What is 1999 in ternary

Question

I am converting decimal to ternary

Answer 1

Ternary is a base 3 system. We can use a type of long division to get the answer.

We are going to divide 1999 by 3 and just note the remainder:

1999/3=666 rem 1. So 1 is the last digit in the ternary number.

Now divide 666 by 3=222 remainder 0. This goes in front of the 1.

222/3=74 remainder 0. That's another 0 to place in front of 01.

74/3=24 remainder 2. The ternary now ends 2001.

24/3=8 remainder 0. Ternary ends 02001.

8/3=2 remainder 2. Ternary ends 202001.

The last quotient is smaller than 3 so we just make it the first digit of the answer:

2202001. To check this we can use the powers of 3: 1, 3₁₀=10₃, 9₁₀=100₃, 27₁₀=1000₃, 81₁₀=10000₃, 243₁₀=100000₃, 729₁₀=1000000₃.

2000000₃ is 2×729=1458₁₀; 200000=2×243=486₁₀; 2000₃=2×27=54₃; 1 is the same in both bases.

So 2202001₃=1458+486+54+1=1999₁₀.