To work effectively with data, it must be represented in a format that a computer can process. Digital computers accomplish this by using digits, which are easily identifiable symbols. These digits, typically binary digits (0s and 1s), form the basis of data representation in computers. This binary system allows computers to store, process, and manipulate data efficiently, enabling various computational tasks and functions. Through binary representation, complex data structures, such as text, images, and sounds, can be encoded and interpreted by digital systems.

## Numbering Systems

Each number system has a base, also known as a radix, which determines the number of unique digits, including zero, that the system uses to represent numbers. For instance, the decimal number system has a base of 10, meaning it uses ten digits (0 through 9). The binary number system has a base of 2, using only two digits (0 and 1). The octal number system has a base of 8, utilizing eight digits (0 through 7). The hexadecimal number system has a base of 16, using sixteen symbols (0 through 9 and A through F).

The concept of varying bases becomes clear when we consider what happens as we count up to the maximum number allowed by a numbering system. In the decimal system (base 10), we count from 0 to 9, giving us ten unique digits. When we reach the highest digit (9), we add another digit to the left, starting again at 0 (e.g., 10, 11, and so on).

In binary (base 2), we count from 0 to 1. When we reach the highest digit (1), we add another digit to the left, starting again at 0 (e.g., 10 in binary equals 2 in decimal). Similarly, in the octal system (base 8), we count from 0 to 7, and upon reaching 7, we add another digit to the left (e.g., 10 in octal equals 8 in decimal).

In the hexadecimal system (base 16), we count from 0 to 9 and then continue with the letters A through F, where A represents 10, B represents 11, up to F, which represents 15. Upon reaching F, we add another digit to the left (e.g., 10 in hexadecimal equals 16 in decimal). This system is particularly useful in computing because it can represent large binary numbers more compactly.

### Binary to Decimal

Binary and decimal are two different number systems used in computing and mathematics. The binary system (base-2) uses only two digits, 0 and 1, whereas the decimal system (base-10) uses ten digits, from 0 to 9. Converting binary numbers to decimal numbers is a fundamental process in computer science.

### Understanding the Binary System

In the binary system, each digit represents a power of 2. The rightmost digit represents $2^0$, the next digit to the left represents $2^1$, the next represents $2^2$, and so on. For example, the binary number `1011`

can be broken down as follows:

### Steps to Convert Binary to Decimal

**Write down the binary number.****Label each bit with its corresponding power of 2**, starting from 0 on the right.**Multiply each bit by its corresponding power of 2**.**Sum all the products**to get the decimal number.

### Example Conversion

Let’s convert the binary number `1101`

to decimal.

$1101$**Write down the binary number:**

$1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$**Label each bit with its corresponding power of 2:**

$1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1$**Multiply each bit by its corresponding power of 2:**

$8 + 4 + 0 + 1 = 13$**Sum all the products:**

So, the binary number `1101`

converts to the decimal number `13`

.

### Another Example

Let’s convert the binary number `10101`

to decimal.

$10101$**Write down the binary number:**

$1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$**Label each bit with its corresponding power of 2:**

$1 \times 16 + 0 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1$**Multiply each bit by its corresponding power of 2:**

$16 + 0 + 4 + 0 + 1 = 21$**Sum all the products:**

So, the binary number `10101`

converts to the decimal number `21`

.

### General Formula

The general formula for converting a binary number to decimal is:

$\text{Decimal} = \sum_{i=0}^{n-1} \text{bit}_i \times 2^i$Where:

- $\text{bit}_i$ is the $i$-th bit in the binary number, starting from the right (0-indexed).
- $n$ is the number of bits in the binary number.

### Practice Problems

- Convert the binary number
`1110`

to decimal. - Convert the binary number
`10001`

to decimal. - Convert the binary number
`110011`

to decimal.

### Solutions

- $1110 \rightarrow 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 4 + 2 + 0 = 14$
- $10001 \rightarrow 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 0 + 0 + 1 = 17$
- $110011 \rightarrow 1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 0 + 0 + 2 + 1 = 51$

Understanding how to convert binary numbers to decimal numbers is crucial for various applications in computer science, including programming, data processing, and digital electronics. By mastering this skill, you gain a deeper insight into how computers process and represent data.

## Binary fraction to decimal

Converting binary fractions to decimal fractions involves a similar process to converting whole binary numbers to decimals but also includes handling the fractional part.

### Understanding Binary Fractions

A binary fraction is a number that has a binary point (similar to a decimal point in base-10 numbers). Each position to the right of the binary point represents a negative power of 2.

For example, in the binary number `101.101`

:

- The whole number part
`101`

is $1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 5$. - The fractional part
`.101`

is $1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3}$.

### Steps to Convert Binary Fractions to Decimal

**Separate the binary number into whole and fractional parts.****Convert the whole part**to decimal using the method for whole binary numbers.**Convert the fractional part**to decimal by multiplying each bit by its corresponding negative power of 2.**Add the results**of the whole and fractional parts.

### Example Conversion

Let’s convert the binary fraction `110.101`

to decimal.

**Separate the binary number into whole and fractional parts:**- Whole part:
`110`

- Fractional part:
`.101`

- Whole part:

$1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6$**Convert the whole part to decimal:**

$1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 0.5 + 0 + 0.125 = 0.625$**Convert the fractional part to decimal:**

$6 + 0.625 = 6.625$**Add the whole and fractional parts:**

So, the binary fraction `110.101`

converts to the decimal number `6.625`

.

### Another Example

Let’s convert the binary fraction `10.011`

to decimal.

**Separate the binary number into whole and fractional parts:**- Whole part:
`10`

- Fractional part:
`.011`

- Whole part:

$1 \times 2^1 + 0 \times 2^0 = 2 + 0 = 2$**Convert the whole part to decimal:**

$0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3} = 0 + 0.25 + 0.125 = 0.375$**Convert the fractional part to decimal:**

$2 + 0.375 = 2.375$**Add the whole and fractional parts:**

So, the binary fraction `10.011`

converts to the decimal number `2.375`

.

### General Formula

The general formula for converting a binary fraction to a decimal is:

$\text{Decimal} = \sum_{i=k}^{n-1} \text{bit}_i \times 2^i$Where:

- $\text{bit}_i$ is the $i$-th bit in the binary number.
- $k$ is the position of the first bit in the whole part (starting from 0 for the rightmost bit of the whole part).
- $n$ is the position of the last bit in the fractional part (negative values for positions right of the binary point).

### Practice Problems

- Convert the binary fraction
`101.11`

to decimal. - Convert the binary fraction
`11.001`

to decimal. - Convert the binary fraction
`0.1011`

to decimal.

### Solutions

- $101.11 \rightarrow 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 + 1 \times 2^{-1} + 1 \times 2^{-2} = 4 + 0 + 1 + 0.5 + 0.25 = 5.75$
- $11.001 \rightarrow 1 \times 2^1 + 1 \times 2^0 + 0 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 2 + 1 + 0 + 0 + 0.125 = 3.125$
- $0.1011 \rightarrow 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4} = 0.5 + 0 + 0.125 + 0.0625 = 0.6875$

Understanding binary fractions and their conversion to decimal is essential in various fields such as computer science, digital electronics, and numerical analysis, where precise calculations with fractional values are often required.

## Converting Decimal fractions to Binary

Converting decimal fractions to binary fractions involves separating the integer part and the fractional part of the decimal number, converting each part individually, and then combining the results.

### Steps to Convert Decimal Fraction to Binary

**Separate the decimal number into integer and fractional parts.****Convert the integer part**to binary using repeated division by 2.**Convert the fractional part**to binary using repeated multiplication by 2.**Combine the binary results**of the integer and fractional parts.

### Example Conversion

Let’s convert the decimal fraction `10.625`

to binary.

#### 1. Separate the Decimal Number

- Integer part:
`10`

- Fractional part:
`0.625`

#### 2. Convert the Integer Part to Binary

To convert the integer part `10`

to binary:

- Divide the number by 2 and record the remainder.
- Continue dividing the quotient by 2 until the quotient is 0.
- The binary representation is the remainders read from bottom to top.

Reading from bottom to top, the integer part `10`

in binary is `1010`

.

#### 3. Convert the Fractional Part to Binary

To convert the fractional part `0.625`

to binary:

- Multiply the fractional part by 2.
- Record the integer part (0 or 1) as the next binary digit.
- Repeat with the new fractional part until it becomes 0 or repeats.

Reading the integer parts from top to bottom, the fractional part `0.625`

in binary is `0.101`

.

#### 4. Combine the Results

Combining the integer part and fractional part, the decimal number `10.625`

in binary is `1010.101`

.

### Another Example

Let’s convert the decimal fraction `7.375`

to binary.

#### 1. Separate the Decimal Number

- Integer part:
`7`

- Fractional part:
`0.375`

#### 2. Convert the Integer Part to Binary

To convert the integer part `7`

to binary:

Reading from bottom to top, the integer part `7`

in binary is `111`

.

#### 3. Convert the Fractional Part to Binary

To convert the fractional part `0.375`

to binary:

Reading the integer parts from top to bottom, the fractional part `0.375`

in binary is `0.011`

.

#### 4. Combine the Results

Combining the integer part and fractional part, the decimal number `7.375`

in binary is `111.011`

.

### Practice Problems

- Convert the decimal fraction
`12.75`

to binary. - Convert the decimal fraction
`3.5`

to binary. - Convert the decimal fraction
`6.125`

to binary.

### Solutions

**12.75**- Integer part:
`12`

in binary is`1100`

. - Fractional part:
`0.75`

in binary is`0.11`

. - Combined:
`12.75`

in binary is`1100.11`

.

- Integer part:
**3.5**- Integer part:
`3`

in binary is`11`

. - Fractional part:
`0.5`

in binary is`0.1`

. - Combined:
`3.5`

in binary is`11.1`

.

- Integer part:
**6.125**- Integer part:
`6`

in binary is`110`

. - Fractional part:
`0.125`

in binary is`0.001`

. - Combined:
`6.125`

in binary is`110.001`

.

- Integer part:

Converting decimal fractions to binary is an essential skill in fields such as computer science and digital electronics. Understanding this process helps in comprehending how computers represent and manipulate fractional numbers.

## Decimal to Octal

The octal number system (base-8) uses digits from 0 to 7. Converting decimal numbers (base-10) to octal involves a process of repeated division for the integer part and repeated multiplication for the fractional part.

### Steps to Convert Decimal to Octal

**Separate the decimal number into integer and fractional parts.****Convert the integer part**to octal using repeated division by 8.**Convert the fractional part**to octal using repeated multiplication by 8.**Combine the octal results**of the integer and fractional parts.

### Example Conversion

Let’s convert the decimal number `83.625`

to octal.

#### 1. Separate the Decimal Number

- Integer part:
`83`

- Fractional part:
`0.625`

#### 2. Convert the Integer Part to Octal

To convert the integer part `83`

to octal:

- Divide the number by 8 and record the remainder.
- Continue dividing the quotient by 8 until the quotient is 0.
- The octal representation is the remainders read from bottom to top.

Reading from bottom to top, the integer part `83`

in octal is `123`

.

#### 3. Convert the Fractional Part to Octal

To convert the fractional part `0.625`

to octal:

- Multiply the fractional part by 8.
- Record the integer part (0 to 7) as the next octal digit.
- Repeat with the new fractional part until it becomes 0 or reaches the desired precision.

Reading the integer parts from top to bottom, the fractional part `0.625`

in octal is `0.5`

.

#### 4. Combine the Results

Combining the integer part and fractional part, the decimal number `83.625`

in octal is `123.5`

.

### Another Example

Let’s convert the decimal number `45.375`

to octal.

#### 1. Separate the Decimal Number

- Integer part:
`45`

- Fractional part:
`0.375`

#### 2. Convert the Integer Part to Octal

To convert the integer part `45`

to octal:

Reading from bottom to top, the integer part `45`

in octal is `55`

.

#### 3. Convert the Fractional Part to Octal

To convert the fractional part `0.375`

to octal:

Reading the integer parts from top to bottom, the fractional part `0.375`

in octal is `0.3`

.

#### 4. Combine the Results

Combining the integer part and fractional part, the decimal number `45.375`

in octal is `55.3`

.

### Practice Problems

- Convert the decimal number
`100.75`

to octal. - Convert the decimal number
`56.125`

to octal. - Convert the decimal number
`29.875`

to octal.

### Solutions

**100.75**- Integer part:
`100`

in octal is`144`

. - Fractional part:
`0.75`

in octal is`0.6`

. - Combined:
`100.75`

in octal is`144.6`

.

- Integer part:
**56.125**- Integer part:
`56`

in octal is`70`

. - Fractional part:
`0.125`

in octal is`0.1`

. - Combined:
`56.125`

in octal is`70.1`

.

- Integer part:
**29.875**- Integer part:
`29`

in octal is`35`

. - Fractional part:
`0.875`

in octal is`0.7`

.

- Integer part:

## Converting Decimal fractions to Octal

**Converting Decimal Fractions to Octal**

Converting decimal fractions to octal involves a methodical process that includes repeated multiplication and extraction of integer parts. The process is different from converting whole numbers, as it focuses on the fractional part of the decimal number. Below is a step-by-step guide along with an example to help illustrate the method.

### Steps for Converting Decimal Fractions to Octal

**Identify the Fractional Part**: Start with the fractional part of the decimal number you want to convert. For example, if you have the number 0.6875, you will be working with 0.6875.**Multiply by 8**: Multiply the fractional part by 8. Note the integer part of the result; this will be the first digit of your octal fraction.**Extract the Integer Part**: Take the integer part of the result from step 2 as the next digit in the octal fraction.**Remove the Integer Part**: Subtract the integer part from the result to get a new fractional part.**Repeat**: Repeat steps 2 to 4 with the new fractional part until you reach the desired level of precision or until the fractional part becomes zero.**Combine the Digits**: Combine all the integer parts you’ve extracted in order to form the octal fraction.

### Example

Let’s convert the decimal fraction 0.6875 to octal.

**Start with 0.6875**

$0.6875 \times 8 = 5.5$**First Multiplication**:- Integer part = 5
- New fractional part = 0.5 (since $5.5 – 5 = 0.5$)

$0.5 \times 8 = 4.0$**Second Multiplication**:- Integer part = 4
- New fractional part = 0.0 (since $4.0 – 4 = 0.0$)

Since the new fractional part is 0, we can stop here.

Combining the integer parts we obtained in the order, we get the octal fraction: **0.54**

### Final Octal Value

Thus, the decimal fraction 0.6875 is equivalent to the octal fraction 0.54.

### Another Example

Let’s consider converting 0.3 to octal for more practice.

**Start with 0.3**

$0.3 \times 8 = 2.4$**First Multiplication**:- Integer part = 2
- New fractional part = 0.4

$0.4 \times 8 = 3.2$**Second Multiplication**:- Integer part = 3
- New fractional part = 0.2

$0.2 \times 8 = 1.6$**Third Multiplication**:- Integer part = 1
- New fractional part = 0.6

$0.6 \times 8 = 4.8$**Fourth Multiplication**:- Integer part = 4
- New fractional part = 0.8

$0.8 \times 8 = 6.4$**Fifth Multiplication**:- Integer part = 6
- New fractional part = 0.4

Notice the fractional part repeats (0.4), indicating a repeating cycle in the octal representation.

Combining the integer parts, we get: **0.23146 (repeating)**

### Observations

- Decimal fractions may result in terminating or repeating octal fractions.
- The precision of the conversion can be adjusted by continuing the process for more steps.

By understanding and practicing this method, converting decimal fractions to octal can become a straightforward and systematic task.

## Decimal to Hexadecimal

**Converting Decimal Fractions to Hexadecimal**

Converting decimal fractions to hexadecimal involves a similar process to converting to octal, but with multiplication by 16 instead of 8. Here’s a detailed guide with an example to illustrate the steps.

### Steps for Converting Decimal Fractions to Hexadecimal

**Identify the Fractional Part**: Start with the fractional part of the decimal number you want to convert. For example, if you have the number 0.625, you will be working with 0.625.**Multiply by 16**: Multiply the fractional part by 16. Note the integer part of the result; this will be the first digit of your hexadecimal fraction.**Extract the Integer Part**: Take the integer part of the result from step 2 as the next digit in the hexadecimal fraction.**Remove the Integer Part**: Subtract the integer part from the result to get a new fractional part.**Repeat**: Repeat steps 2 to 4 with the new fractional part until you reach the desired level of precision or until the fractional part becomes zero.**Combine the Digits**: Combine all the integer parts you’ve extracted in order to form the hexadecimal fraction.

### Example

Let’s convert the decimal fraction 0.625 to hexadecimal.

**Start with 0.625**

$0.625 \times 16 = 10.0$**First Multiplication**:- Integer part = A (10 in decimal is A in hexadecimal)
- New fractional part = 0.0 (since $10.0 – 10 = 0.0$)

Since the new fractional part is 0, we can stop here.

Combining the integer parts we obtained in order, we get the hexadecimal fraction: **0.A**

### Final Hexadecimal Value

Thus, the decimal fraction 0.625 is equivalent to the hexadecimal fraction 0.A.

### Another Example

Let’s consider converting 0.1 to hexadecimal for more practice.

**Start with 0.1**

$0.1 \times 16 = 1.6$**First Multiplication**:- Integer part = 1
- New fractional part = 0.6

$0.6 \times 16 = 9.6$**Second Multiplication**:- Integer part = 9
- New fractional part = 0.6

$0.6 \times 16 = 9.6$**Third Multiplication**:- Integer part = 9
- New fractional part = 0.6

Notice the fractional part repeats (0.6), indicating a repeating cycle in the hexadecimal representation.

Combining the integer parts, we get: **0.1999… (repeating)**

### Observations

- Decimal fractions may result in terminating or repeating hexadecimal fractions.
- The precision of the conversion can be adjusted by continuing the process for more steps.

By following this method, converting decimal fractions to hexadecimal becomes a systematic process that can be applied to any fractional decimal number.

## Converting Decimal fractions to Hexadecimal

**Converting Decimal Fractions to Hexadecimal**

Converting decimal fractions to hexadecimal involves a methodical process of repeated multiplication and extraction of integer parts. This guide explains the steps and provides examples to help illustrate the method.

### Steps for Converting Decimal Fractions to Hexadecimal

**Identify the Fractional Part**: Start with the fractional part of the decimal number you want to convert. For example, if you have the number 0.6875, you will be working with 0.6875.**Multiply by 16**: Multiply the fractional part by 16. Note the integer part of the result; this will be the first digit of your hexadecimal fraction.**Extract the Integer Part**: Take the integer part of the result from step 2 as the next digit in the hexadecimal fraction.**Remove the Integer Part**: Subtract the integer part from the result to get a new fractional part.**Repeat**: Repeat steps 2 to 4 with the new fractional part until you reach the desired level of precision or until the fractional part becomes zero.**Combine the Digits**: Combine all the integer parts you’ve extracted in order to form the hexadecimal fraction.

### Example

Let’s convert the decimal fraction 0.6875 to hexadecimal.

**Start with 0.6875**

$0.6875 \times 16 = 11.0$**First Multiplication**:- Integer part = B (11 in decimal is B in hexadecimal)
- New fractional part = 0.0 (since $11.0 – 11 = 0.0$)

Since the new fractional part is 0, we can stop here.

Combining the integer parts we obtained in order, we get the hexadecimal fraction: **0.B**

### Final Hexadecimal Value

Thus, the decimal fraction 0.6875 is equivalent to the hexadecimal fraction 0.B.

### Another Example

Let’s consider converting 0.1 to hexadecimal for more practice.

**Start with 0.1**

$0.1 \times 16 = 1.6$**First Multiplication**:- Integer part = 1
- New fractional part = 0.6

$0.6 \times 16 = 9.6$**Second Multiplication**:- Integer part = 9
- New fractional part = 0.6

$0.6 \times 16 = 9.6$**Third Multiplication**:- Integer part = 9
- New fractional part = 0.6

Notice the fractional part repeats (0.6), indicating a repeating cycle in the hexadecimal representation.

Combining the integer parts, we get: **0.1999… (repeating)**

### Observations

- Decimal fractions may result in terminating or repeating hexadecimal fractions.
- The precision of the conversion can be adjusted by continuing the process for more steps.

By understanding and practicing this method, converting decimal fractions to hexadecimal can become a straightforward and systematic task.

## Hexadecimal to Decimal

**Converting Hexadecimal Fractions to Decimal**

Converting hexadecimal fractions to decimal involves understanding the place value of each digit in the hexadecimal system. Hexadecimal is a base-16 system, where each digit represents a power of 16. This guide explains the steps and provides an example to illustrate the method.

### Steps for Converting Hexadecimal Fractions to Decimal

**Identify the Hexadecimal Digits**: Each digit in a hexadecimal number represents a power of 16. The digits include 0-9 and A-F, where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.**Place Value**: The position of each digit to the right of the hexadecimal point (the equivalent of the decimal point) represents a negative power of 16. The first place to the right is $16^{-1}$, the second place is $16^{-2}$, and so on.**Multiply and Sum**: Multiply each hexadecimal digit by its place value (a power of 16) and sum the results to get the decimal equivalent.

### Example

Let’s convert the hexadecimal fraction 0.BA to decimal.

**Identify the Hexadecimal Digits**:- B = 11
- A = 10

**Place Value**:- B is in the $16^{-1}$ position
- A is in the $16^{-2}$ position

**Multiply and Sum**:- $B \times 16^{-1} = 11 \times 16^{-1} = 11 \times 0.0625 = 0.6875$
- $A \times 16^{-2} = 10 \times 16^{-2} = 10 \times 0.00390625 = 0.0390625$

Sum these values:

$0.6875 + 0.0390625 = 0.7265625$

Thus, the hexadecimal fraction 0.BA is equivalent to the decimal fraction 0.7265625.

### Another Example

Let’s convert the hexadecimal fraction 0.1C to decimal.

**Identify the Hexadecimal Digits**:- 1 = 1
- C = 12

**Place Value**:- 1 is in the $16^{-1}$ position
- C is in the $16^{-2}$ position

**Multiply and Sum**:- $1 \times 16^{-1} = 1 \times 0.0625 = 0.0625$
- $C \times 16^{-2} = 12 \times 16^{-2} = 12 \times 0.00390625 = 0.046875$

Sum these values:

$0.0625 + 0.046875 = 0.109375$

Thus, the hexadecimal fraction 0.1C is equivalent to the decimal fraction 0.109375.

### Observations

- Each digit in a hexadecimal fraction represents a specific power of 16.
- The conversion involves multiplying each digit by its respective place value and summing the results.
- This method ensures an accurate conversion from hexadecimal to decimal, accommodating both terminating and repeating hexadecimal fractions.

By understanding and practicing this method, converting hexadecimal fractions to decimal can become a straightforward and systematic task.

## Binary to Hexadecimal

**Converting Binary to Hexadecimal**

Converting binary numbers to hexadecimal is a straightforward process because both systems are base powers of 2. This makes it easy to group binary digits and convert them directly to hexadecimal digits. This guide explains the steps and provides an example to illustrate the method.

### Steps for Converting Binary to Hexadecimal

**Group Binary Digits**: Start from the binary point and group the binary digits into sets of four. If there are not enough digits to form a complete group, add leading zeros to the left of the binary number.**Convert Each Group**: Convert each group of four binary digits to their hexadecimal equivalent.**Combine the Hexadecimal Digits**: Write down the hexadecimal digits in the order they were grouped.

### Binary to Hexadecimal Conversion Table

Binary | Hexadecimal |
---|---|

0000 | 0 |

0001 | 1 |

0010 | 2 |

0011 | 3 |

0100 | 4 |

0101 | 5 |

0110 | 6 |

0111 | 7 |

1000 | 8 |

1001 | 9 |

1010 | A |

1011 | B |

1100 | C |

1101 | D |

1110 | E |

1111 | F |

### Example

Let’s convert the binary number 10111001 to hexadecimal.

**Group Binary Digits**:- Group the digits into sets of four: 1011 and 1001.

**Convert Each Group**:- 1011 in binary is B in hexadecimal.
- 1001 in binary is 9 in hexadecimal.

**Combine the Hexadecimal Digits**:- The binary number 10111001 converts to the hexadecimal number B9.

### Final Hexadecimal Value

Thus, the binary number 10111001 is equivalent to the hexadecimal number B9.

### Another Example

Let’s convert the binary number 110011 to hexadecimal.

**Group Binary Digits**:- First, add leading zeros to make a full group: 0011 0011.

**Convert Each Group**:- 0011 in binary is 3 in hexadecimal.
- 0011 in binary is 3 in hexadecimal.

**Combine the Hexadecimal Digits**:- The binary number 00110011 converts to the hexadecimal number 33.

### Final Hexadecimal Value

Thus, the binary number 110011 is equivalent to the hexadecimal number 33.

### Observations

- Binary digits are grouped in sets of four for easy conversion to hexadecimal.
- Each group of four binary digits directly maps to a single hexadecimal digit.
- This method ensures an accurate and efficient conversion from binary to hexadecimal.

By following this method, converting binary numbers to hexadecimal becomes a straightforward and systematic process.

## Hexadecimal to Binary

**Converting Hexadecimal to Binary**

Converting hexadecimal numbers to binary is a straightforward process due to the direct relationship between the two systems. Each hexadecimal digit translates to a specific 4-bit binary equivalent. This guide explains the steps and provides examples to illustrate the method.

### Steps for Converting Hexadecimal to Binary

**Identify Hexadecimal Digits**: Each digit in a hexadecimal number represents a specific value from 0 to F (0-15).**Convert Each Hexadecimal Digit**: Convert each hexadecimal digit to its 4-bit binary equivalent using a conversion table.**Combine the Binary Digits**: Write down the binary equivalents in the same order as the hexadecimal digits to form the complete binary number.

### Hexadecimal to Binary Conversion Table

Hexadecimal | Binary |
---|---|

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

A | 1010 |

B | 1011 |

C | 1100 |

D | 1101 |

E | 1110 |

F | 1111 |

### Example

Let’s convert the hexadecimal number 2F to binary.

**Identify Hexadecimal Digits**:- The digits are 2 and F.

**Convert Each Hexadecimal Digit**:- 2 in hexadecimal is 0010 in binary.
- F in hexadecimal is 1111 in binary.

**Combine the Binary Digits**:- The hexadecimal number 2F converts to the binary number 00101111.

### Final Binary Value

Thus, the hexadecimal number 2F is equivalent to the binary number 00101111.

### Another Example

Let’s convert the hexadecimal number A3 to binary.

**Identify Hexadecimal Digits**:- The digits are A and 3.

**Convert Each Hexadecimal Digit**:- A in hexadecimal is 1010 in binary.
- 3 in hexadecimal is 0011 in binary.

**Combine the Binary Digits**:- The hexadecimal number A3 converts to the binary number 10100011.

### Final Binary Value

Thus, the hexadecimal number A3 is equivalent to the binary number 10100011.

### Observations

- Each hexadecimal digit translates directly to a 4-bit binary sequence.
- The conversion process involves simple lookup and replacement using the conversion table.
- This method ensures accurate and efficient conversion from hexadecimal to binary.

By understanding and practicing this method, converting hexadecimal numbers to binary becomes a straightforward and systematic task.

## Binary to Octal and Octal to Binary

**Converting Binary to Octal and Octal to Binary**

Binary and octal systems are closely related because both are based on powers of 2. Converting between these systems involves grouping binary digits for binary to octal and expanding octal digits for octal to binary. This guide explains both conversions with examples.

### Converting Binary to Octal

**Group Binary Digits**: Start from the binary point and group the binary digits into sets of three. Add leading zeros if necessary to make a complete group.**Convert Each Group**: Convert each group of three binary digits to their octal equivalent.**Combine the Octal Digits**: Write down the octal digits in the order they were grouped.

### Binary to Octal Conversion Table

Binary | Octal |
---|---|

000 | 0 |

001 | 1 |

010 | 2 |

011 | 3 |

100 | 4 |

101 | 5 |

110 | 6 |

111 | 7 |

#### Example: Binary to Octal

Let’s convert the binary number 110010 to octal.

**Group Binary Digits**:- Add leading zeros: 110010 becomes 011 001 010.

**Convert Each Group**:- 011 in binary is 3 in octal.
- 001 in binary is 1 in octal.
- 010 in binary is 2 in octal.

**Combine the Octal Digits**:- The binary number 110010 converts to the octal number 312.

### Final Octal Value

Thus, the binary number 110010 is equivalent to the octal number 312.

### Converting Octal to Binary

**Expand Each Octal Digit**: Convert each octal digit to its 3-bit binary equivalent using a conversion table.**Combine the Binary Digits**: Write down the binary equivalents in the same order as the octal digits to form the complete binary number.

### Octal to Binary Conversion Table

Octal | Binary |
---|---|

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

#### Example: Octal to Binary

Let’s convert the octal number 537 to binary.

**Expand Each Octal Digit**:- 5 in octal is 101 in binary.
- 3 in octal is 011 in binary.
- 7 in octal is 111 in binary.

**Combine the Binary Digits**:- The octal number 537 converts to the binary number 101011111.

### Final Binary Value

Thus, the octal number 537 is equivalent to the binary number 101011111.

### Observations

**Binary to Octal**: Group binary digits in sets of three, convert each group to octal.**Octal to Binary**: Expand each octal digit to its 3-bit binary equivalent.- The process is efficient and straightforward due to the direct relationship between binary (base 2) and octal (base 8).

By following these methods, converting between binary and octal becomes a systematic and easy task.

## Hexadecimal to Octal and Octal to Hexadecimal

**Converting Hexadecimal to Octal and Octal to Hexadecimal**

Converting between hexadecimal and octal involves using binary as an intermediate step because both hexadecimal and octal can be easily converted to and from binary. This guide explains both conversions with examples.

### Converting Hexadecimal to Octal

**Convert Hexadecimal to Binary**: Convert each hexadecimal digit to its 4-bit binary equivalent.**Group Binary Digits**: Once you have the binary equivalent, group the binary digits into sets of three, starting from the binary point. Add leading zeros if necessary to make a complete group.**Convert Binary to Octal**: Convert each group of three binary digits to their octal equivalent.

### Example: Hexadecimal to Octal

Let’s convert the hexadecimal number 2F to octal.

**Convert Hexadecimal to Binary**:- 2 in hexadecimal is 0010 in binary.
- F in hexadecimal is 1111 in binary.
- So, 2F in binary is 0010 1111.

**Group Binary Digits**:- Group into sets of three: 001 011 111 (add a leading zero to the left).

**Convert Binary to Octal**:- 001 in binary is 1 in octal.
- 011 in binary is 3 in octal.
- 111 in binary is 7 in octal.

Combining these, the binary number 001011111 converts to the octal number 137.

### Final Octal Value

Thus, the hexadecimal number 2F is equivalent to the octal number 137.

### Converting Octal to Hexadecimal

**Convert Octal to Binary**: Convert each octal digit to its 3-bit binary equivalent.**Group Binary Digits**: Once you have the binary equivalent, group the binary digits into sets of four, starting from the binary point. Add leading zeros if necessary to make a complete group.**Convert Binary to Hexadecimal**: Convert each group of four binary digits to their hexadecimal equivalent.

### Example: Octal to Hexadecimal

Let’s convert the octal number 537 to hexadecimal.

**Convert Octal to Binary**:- 5 in octal is 101 in binary.
- 3 in octal is 011 in binary.
- 7 in octal is 111 in binary.
- So, 537 in binary is 101 011 111.

**Group Binary Digits**:- Group into sets of four: 0101 0111 1 (add a leading zero to the left and trailing zeros to the right if necessary).

**Convert Binary to Hexadecimal**:- 0101 in binary is 5 in hexadecimal.
- 0111 in binary is 7 in hexadecimal.

Combining these, the binary number 01010111 converts to the hexadecimal number 57.

### Final Hexadecimal Value

Thus, the octal number 537 is equivalent to the hexadecimal number 57.

### Observations

**Hexadecimal to Octal**: Convert each hexadecimal digit to binary, group binary digits in sets of three, then convert to octal.**Octal to Hexadecimal**: Convert each octal digit to binary, group binary digits in sets of four, then convert to hexadecimal.- Using binary as an intermediate step simplifies the conversion process between hexadecimal and octal.

By understanding and practicing these methods, converting between hexadecimal and octal becomes a straightforward and systematic task.

## Binary Representation of Integers

**Binary Representation of Integers**

Binary representation is a fundamental concept in computing and digital electronics, where numbers are expressed using only two digits: 0 and 1. Each digit in a binary number represents a power of 2, starting from the rightmost digit as $2^0$. Here’s how integers are represented in binary and some examples to illustrate the concept.

### Basics of Binary Representation

In binary representation:

- Each digit (bit) can be either 0 or 1.
- The rightmost bit (least significant bit) represents $2^0$, the next bit to the left represents $2^1$, then $2^2$, and so on.
- Numbers in binary are read from right to left, with the rightmost bit being the least significant.

### Example: Binary Representation

Let’s represent the decimal number 42 in binary:

**Divide by 2 Method**: Divide the number by 2 continuously and keep track of the remainders.- Divide 42 by 2, quotient is 21, remainder is 0.
- Divide 21 by 2, quotient is 10, remainder is 1.
- Divide 10 by 2, quotient is 5, remainder is 0.
- Divide 5 by 2, quotient is 2, remainder is 1.
- Divide 2 by 2, quotient is 1, remainder is 0.
- Divide 1 by 2, quotient is 0, remainder is 1.

Reading remainders from bottom to top gives us: 101010 in binary.

**Binary to Decimal Conversion**: To convert from binary to decimal, each bit is multiplied by $2^n$ (where $n$ is the position from the right starting at 0) and then summed up.

## One’s Complement Method

**One’s Complement Method**

In computing, the one’s complement method is a binary arithmetic operation used for binary numbers. It involves flipping all the bits (0s become 1s and 1s become 0s) to compute the inverse of a binary number. This method has implications in digital systems, particularly in representing negative numbers and performing arithmetic operations. Here’s a detailed explanation of the one’s complement method:

### Understanding One’s Complement

**Binary Representation**: In binary representation, each bit can either be 0 or 1. For example, the binary number`0101`

represents the decimal number`5`

.**Finding One’s Complement**: To find the one’s complement of a binary number, you flip all the bits. That is, every`0`

becomes`1`

and every`1`

becomes`0`

.**Why Use One’s Complement?**: One’s complement is primarily used in digital systems to represent negative numbers and simplify arithmetic operations, such as addition and subtraction.

### Examples of One’s Complement

Let’s take a binary number `0101`

as an example:

- Original binary number:
`0101`

- One’s complement: Flip all bits to get
`1010`

#### Steps to Compute One’s Complement

**Step 1**: Start with the binary number`0101`

.**Step 2**: Flip all the bits:`0`

becomes`1`

`1`

becomes`0`

**Step 3**: The one’s complement of`0101`

is`1010`

.

### Applications

**Negative Numbers**: In one’s complement representation, the leftmost bit (most significant bit) indicates the sign of the number.`0`

typically represents a positive number, and`1`

represents a negative number.**Arithmetic Operations**: One’s complement simplifies arithmetic operations like addition and subtraction because the same rules apply for both positive and negative numbers.

### Important Considerations

**Range of Numbers**: In an 8-bit system, for example, the range of numbers represented in one’s complement is from`-127`

to`127`

, where`-0`

and`0`

both represent the number`0`

.**Overflow**: One’s complement arithmetic can lead to overflow situations, where the carry out of the most significant bit (when performing addition) has to be added back to the result.

The one’s complement method is a fundamental concept in digital systems for representing and manipulating binary numbers, particularly in contexts involving negative numbers and arithmetic operations. It provides a straightforward way to invert binary numbers, which is crucial for various computational tasks in digital computing.

## Two’s Complement method

**Two’s Complement Method**

The two’s complement method is widely used in computing to represent signed integers and perform binary arithmetic. It provides a way to encode negative numbers and simplifies arithmetic operations such as addition and subtraction. This method has become the standard for integer representation in modern computers. Here’s a detailed explanation of the two’s complement method:

### Understanding Two’s Complement

**Binary Representation**: Each bit in a binary number represents a power of 2, starting from the rightmost bit (least significant bit). For example, the binary number`0101`

represents the decimal number`5`

.**Two’s Complement**: To find the two’s complement of a binary number, you follow these steps:- Find the one’s complement (invert all the bits).
- Add 1 to the least significant bit of the one’s complement.

**Why Use Two’s Complement?**: Two’s complement simplifies binary arithmetic and eliminates the ambiguity of representing zero with both positive and negative signs, which is an issue in the one’s complement method.

### Examples of Two’s Complement

Let’s take the binary number `0101`

as an example:

- Original binary number:
`0101`

- One’s complement: Flip all bits to get
`1010`

- Add 1 to the least significant bit of
`1010`

to get`1011`

Thus, the two’s complement of `0101`

is `1011`

.

#### Steps to Compute Two’s Complement

**Start with the Binary Number**: For example,`0101`

.**Find the One’s Complement**:`0`

becomes`1`

`1`

becomes`0`

- So,
`0101`

becomes`1010`

.

**Add 1 to the One’s Complement**:- Add
`1`

to`1010`

:`1010`

`+ 1`

`----`

`1011`

- Add

The two’s complement of `0101`

is `1011`

.

### Using Two’s Complement to Represent Negative Numbers

In two’s complement representation, the most significant bit (leftmost bit) indicates the sign of the number:

`0`

represents a positive number.`1`

represents a negative number.

For example, in an 8-bit system:

- The decimal number
`5`

is represented as`00000101`

. - The decimal number
`-5`

is represented as`11111011`

(two’s complement of`00000101`

).

### Converting Two’s Complement Back to Decimal

To convert a two’s complement binary number back to decimal:

- If the number is positive (most significant bit is
`0`

), convert it as a normal binary number. - If the number is negative (most significant bit is
`1`

), compute the two’s complement again to find its positive counterpart, then add a negative sign.

Example: Convert `11111011`

to decimal:

- Find the two’s complement:
- Invert bits:
`00000100`

- Add
`1`

:`00000101`

- Invert bits:
- Convert
`00000101`

to decimal:`5`

- Original number is negative, so the decimal value is
`-5`

.

### Benefits of Two’s Complement

**Single Zero Representation**: Unlike one’s complement, which has two representations for zero (positive and negative zero), two’s complement has only one representation for zero.**Simplifies Arithmetic Operations**: Addition, subtraction, and multiplication operations are straightforward with two’s complement.**Efficient Hardware Implementation**: Modern CPUs are optimized to perform arithmetic operations using two’s complement.

The two’s complement method is a crucial concept in digital systems for representing and manipulating signed integers. It simplifies arithmetic operations and provides a consistent and efficient way to handle negative numbers, making it the standard for integer representation in modern computing.

## Representing Characters

**Representing Characters in Computer Science**

In computer science, characters are represented using various encoding schemes that map characters to numerical values that computers can process. Understanding how characters are represented is fundamental to working with text in computing. Here, we discuss the most common character encoding schemes: ASCII, Unicode, and UTF-8, and provide examples for each.

**1. ASCII (American Standard Code for Information Interchange)**

ASCII is one of the oldest and simplest character encoding schemes, using 7 bits to represent each character. It includes 128 characters, which cover:

- Control characters (e.g., NULL, BEL, etc.)
- Printable characters (e.g., digits 0-9, lowercase and uppercase letters, punctuation marks)

Each character is assigned a numerical value between 0 and 127.

Example:

- The character ‘A’ is represented by the ASCII code 65.
- The character ‘a’ is represented by the ASCII code 97.
- The character ‘0’ is represented by the ASCII code 48.

To see how ‘A’ is represented in binary:

- ASCII code for ‘A’ is 65.
- In binary, 65 is 01000001.

**2. Unicode**

Unicode is a comprehensive character encoding standard that aims to represent all characters from all writing systems in the world. Unicode supports over 143,000 characters across multiple languages and scripts. It uses different encoding forms such as UTF-8, UTF-16, and UTF-32.

Unicode provides a unique code point for each character, written in the format U+xxxx, where “xxxx” is a hexadecimal number.

Example:

- The character ‘A’ has the Unicode code point U+0041.
- The character ‘あ’ (Hiragana letter A) has the Unicode code point U+3042.
- The character ‘😊’ (smiling face with smiling eyes) has the Unicode code point U+1F60A.

**3. UTF-8 (Unicode Transformation Format – 8-bit)**

UTF-8 is a variable-length character encoding for Unicode. It uses one to four bytes to represent characters:

- Characters from the ASCII set (U+0000 to U+007F) are represented with one byte, identical to their ASCII values.
- Characters from U+0080 to U+07FF are represented with two bytes.
- Characters from U+0800 to U+FFFF are represented with three bytes.
- Characters from U+10000 to U+10FFFF are represented with four bytes.

UTF-8 is backward compatible with ASCII and is widely used on the internet.

Example:

- The character ‘A’ (U+0041) in UTF-8 is 01000001 (1 byte, same as ASCII).
- The character ‘€’ (Euro sign, U+20AC) in UTF-8 is 11100010 10000010 10101100 (3 bytes).
- The character ‘😊’ (smiling face with smiling eyes, U+1F60A) in UTF-8 is 11110000 10011111 10011000 10101010 (4 bytes).

**Example of Converting Characters to Binary**

Let’s convert the string “Hello” to binary using ASCII and UTF-8:

For ASCII:

- ‘H’ -> ASCII 72 -> Binary 01001000
- ‘e’ -> ASCII 101 -> Binary 01100101
- ‘l’ -> ASCII 108 -> Binary 01101100
- ‘l’ -> ASCII 108 -> Binary 01101100
- ‘o’ -> ASCII 111 -> Binary 01101111

In ASCII, “Hello” is represented as: 01001000 01100101 01101100 01101100 01101111

For UTF-8 (same as ASCII for these characters):

- ‘H’ -> UTF-8 72 -> Binary 01001000
- ‘e’ -> UTF-8 101 -> Binary 01100101
- ‘l’ -> UTF-8 108 -> Binary 01101100
- ‘l’ -> UTF-8 108 -> Binary 01101100
- ‘o’ -> UTF-8 111 -> Binary 01101111

In UTF-8, “Hello” is represented as: 01001000 01100101 01101100 01101100 01101111

**Character Encoding Table**

Character | ASCII Code | Binary (ASCII) | Unicode Code Point | Binary (UTF-8) |
---|---|---|---|---|

A | 65 | 01000001 | U+0041 | 01000001 |

a | 97 | 01100001 | U+0061 | 01100001 |

0 | 48 | 00110000 | U+0030 | 00110000 |

€ | – | – | U+20AC | 11100010 10000010 10101100 |

😊 | – | – | U+1F60A | 11110000 10011111 10011000 10101010 |

Representing characters in computer science is essential for text processing and data exchange. ASCII, Unicode, and UTF-8 are key encoding schemes that enable computers to handle a wide range of characters from different languages and symbols. Understanding these encoding systems is fundamental for working with text data in various applications.