Binary, Octal, and Hexadecimal
In modern computing and digital electronics, the most commonly used bases are decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). If we are converting between two bases other than decimal, we typically have to convert the number to base 10 first, and then convert that number to the second base. However, we can easily convert directly from binary to octal, and vice versa, and from binary to hexadecimal, and vice versa. This video gives a basic introduction to these conversions: https://youtu.be/5sS7w-CMHkU For another description, this one is more like a math lecture: https://youtu.be/2UwxdCLFW70 For further clarification, recall that the numbers 0 through 7 can be represented by up to three digits in base two. In base eight, these numbers are represented by a single digit.Base 2 (binary) number | Base 10 (decimal) equivalent | Base 8 (octal) number |
---|---|---|
000 | 0 | 0 |
001 | 1 | 1 |
010 | 2 | 2 |
011 | 3 | 3 |
100 | 4 | 4 |
101 | 5 | 5 |
110 | 6 | 6 |
111 | 7 | 7 |
Base 2 number | Base 10 equivalent | Base 8 number |
---|---|---|
1000 | 8 | 10 = 1 × 8 + 0 × 1 |
1001 | 9 | 11 = 1 × 8 + 1 × 1 |
1010 | 10 | 12 = 1 × 8 + 2 × 1 |
… | … | … |
111100 | 60 | 74 = 7 × 8 + 4 × 1 |
111101 | 61 | 75 = 7 × 8 + 5 × 1 |
111110 | 62 | 76 = 7 × 8 + 6 × 1 |
111111 | 63 | 77 = 7 × 8 + 7 × 1 |
- Convert the number 61578 to base 2. We split each digit in base 8 to three digits in base 2, using the three digit base 2 equivalent, so 68 = 1102, 18 = 0012, etc.
- Convert the number 101110110010102 to base 8. Split this number into sets of three, starting with the right-most digit, then convert each set of three to its equivalent in base 8.