Number systems ( decimal, binary, octal, hexadecimal) and their conversions

Here is the explanation of these number systems:

1. The Binary Number System: The binary number system is a base-2 number system with only two symbols: 0 and 1. A binary number's digits each represent a power of two. The binary number 1011, for example, is equivalent to (1 * 23) + (0 * 22) + (1 * 21) + (1 * 20), which equals 11 in decimal.

2. Octal Number System: The octal number system is a base-8 number system that employs digits ranging from 0 to 7. Each octal digit corresponds to three binary bits. Octal is frequently used in computing, particularly in older systems, to more succinctly express groups of binary digits. For example, the octal number 36 corresponds to the binary value 011 110 and the decimal number 30.

3. Hexadecimal Number System: The hexadecimal number system is a base-16 number system that employs numbers 0 through 9 and letters A through F to represent values 0 through 15. In computers, hexadecimal is frequently used to represent memory addresses, colors, and other binary-coded information in a more compact and human-readable format. Each hexadecimal numeral represents four binary bits. The hexadecimal number 1A3, for example, is equivalent to the binary number 0001 1010 0011 and the decimal value 419.

These number systems are useful in many domains, particularly computer science and digital electronics, since they provide efficient ways to encode and process binary data.

Converting between number systems entails translating a number's value from one base to another. Here are some examples of conversions:

1. Conversion from decimal to binary: To convert a decimal number to binary, divide it by 2 several times and record the remainders. The binary representation is then obtained by reading the remainders in reverse order. To convert 25 to binary, for example:

   - 25 ÷ 2 = 12 remainder 1

   - 12 ÷ 2 = 6 remainder 0

   - 6 ÷ 2 = 3 remainder 0

   - 3 ÷ 2 = 1 remainder 1

   - 1 ÷ 2 = 0 remainder 1

   So, 25 in binary is 11001.

2. Conversion from Decimal to Octal: To convert a decimal number to an octal number, repeat the process but divide by 8 instead of 2. To convert 57 to octal, for example:

   - 57 ÷ 8 = 7 remainder 1

   - 7 ÷ 8 = 0 remainder 7

   So, 57 in octal is 71.

3. Converting Decimal to Hexadecimal: Divide a decimal number by 16 and write down the remainders, using numbers 0-9 and A-F for values 10-15. To convert 255 to hexadecimal, for example:

   - 255 ÷ 16 = 15 remainder 15 (F)

   - 15 ÷ 16 = 0 remainder 15 (F)

   So, 255 in hexadecimal is FF.

4. Conversion from Binary to Decimal: To convert a binary number to a decimal number, multiply each binary digit by 2 raised to the power of its location (beginning with 0) and add the results. To convert 11001 to decimal, for example:

   - (1 * 2^4) + (1 * 2^3) + (0 * 2^2) + (0 * 2^1) + (1 * 2^0) = 16 + 8 + 0 + 0 + 1 = 25.

5. Converting from Octal to Decimal: To convert an octal number to a decimal number, multiply each digit by 8 raised to the power of its location and add the results.

6. Convert Hexadecimal to Decimal: To convert a hexadecimal integer to decimal, multiply each digit by 16 raised to the power of its location and add the results.

These are the fundamental concepts for converting between number systems. The key is to understand the basis of the number system you're converting from as well as the destination base.