SOLUTION: Convert 20220 base 3 to a Hindu-Arabic numeral (base 10)

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Question 1138448: Convert 20220 base 3 to a Hindu-Arabic numeral (base 10)
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52795) About Me  (Show Source):
You can put this solution on YOUR website!
.
20220_base_3 = 0%2A3%5E0+%2B+2%2A3%5E1+%2B+2%2A3%5E2+%2B+0%2A3%5E3+%2B+2%2A3%5E4 = 186_base_10.

Solved and explained.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


20220 (base b) = 2b^4+0b^3+2b^2+2b^1+0b^0 = 2b^4+0b^3+2b^2+2b+0

20220 (base 3) = 2(3^4)+0(3^3)+2(3^2)+2(3^1)+0(3^0) = 2(81)+0+2(9)+2(3)+0 = 162+18+6 = 186

Here is a different method for performing the conversion, without having to know all the powers of 3. This method becomes far more efficient than the above method as the number of digits becomes larger.

Starting at the left, multiply each digit by the base and add the next digit; repeat until there are no more digits. For this example:

2*3=6; 6+0=6
6*3=18; 18*2 = 20
20*3 = 60; 60+2 = 62
62*3 = 186; 186+0 = 186

Algebraically, here is what this method is doing to perform the conversion from base b to base 10:

2*b = 2b; +0 = 2b+0
b(2b+0) = 2b^2+0b; +2 = 2b^2+0b+2
b(2b^2+0b+2) = 2b^3+0b^2+2b; +2 = 2b^3+0b^2+2b+2
b(2b^3+0b^2+2b+2) = 2b^4+0b^3+2b^2+2b; +0 = 2b^4+0b^3+2b^2+2b+0

And that last algebraic expression is exactly the meaning of 20220 in base b, as shown at the beginning of my response.