A number base is a system for writing quantities using place values that are powers of a chosen base. Decimal uses base 10, but computers often use binary, octal, and hexadecimal because they match digital on and off states neatly. Learning number bases helps students see that the value of a digit depends on both the digit and its position.
It also builds a stronger understanding of place value, arithmetic, and how data is represented in computing.
In any base b, the rightmost place is b^0, then b^1, b^2, and so on moving left. To convert a number to decimal, multiply each digit by its place value and add the results. To convert from decimal to another base, repeatedly divide by the base and read the remainders from last to first.
Binary, octal, and hexadecimal are closely related because 8 = 2^3 and 16 = 2^4, so groups of 3 binary bits make one octal digit and groups of 4 binary bits make one hex digit.
Key Facts
- In base b, place values are powers of b: ..., b^3, b^2, b^1, b^0.
- The decimal value of digits d_n...d_1d_0 in base b is d_n b^n + ... + d_1 b + d_0.
- Binary is base 2 and uses digits 0 and 1.
- Octal is base 8 and uses digits 0 through 7.
- Hexadecimal is base 16 and uses digits 0 through 9 and A through F, where A = 10 and F = 15.
- Binary grouping rules: 3 bits convert to 1 octal digit, and 4 bits convert to 1 hexadecimal digit.
Vocabulary
- Base
- The base is the number of different digit symbols used in a place-value number system.
- Place value
- Place value is the value assigned to a digit because of its position in a number.
- Binary
- Binary is the base 2 number system that uses only the digits 0 and 1.
- Octal
- Octal is the base 8 number system that uses the digits 0 through 7.
- Hexadecimal
- Hexadecimal is the base 16 number system that uses digits 0 through 9 and letters A through F.
Common Mistakes to Avoid
- Using digits that are not allowed in a base, such as writing 8 in octal. Each base only permits digits from 0 up to one less than the base.
- Reading a nondecimal number as if it were decimal, such as treating 1010 base 2 as one thousand ten. The same written digits can represent different values in different bases.
- Writing division remainders in the wrong order when converting from decimal. Remainders must be read from last to first because the final remainder becomes the leftmost digit.
- Forgetting that hexadecimal letters are digit values, not variables. A, B, C, D, E, and F stand for 10, 11, 12, 13, 14, and 15.
Practice Questions
- 1 Convert 110101 base 2 to decimal.
- 2 Convert 345 decimal to hexadecimal using repeated division.
- 3 Explain why grouping binary digits into sets of 4 makes hexadecimal conversion easier than converting the whole binary number directly to decimal first.