Question

Why is 2031 base 4 = 141?

Answer 1

2031 base 4 = 1 + 3*4^1 + 0 + 2*4*3
= 1 + 12 + 2*64
= 13 + 128 = 141

Answer 2

each position is a different power of 4
the lowest power (0) is on the right:

4^3 4^2 4^1 4^0
2 0 3 1

so 2031 is short hand for 2*4^3 + 0 * 4^2 +3*4^1 +1* 4^0

Answer 3

Thank you both....but I am trying to figure out a way to explain this to an elementary school student using base 4 blocks, longs and units...not quite sure how to do that.

Answer 4

I assume you have explained and it did not "click"? Not surprising because the idea of using position and powers (of 10) is a relatively recent invention. Greeks and Romans did not use it.

There are three ideas. One is the idea that different coins represent different values, e.g. pennies, nickels, quarters

The second idea is the "type of coin" is defined by its position when written down
1 is different from 10, which is different from 100

The third idea is the "amount of each coin" is a power of 4 (or whatever the base).
That idea may not be easy to get across.

And maybe a fourth idea: the number of "coins" of each type is limited to 0 to 3 (in base 4)
If you get 4 of the ones, you have to trade it in for 1 of the 4^1 coins.

Good Luck!

Answer 5

I really like your last idea...we'll start there. Thank you very much!