Computer Number Systems

All digital computers – from supercomputers to your smartphone – 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 (of false) when the signal is off. Larger values can be stored by a group of bits. For example, 3 bits together can take on 8 different values.

Computer scientists use the binary number system (that is, base 2) to represent the values of bits. Proficiency in the binary number system is essential to understanding how numbers and information is represented in a computer. Since binary numbers representing moderate values quickly become rather lengthy, bases eight (octal) and sixteen (hexadecimal) are frequently used as shorthand. In octal, groups of 3 bits form a single octal digit; in hexadecimal, group of 4 bits form a single hex digit.

In this category, we will focus on conversion between binary, octal, decimal, and hexadecimal numbers. There may be some arithmetic in these bases, and occasionally, a number with a fractional part. We will not cover how negative numbers or floating point numbers are represented in binary.

Resources

Ryan's Tutorials covers this topic in beautifully. Rather than trying to duplicate that work, we'll point you to the different sections:

1. Number Systems - An introduction to what numbers systems are all about, with emphasis on decimal, binary, octal, and hexadecimal. ACSL will typically identify the base of a number using a subscript. For example, $123_8$ is an octal number, whereas $123_{16}$ is a hexadecimal number.

2. Binary Conversions - This section shows how to convert between binary, decimal, hexadecimal and octal numbers. In the Activities section, you can practice converting numbers.

3. Binary Arithmetic - Describes how to perform various arithmetic operations (addition, subtraction, multiplication, and division) with binary numbers. ACSL problems will also cover basic arithmetic in other bases, such as adding and subtracting together 2 hexadecimal numbers. ACSL problems will not cover division in other bases.

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

5. Binary Fractions and Floating Point - The first part of this section is relevant to ACSL: fractions in other bases. ACSL will not cover floating point numbers in other basis. So, focus on the section Converting to a Binary Fraction, but keep in mind that ACSL problems may also cover octal and hexadicmal fractions.

The CoolConversion.com online calculator is another online app for practicing conversion from/to decimal, hexadecimal, octal and binary; this tool shows the steps that one goes through in the conversion.

Basics

Evaluation of binary, octal, and hex numbers

The binary number system consists of two symbols, 0 and 1. The value of a binary number is found by looking at the bit in each position and multiplying it by the value of that position. For example, the binary number 101011 has a decimal value of $$ {\bf 1} \cdot 2^5 + {\bf 0} \cdot 2^4 + {\bf 1} \cdot 2^3 + {\bf 0} \cdot 2^2 +{\bf 1} \cdot 2^1 + {\bf 1} \cdot 2^0 = 32+ 0+8 + 0 +2 + 1 = 43$$

The base 8 number system, octal, uses the symbols 0, 1, 2, 3, 4, 5, 6 and 7. To convert an octal number into decimal, simply multiply each digit by the value of that position. For example, the octal number 4170 has decimal value of $$ {\bf 4} \cdot 8^3+ {\bf 1} \cdot 8^2 + {\bf 7} \cdot 8^1 + {\bf 0} \cdot 8^0 = 4\cdot 512 + 1\cdot 64 + 7\cdot 8 + 0\cdot 1 = 2048+64+56+0=2168$$

The base 16 number system, hexadecimal, uses 16 symbols: the symbol 0 through 9, A, B, C, D, E, and F. The decimal value of a hex number follows the same pattern as we've just seen for determining the decimal value of a binary or octal number.

• Step 1: Write down the hexadecimal number: $ 3AF _{16}$
• Step 2: Write each hexit as an increasing power of 16: $ 3 \cdot 16^2 + A \cdot 16^1 + F \cdot 16^0 $
• Step 3: Convert each hexit to decimal: $ 3\cdot 256 + 10\cdot 16 + 15\cdot 1 $
• Step 4: Perform the math: $ 768 + 160 + 15 = 943$

Converting from decimal to binary, octal, and hex

The are fundamentally two ways to convert a base 10 into another base: finding powers and repeated division. Here is the conversion of 1375 into octal using these two methods.

Method 1: Finding powers. Find how many times each decreasing power of that base can be divided evenly into the number and repeating the process with the remainder. The powers of 8 are: 1, 8, 64, 512, 4096, .... SInce 4096 is larger than 1375, we know that the form of the octal number will be $a \cdot 512 + b \cdot 64 + c \cdot 8 + d$.

Step 1: How many times does the largest power of 8 go into the number: 1375 / 512 = 2, remainder of 351.

Step 2: How many times does the next smaller power of 8 go into the remainder: 351 / 64 = 5, remainder of 31.

Step 2 (again): How many times does the next smaller power of 8 go into the remainder: 31 / 8 = 3, remainder of 7.

Step 2 (again): How many times does the next smaller power of 8 go into the remainder: 7 / 1 = 7, remainder of 0.

At this point, the number can be read off: 2 * 512 + 5 * 64 + 3*8 + 7 = 2537 (base 8)

Method 2: Repeated Division.

Step 1: Divide (1375)10 successively by 8 until the quotient is 0:

1375/8 = 171, remainder is 7

171/8 = 21, remainder is 3

21/8 = 2, remainder is 5

2/8 = 0, remainder is 2

Step 2: Read from the bottom to top as 2537. This is the octal equivalent of decimal number 1375.

Adding in bases other than 10 means that you must carry the value of that base and subtracting in bases other than 10 means that you must borrow the value of that base if necessary. For example:

since D=13 and 13+3=16 so leave the 0 and carry the 16 as a 1. Then E=14 and A=10 so 1+14+10 = 25 so leave the 9 and carry the 16 as a 1. E=14 so 1+14+9=24 so leave the 8 and carry the 16 as 1. Finally, F=15 so 1+15=16 which is 10.

Subtracting in base 8 is as follows:

Borrow 1=8 from the 7 since 2+8-6=4. Therefore, 6–5=1. Then, borrow 1=8 from the 4 since 5+8–7=6. Then the last digit on the left is a 3.

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 fact cold:

1. The decimal values of hex digits A, B, C, D, E, F
2. The binary value of hex digits A, B, C, D, E, F
3. Powers of 2, up to 4096
4. Powers of 8, up to 4096
5. Powers of 16, up to 65,536

Other Resources

There are lots and lots and lots of YouTube videos about computer number systems. Here are a handful that cover the topics, without too many ads:

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