How can I convert 0.4444.... in base 6 to number in base 10??

Question

This is one problem, but could anyone give me a general solution for any numbers (rational and irrational) in any bases...to other bases???

Answer 1

In decimals it's 4 / 5 = 0.8 . Here's how to show this :

Just remember that for numbers written in base 6, the first place past the ' . ' sign represents 1/6 in decimals, the second place represents 1/6^2, ... the rth place represents 1/6^r, ... etc.

So: first factor out the ' 4 '. You now have 4 (0.111...). Then:

In base 10 it's 4 ( 1/6 + (1/6)^2 + (1/6)^3 + ...

= 4 / 5 = 0.8 QED

(You can use a geometric series sum to add up the infinite series here. In fact, what I used was the knowledge, embedded in my head for over 55 years, that the corresponding infinite sum with ' 6 ' replaced by ' n ' is : simply 1/ (n - 1). Pretty neat, huh ?!)

That gives you a pattern to follow in converting any "enimal" into some other "emimal" (these are generalizations of "decimal.") You have to interpret the successive digits in terms of representing the successive amounts 1/n, 1/n^2, 1/n^3 ... . There's no easy way to do it from one system to another, in general. You could be in for quite a hard slog!

Live long and prosper.

Answer 2

Well, you remember the meaning of the radix point in other bases - it simply means that digits after the point are taken to negative powers of the base. So, in base 6, 0.44444... means 4*6^(-1) + 4*6^(-2) + 4*6^(-3) + 4*6^(-4)... and so on. To find the decimal equivalent for an arbitrary irrational number, there is nothing left to do but sum the infinite series. In this case, though, the number is rational, which means that by grouping together the repeating sequence, you obtain a geometric series which may be summed easily (this is the case in general for rational numbers). Thus, in this case, we have:

[k=1, ∞]∑4/6^k

Which is the same as:

4/6 * [k=0, ∞]∑1/6^k

By the formula for summing a geometric series, this is:

4/6 * 1/(1-1/6)
4/6 * 1/(5/6)
4/6 * 6/5
4/5

Which has the obvious decimal representation of 0.8 .

Answer 3

To convert 0.4444.... base 6 to base 10 you sum the series 4(1/6 + 1/6^2 + 1/6^3 + . . . ) =
lim{4[(1-1/6^n)/(1 - 1/6) - 1]} = 0.8
n→∞

the reciprocal of 0.4444.... base 6 = 5/4 base 6 = 5/4 base 10. 4/5 = 0.8

For rational numbers the general method would be find the equivalent fraction, convert numerator and denominator to the desired base, and calculate the "decimal" quantity in that base.

For irrational numbers, you are pretty much stuck with a summation of digit-by-digit conversions.

Answer 4

to transform a large determination from base 6 to base ten write the huge style down on a sheet of paper 4 5 a million 6 Now think of of it this variety: The 6 represents single digits The a million represents instruments of six The 5 represents instruments of thirty-six The 4 represents instruments of 200 sixteen then you quite could convert and upload: 6 + 6 + one hundred eighty + 864 = 1056 this means that 4516 base6 = 1056 base10

Answer 5

0.8