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Understanding Integer Bases

Introduction

In the world of mathematics and computer science, the concept of number representation is fundamental. One aspect of this is the idea of integer bases, which determines how numbers are expressed using different sets of digits. The most commonly used integer bases are binary (base 2), decimal (base 10), hexadecimal (base 16), and octal (base 8).

Binary (Base 2)

Binary is the simplest integer base, using only two digits: 0 and 1. In binary, each digit, called a bit, represents a power of 2. The rightmost bit represents 2^0 (or 1), the next bit to the left represents 2^1 (or 2), and so on. For example, the binary number 1010 represents the decimal number 10, as (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0) = 8 + 0 + 2 + 0 = 10.

Decimal (Base 10)

Decimal, or base 10, is the integer base most familiar to us. It uses the digits 0 through 9, with each digit representing a power of 10. The rightmost digit represents 10^0 (or 1), the next digit to the left represents 10^1 (or 10), and so on. For example, the decimal number 1234 represents (1 × 10^3) + (2 × 10^2) + (3 × 10^1) + (4 × 10^0) = 1000 + 200 + 30 + 4 = 1234.

Hexadecimal (Base 16)

Hexadecimal, or base 16, is a commonly used integer base in computer science and engineering. It uses the digits 0 through 9, as well as the letters A through F to represent the values 10 through 15, respectively. Each hexadecimal digit represents a power of 16. The rightmost digit represents 16^0 (or 1), the next digit to the left represents 16^1 (or 16), and so on. For example, the hexadecimal number 1A3F represents the decimal number 6719, as (1 × 16^3) + (10 × 16^2) + (3 × 16^1) + (15 × 16^0) = 4096 + 1600 + 48 + 15 = 6719.

Octal (Base 8)

Octal, or base 8, is another integer base used in computer science and engineering. It uses the digits 0 through 7, with each digit representing a power of 8. The rightmost digit represents 8^0 (or 1), the next digit to the left represents 8^1 (or 8), and so on. For example, the octal number 1234 represents the decimal number 668, as (1 × 8^3) + (2 × 8^2) + (3 × 8^1) + (4 × 8^0) = 512 + 128 + 24 + 4 = 668.

Conversions

Conversion between integer bases is a common task. To convert a number from one base to another, you can use the following steps:

  1. Convert the number to decimal.
  2. Convert the decimal number to the desired base.

For example, to convert the binary number 1010 to hexadecimal:

  1. Convert 1010 to decimal: (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0) = 8 + 0 + 2 + 0 = 10.
  2. Convert 10 to hexadecimal: 10 in hexadecimal is A.

Conclusion

Understanding integer bases is crucial in various fields, from computer programming to digital electronics. By knowing how to represent and convert numbers between different bases, you can better understand and manipulate data in these domains.