How do you count to 2(in base .75 number system)?

Question

curious as to decimal number base systems

Answer 1

Please ignore the other answerers. They are all misinformed.

http://en.wikipedia.org/wiki/Non-integer_representation

To use a non-integer base, you use the SAME digits as base 10 (0 through 9). You just have to write 2 as a sum of powers of 0.75.

For example, try:

18/16 = 2(9/16) = 1× 0.75 ² = 100*

Where the * means base 0.75.

So you want:

2 = a (0.75²) + b (0.75) + c (0.75^0) + d (0.75^(-1)) +...

Just like in any other number base. I'm assuming 2 in that base looks something like:

abc.defgh...*

Well, what is the h-place worth? In base 10, it's worth 10^(-5) = 1/100000, but base 0.75, it's worth 0.75^(-5) = 4.213...

That's right, the h's place is actually MORE than the a or b's place. That's because our base is less than 1. So we can actually figure out that the e's place is worth 2.37037.

So e=f=g=...= 0

So we want to know what b and a are. In fact, we might worry that we'll really need something like this:


...hgfedcb.ba*

Why? How does this make sense? How can a number have ... on the LEFT?? Well, think about base 1/10. How do you write 2 base 1/10. Simple: .2. Watch this:

base 10:
45
base 1/10:
5.4

base 10:
4865
base 1/10:
5.684

base 10:
4/3 = 1.333333
base 1/10:
...3333.1000...

It just flips around the 1's place. That works for any base b and the corresponding base 1/b. That means we can have a finite number that has an infinite number of non-zero digits to the left of the decimal place!

So let's get back to our question:

2 = ....gfedc.ba

How much are the a's worth?

(4/3)³ = 16/9 = 1.77777

a=1, and we have .22222 left to use.

b's are worth 4/3 (too much)
c's are worth 1 (too much)
d's are worth 3/4 (too much)
e's are worth 9/16 (too much)
f's are worth 27/64 (too much)
etc..

i's are the next digit we can use, since i's are worth 0.75^6 approximately .316406.

so we get i=1 and we're left with:

2 - (0.75)^(-2) - (0.75)^6 = .044237

The next digit we can use is going to be 5 more digits down, because at that point, digits are worth:

0.75^11 = .0422351

This becomes very hard to compute, and there may not even be a pattern to it! This is disappointing, but we can write:

...0100001000000.01* = 2

HOWEVER like any base system, there is more than one way to write any number. For example, in base 10, we know:

2 = 1.999999....(repeating 9)

What's crazy is that for any base, the 1s place is STILL the 1s place. That means that 2* = 2.

So while there are many ways to write 2 base 0.75, some of which I've shown you how to compute, the easiest is simply:

2* = 2

Answer 2

Good question. If we have base 0.75, then each numerical place corresponds to 0.75 to a different integer exponent.

If we have the following places:
_ _ _ . _ _ _

(that's three places, a decimal point, then three more places), the unit value of each place would be written:

9/16, 3/4, 1, decimal point, 4/3, 16/9, 64/27

However, in a non-integer base system, there is no definition for what characters to write in each place. If we write a 1 in the 3/4s place, what does it mean? Does it mean 3/4? In base 10, you only have ten characters to work with: 0 through 9. In base 0.75, how many characters do you have to work with? Less than one?

In short, it is not defined. There is no way to write this, because you cannot have a non-integer number of discrete objects (the characters used for writing).

Update:

Yeah, the guy below me is correct. I forgot about the "binary" representation using zeros and ones. In case you're curious about the sequence of digits, the first thousand digits, in reverse order, are

10.
00000010000100000000001000
00001000010000000010000010
00000001000010000100000000
00010000100000010000010000
10000001000001000010000100
00000010000010000000000000
00000000001000010000001000
00010000010000000100000000
10000000000010000010000000
01000000000010000000000100
00001000001000001000010000
00000000010000001000010000
10000000010000001000010000
10000010000010001000000000
00010000000100000000000100
00001000000001000000000000
01000000010000001000000100
00010000100000001000010000
00100000001000000001000000
00000000010000000100001000
00010000010000000000000000
01000000100000000010000000
00001000010001000000000001
00000100010000000000100000
00100000000100000100000000
10000010000010001000000000
01000001000000001000010000
00100000001000100000000000
00100000000001000000001000
00010000100001000000100001
00000001000000001000010000
00000100001000010001000000
00000000010000001000000000
00010000000010000000100001
00001000001000001000001000
00000100001000001000001000
00010000100000000100000000
00100000010000000000001000
010000100000

Answer 3

No such thing.

base 2 is the lowest base.

Answer 4

slowly ... since it is 2.66666666666....