the decimal value of a decimal number is simply the sum of each digit multiplied by its place value:

$12345 = 1 \times 10^4 + 2 \times 10^3 + 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 = 10000 + 2000 + 300 + 40 + 5 = 12345$

$3079 = 3 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 9 \times 10^0 = 3000 + 70 + 9 = 3079$

Analogously, the decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:

$1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13_{10}$

$175_8 = 1 \times 8^2 + 7 \times 8^1 + 5 \times 8^0 = 1 \times 64 + 7 \times 8 + 5 \times 1 = 64 + 56 + 5 = 125_{10}$

$A5E_{16} = 10 \times 16^2 + 5 \times 16^1 + 14 \times 16^0 = 10 \times 256 + 5 \times 16 + 14 \times 1 = 2560 + 80 + 14 = 2654_{10}$

Converting from Decimal

The algorithm to convert a number from an arbitrary base requires finding how many times successive powers of the base go into the number, starting with the largest power of the base less than the starting number. For example, converting 3306 from base 10 to octal proceeds as follows:

1. The largest power of 8 that is less than or equal to 3306 is 512 ($8^3$). Divide 3306 by 512:

$3306 = 6 \times 8^3 + 234$

2. The next power of 8 is 64 ($8^2$). Divide 234 by 64:

$234 = 3 \times 8^2 + 42$

3. The next smaller power of 8 is 8 ($8^1$). Divide 42 by 8:

$42 = 5 \times 8^1 + 2$

4. Finally, the next smaller power of 8 is 1 ($8^0$). Divide 2 by 1:

$2 = 2 \times 8^0$

The answer is $6352_8$.

Converting between Binary, Octal, and Hexadecimal

Converting from octal to binary is simple: replace each octal digit by its corresponding 3 binary bits. For example:

$375_8 = 011\ \ \ 111\ \ \ 101_2 = 11111101_2$

Converting from hex to binary is also simple: replace each hex digit by its corresponding 4 binary bits. For example:

$FD_{16} = 1111\ \ \ 1101_2 = 11111101_2$

Converting from binary to either octal or hex is pretty simple as well: group the bits by 3s or 4s (starting at the right), and convert each group:

$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101\ 100 = 2017054_8$

$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$

Converting between base 8 and 16 is easy by first expressing the number in base 2 and then converting from base 2. This is shown in Sample Problem #1.

Using Hexadecimal Numbers to Represent Colors

Computers use hexadecimal numbers to represent various colors in computer graphics because all computer screens use combinations of red, green, and blue light (RGB) to represent thousands of different colors. Two digits are used for each, so the hexadecimal number #FF0000 represents red, #00FF00 represents green, and #0000FF represents blue. The color black is #000000 and white is #FFFFFF.

$FF_{16} = 15 \times 16 + 15 \times 1 = 240 + 15 = 255_{10}$, so there are 0 to 255, or 256 different shades of each color, giving $256^3 = 16{,}777{,}216$ different colors.

The following web site has nearly every color name, along with its hex code and decimal values: https://www.rapidtables.com/web/color/RGB_Color.html

For example, “salmon” is #FA8072, which represents the decimal numbers 250 (hex FA), 128 (hex 80), and 114 (hex 72).

Resources

Ryan’s Tutorials covers this topic beautifully. Here are links to the relevant sections:

• 1. Number Systems — An introduction to number systems, with emphasis on decimal, binary, octal, and hexadecimal. ACSL identifies the base of a number using a subscript, e.g. $123_8$ is octal and $123_{16}$ is hexadecimal.

• 2. Binary Conversions — How to convert between binary, decimal, hexadecimal and octal numbers, with practice activities.

• 3. Binary Arithmetic — Addition, subtraction, multiplication, and division with binary numbers. ACSL also covers basic arithmetic in other bases. Division in other bases is not covered.

• 4. Negative Numbers — ACSL problems will not cover how negative numbers are represented in binary.

• 5. Binary Fractions and Floating Point — The first part on fractions in other bases is relevant to ACSL. Focus on Converting to a Binary Fraction, but keep in mind ACSL may also cover octal and hexadecimal fractions. Floating point numbers are not covered.

The CoolConversion.com online calculator is another tool for practicing conversion from/to decimal, hexadecimal, octal and binary; it shows the steps involved in the conversion.

The AskNumbers.com site has a nice description of the conversions between binary, octal, decimal and hexadecimal numbers.

Format of ACSL Problems

The problems in this category will focus on converting between binary, octal, decimal, and hexadecimal, basic arithmetic of numbers in those bases, and, occasionally, fractions in those bases.

To be successful in this category, you must know the following facts cold:

1. The binary value of each octal digit 0, 1, …, 7
2. The binary value of each hex digit 0, 1, …, 9, A, B, C, D, E, F
3. The decimal value of each hex digit 0, 1, …, F
4. Powers of 2, up to 4096
5. Powers of 8, up to 4096
6. Powers of 16, up to 65,536

Sample Problems

Sample Problem 1

Solve for $x$ where $x_{16} = 3676_8$.

Solution: One method is to convert $3676_8$ into base 10, and then convert that number into base 16. An easier solution, less prone to arithmetic mistakes, is to convert through the binary representation:

$$\begin{align} 3676_8 &= 011\ 110\ 111\ 110_2 & \text{convert each octal digit into base 2} \\ &= 0111\ 1011\ 1110_2 & \text{group by 4 bits, right-to-left} \\ &= 7BE_{16} & \text{convert each group of 4 bits into a hex digit} \end{align}$$

Sample Problem 2

Solve for $x$ in the following hexadecimal equation: $x = F5AD_{16} - 69EB_{16}$

Solution: Working directly in base 16 from right to left:

• The rightmost digit: D − B = 2.
• Next column: A − E requires borrowing; 1A − E = C.
• Next column: 4 − 9 requires borrowing; 14 − 9 = B (borrowing from the next column, so 5 becomes 4).
• Leftmost column: E − 6 = 8.

Combining the results: $x = 8BC2_{16}$.

Sample Problem 3

How many numbers from 100 to 200 in base 10 consist of distinct ascending digits and also have distinct ascending hex digits when converted to base 16?

Solution: There are 13 such numbers. They are (in base 10): 123 (7B), 124, 125, 126, 127 (7F), 137 (89), 138, 139 (8B), 156 (9C), 157, 158, 159 (9F), 189 (BD).

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