We study positional numeral systems other than base 10. In computers, binary and hexadecimal are important.
How Base-n Numeration Works
In base n, each place has the weight of a power of n:
(dkdk-1…d1d0)n = dk×nk+…+d1×n+d0
Example ① (Base n → Base 10)
Convert (1101)2 to base 10.
1×2³ + 1×2² + 0×2 + 1 = 8+4+0+1 = 13
Convert (1101)2 to base 10.
1×2³ + 1×2² + 0×2 + 1 = 8+4+0+1 = 13
Example ② (Base 10 → Base n)
Convert 45 to base 2.
45÷2=22…1
22÷2=11…0
11÷2=5…1
5÷2=2…1
2÷2=1…0
1÷2=0…1
Read the remainders from bottom to top → (101101)2
Convert 45 to base 2.
45÷2=22…1
22÷2=11…0
11÷2=5…1
5÷2=2…1
2÷2=1…0
1÷2=0…1
Read the remainders from bottom to top → (101101)2
Example ③ (Base 5)
Convert (234)5 to base 10, and then to base 8.
2×25+3×5+4 = 50+15+4 = 69（base 10）
69÷8=8…5、8÷8=1…0、1÷8=0…1 → (105)8
Convert (234)5 to base 10, and then to base 8.
2×25+3×5+4 = 50+15+4 = 69（base 10）
69÷8=8…5、8÷8=1…0、1÷8=0…1 → (105)8
Key Points
- Base n→base 10: add each digit×nk
- Base 10→base n: repeatedly divide by n and read the remainders from bottom to top
- The largest k-digit number in base n = nk−1
- Binary↔hexadecimal conversion is done in groups of four digits (computers)
Practice Problems
- Convert (10110)2 to base 10.
- Convert 100 to base 3.
- Convert (3201)4 to base 10.
Answers and Explanations
- 16+4+2=22
- 100÷3=33…1、33÷3=11…0、11÷3=3…2、3÷3=1…0、1÷3=0…1 → (10201)3
- 3×64+2×16+0×4+1=192+32+0+1=225