What is the value of "pi" in base three and base seven?

Answers:



In base-3, pi = 10.0102...



In base-7, pi = 3.0664...



Explanation:



Pi is an irrational number, so to make things manageable we'll round it to 3.1416.



Our normal decimal system is based on powers of ten. For example, the digits in pi correspond to:



10^1 =%26gt; tens =%26gt; 0

10^0 =%26gt; ones =%26gt; 3

10^-1 =%26gt; tenths =%26gt; 1

10^-2 = hundredths =%26gt; 4

10^-3 = thousandths =%26gt; 1

10^-4 = ten-thousandths =%26gt; 6



So our approximation of pi can be thought of as the sum of 3 ones, 1 tenth, 4 hundredths, 1 thousandth, and 6 ten-thousandths...3.1416



Base-3 works on powers of 3 instead of ten.



3^1 =%26gt; threes

3^0 =%26gt; ones

3^-1 =%26gt; thirds

3^-2 =%26gt; ninths

3^-3 =%26gt; twenty-sevenths

3^-4 =%26gt; eighty-firsts



So what's in 3.1416?



When you write numbers in bases other than 10, think of it as zooming in on the actual value by dividing more and more precise numbers.



Let's start with threes. Pi is greater than 3, but not greater than 6, so the threes place will have a 1 in it. What's left? 0.1416.



Now let's go to the ones place. The remainder isn't greater than 1, so there are no ones (meaning you put a zero in the ones place).



Let's go to thirds. 1/3 = 0.333... The remainder isn't greater than 1/3, so there's a zero in the thirds place.



Moving on to ninths. 1/9 = 0.111... The remainder IS greater than 1/9, but it's NOT greater than 2/9, so we place a 1 in the ninths place. Subtract 1/9 from the remainder and you get 0.0305.



Moving on to twenty-sevenths. 1/27 = 0.0370... The remainder isn't greater than that, so you place a zero in the twenty-sevenths place.



Moving on to eighty-firsts. 1/81 = 0.0123... The remainder IS greater than 0.0123. In fact, the remainder is greater than 2/81 (0.0247...), so you can put a 2 in the eighty-firsts place.



So, pi in base-3 is approximately 10.0102...



Now, how about base seven?



7^1 = sevens

7^0 = ones

7^-1 = sevenths

7^-2 = forty-ninths

7^-3 = 343rds

7^-4 = 2401sts



Start with 3.1416. Pi is less than 7, so put a zero in the sevens place and move on to the ones place.



One goes into pi 3 times with a remainder of 0.1416, so put a 3 in the ones place.



Next, the sevenths place. 1/7 = 0.1429... Our remainder isn't greater than 1/7, so we put a zero in the sevenths place.



Next, the 49ths place. 1/49 = 0.0204. This goes into our remainder 6 times (with 0.0192 left over), so put a 6 in the 49ths place.



Next, the 343rds place. 1/343 = 0.002915... This goes into our remainder (0.0192...) 6 times and leaves 0.00171 behind, so place a 6 in the 343rds place.



Next, the 2401sts place. 1/2401 = 0.0004165... This goes into our remainder (0.00171) 4 times (and leaves a remainder, but we'll stop here), so place a 4 in the 2401sts place.



So pi in base-7 is 3.0664...



I hope that helps. Good luck!|||10.010211012222..._3


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