Create your own number system of base 4, and represent numbers from 1 to 16.
Base-4 Number System:
Digits used: 0, 1, 2, 3
Landmark numbers (powers of 4): 1, 4, 16, 64, ...
Numbers 1-16 in Base-4:
1 = 1 2 = 2 3 = 3 4 = 10 5 = 11 6 = 12 7 = 13 8 = 20 9 = 21 10 = 22 11 = 23 12 = 30 13 = 31 14 = 32 15 = 33 16 = 100
Alternative Symbol System:
- for 0 | for 1 || for 2 ||| for 3
Positions represent powers of 4.
Explanation
The chapter discusses how to create number systems with different bases. A base-4 system follows the same principles as the base-10 (Hindu) and base-5 systems discussed. The landmark numbers are 4⁰=1, 4¹=4, 4²=16, 4³=64, etc. Each position can have digits 0-3, and when a position reaches 4, it carries to the next position.
Solution Steps
Step 1: Define landmark numbers: 1, 4, 16, 64 (powers of 4)
Step 2: Define symbols for digits 0, 1, 2, 3
Step 3: Convert each number 1-16 to base-4 by expressing as sum of powers of 4
Step 4: 1=1, 2=2, 3=3, 4=10 (one 4, zero 1s), 5=11, ..., 16=100 (one 16, zero 4s, zero 1s)