Number Base Equations and Applications for SS 1

Number Base System (2)

Performance Objectives

By the end of the lesson, students should be able to:

1. display basic computer programming skills (coding);
2. using binary numbers for simple coding;
3. use hexadecimal (base 16) for simple programme.

Introduction to Number Bases in Computing

In earlier studies, the concept of number base systems was introduced, including how to convert numbers between different bases like base 10 (decimal), base 2 (binary), base 8 (octal), and other bases. These number base systems are not just abstract mathematical concepts; they have practical applications, especially in the world of computers and digital technology.

Computers operate using electrical signals that are either “on” or “off”. This fundamental characteristic makes the binary number system, which uses only two digits (0 and 1), the natural language for computers. While computers understand binary, humans often find long strings of binary digits difficult to read and work with. This is where other number bases, like hexadecimal (base 16), become useful as a more compact way to represent binary information.

Binary Numbers (Base 2) in Computer Programming

The binary number system is a base-2 system, meaning it uses only two digits: 0 and 1. These digits are called “bits” (short for binary digits). In a computer, a 0 typically represents an “off” state or a low voltage, while a 1 represents an “on” state or a high voltage.

Every piece of information processed by a computer, from text and images to sounds and instructions, is ultimately converted into binary code. This process of representing information using 0s and 1s is a form of “coding” at the most basic level of computer operation.

Converting Decimal Numbers to Binary

To understand how computers use binary, it is important to be able to convert numbers from the familiar decimal (base 10) system to binary (base 2). This is typically done using repeated division by 2.

The procedure involves the following steps:

1. Divide the decimal number by 2.
2. Note the remainder (which will be either 0 or 1).
3. Divide the quotient from the previous step by 2.
4. Repeat the process until the quotient becomes 0.
5. Write the remainders in reverse order (from bottom to top) to get the binary equivalent.

Example 1: Convert the decimal number 13 to binary.

Solution:

$$
\begin{aligned}
13 \div 2 &= 6 \text{ remainder } 1 \\
6 \div 2 &= 3 \text{ remainder } 0 \\
3 \div 2 &= 1 \text{ remainder } 1 \\
1 \div 2 &= 0 \text{ remainder } 1
\end{aligned}
$$

Reading the remainders from bottom to top, $13_{10} = 1101_2$.

Example 2: Convert the decimal number 25 to binary.

Solution:

$$
\begin{aligned}
25 \div 2 &= 12 \text{ remainder } 1 \\
12 \div 2 &= 6 \text{ remainder } 0 \\
6 \div 2 &= 3 \text{ remainder } 0 \\
3 \div 2 &= 1 \text{ remainder } 1 \\
1 \div 2 &= 0 \text{ remainder } 1
\end{aligned}
$$

Reading the remainders from bottom to top, $25_{10} = 11001_2$.

Converting Binary Numbers to Decimal

Converting binary numbers back to decimal helps in understanding the value represented by a binary sequence.

The procedure involves the following steps:

1. Assign place values to each digit in the binary number, starting from the rightmost digit with $2^0$, then $2^1$, $2^2$, and so on.
2. Multiply each binary digit by its corresponding place value.
3. Sum the results of these multiplications.

Example 3: Convert the binary number $1011_2$ to decimal.

Solution:

$$
\begin{aligned}
1011_2 &= (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) \\
&= (1 \times 8) + (0 \times 4) + (1 \times 2) + (1 \times 1) \\
&= 8 + 0 + 2 + 1 \\
&= 11_{10}
\end{aligned}
$$

So, $1011_2 = 11_{10}$.

Example 4: Convert the binary number $11010_2$ to decimal.

Solution:

$$
\begin{aligned}
11010_2 &= (1 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) \\
&= (1 \times 16) + (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) \\
&= 16 + 8 + 0 + 2 + 0 \\
&= 26_{10}
\end{aligned}
$$

So, $11010_2 = 26_{10}$.

Simple Binary Coding

In computer programming, binary numbers are used to represent various pieces of information. For example, a simple “on/off” switch can be represented by 1 and 0. More complex information, like characters or instructions, are assigned unique binary codes.

Imagine a simple system where each binary digit represents a state:

1. $0_2$ could mean “Light Off”
2. $1_2$ could mean “Light On”

For a slightly more complex example, consider representing simple choices:

1. $00_2$ could mean “No”
2. $01_2$ could mean “Yes”
3. $10_2$ could mean “Maybe”
4. $11_2$ could mean “Unsure”

This is a basic form of coding, where specific binary patterns are assigned to represent different meanings or actions. In real computers, standard codes like ASCII (American Standard Code for Information Interchange) use specific binary patterns to represent letters, numbers, and symbols. For instance, the uppercase letter ‘A’ is represented by $01000001_2$ in ASCII.

Hexadecimal Numbers (Base 16) in Computer Programming

The hexadecimal number system is a base-16 system. It uses 16 distinct symbols to represent numbers. These symbols are the digits 0 through 9, and the letters A through F. Each letter represents a decimal value:

1. A = 10
2. B = 11
3. C = 12
4. D = 13
5. E = 14
6. F = 15

Hexadecimal is widely used in computer programming because it provides a more compact and human-readable way to represent binary data. Since $16 = 2^4$, one hexadecimal digit can represent exactly four binary digits (bits). This makes it much easier for programmers to read and write long binary sequences.

Relationship between Binary and Hexadecimal

The direct relationship between binary and hexadecimal is key to its use in computing. Every group of four binary digits can be directly converted into a single hexadecimal digit.

The table below shows the relationship between decimal, binary, and hexadecimal for values 0 to 15:

The relationship between decimal, 4-bit binary, and hexadecimal numbers is:

1. Decimal 0 = Binary 0000 = Hexadecimal 0
2. Decimal 1 = Binary 0001 = Hexadecimal 1
3. Decimal 2 = Binary 0010 = Hexadecimal 2
4. Decimal 3 = Binary 0011 = Hexadecimal 3
5. Decimal 4 = Binary 0100 = Hexadecimal 4
6. Decimal 5 = Binary 0101 = Hexadecimal 5
7. Decimal 6 = Binary 0110 = Hexadecimal 6
8. Decimal 7 = Binary 0111 = Hexadecimal 7
9. Decimal 8 = Binary 1000 = Hexadecimal 8
10. Decimal 9 = Binary 1001 = Hexadecimal 9
11. Decimal 10 = Binary 1010 = Hexadecimal A
12. Decimal 11 = Binary 1011 = Hexadecimal B
13. Decimal 12 = Binary 1100 = Hexadecimal C
14. Decimal 13 = Binary 1101 = Hexadecimal D
15. Decimal 14 = Binary 1110 = Hexadecimal E
16. Decimal 15 = Binary 1111 = Hexadecimal F

Converting Binary to Hexadecimal

To convert a binary number to hexadecimal, group the binary digits into sets of four, starting from the right. If the leftmost group has fewer than four digits, add leading zeros to complete the group. Then, convert each 4-bit group into its corresponding hexadecimal digit.

Example 5: Convert the binary number $11010110_2$ to hexadecimal.

Solution:

1. Group the binary digits into sets of four from the right: $1101 \ 0110_2$.
2. Convert each group to its hexadecimal equivalent:
   1. $1101_2 = D_{16}$
   2. $0110_2 = 6_{16}$
3. Combine the hexadecimal digits: $D6_{16}$.

So, $11010110_2 = D6_{16}$.

Example 6: Convert the binary number $1011101_2$ to hexadecimal.

Solution:

1. Group the binary digits into sets of four from the right, adding leading zeros if necessary: $0101 \ 1101_2$.
2. Convert each group to its hexadecimal equivalent:
   1. $0101_2 = 5_{16}$
   2. $1101_2 = D_{16}$
3. Combine the hexadecimal digits: $5D_{16}$.

So, $1011101_2 = 5D_{16}$.

Converting Hexadecimal to Binary

To convert a hexadecimal number to binary, convert each hexadecimal digit into its 4-bit binary equivalent.

Example 7: Convert the hexadecimal number $A7_{16}$ to binary.

Solution:

1. Convert each hexadecimal digit to its 4-bit binary equivalent:
   1. $A_{16} = 1010_2$
   2. $7_{16} = 0111_2$
2. Combine the binary groups: $10100111_2$.

So, $A7_{16} = 10100111_2$.

Example 8: Convert the hexadecimal number $3F_{16}$ to binary.

Solution:

1. Convert each hexadecimal digit to its 4-bit binary equivalent:
   1. $3_{16} = 0011_2$
   2. $F_{16} = 1111_2$
2. Combine the binary groups: $00111111_2$. (The leading zeros are often kept for clarity in programming contexts, but can be omitted if not needed for a specific bit length).

So, $3F_{16} = 00111111_2$.

Converting Hexadecimal to Decimal

To convert a hexadecimal number to decimal, multiply each hexadecimal digit by its corresponding power of 16 and sum the results.

Example 9: Convert the hexadecimal number $2B_{16}$ to decimal.

Solution:

$$
\begin{aligned}
2B_{16} &= (2 \times 16^1) + (B \times 16^0) \\
&= (2 \times 16) + (11 \times 1) \quad (\text{since } B = 11) \\
&= 32 + 11 \\
&= 43_{10}
\end{aligned}
$$

So, $2B_{16} = 43_{10}$.

Simple Hexadecimal Applications in Programming

Hexadecimal numbers are used in various aspects of computer programming and digital systems because they offer a concise way to represent binary data. This makes it easier for humans to read, write, and debug code.

Common applications include:

1. Colour Codes: In web design and graphics, colours are often represented using hexadecimal values. For example, a colour code like `#FF0000` represents pure red. Here, `FF` represents the maximum intensity for the red component, while `00` represents no intensity for green and blue. Each pair of hexadecimal digits corresponds to 8 bits (a byte) of colour information.
2. Memory Addresses: Computer memory locations are often referred to using hexadecimal addresses. For instance, a memory address like `0xAF4B` is much shorter and easier to manage than its binary equivalent. The prefix `0x` is a common notation in programming to indicate a hexadecimal number.
3. Error Codes and Debugging: When a computer program encounters an error, the error code is often displayed in hexadecimal. This compact representation helps programmers quickly identify the type of error by looking up the hexadecimal code in documentation.
4. MAC Addresses: Media Access Control (MAC) addresses, which uniquely identify network interfaces (like your computer’s Wi-Fi card), are typically displayed in hexadecimal format, such as `00:1A:2B:3C:4D:5E`.

These applications demonstrate how hexadecimal simplifies the representation of underlying binary data, making it a valuable tool for anyone working with computers at a programming level.