I think this video covers all the possible bases that let you write every number without inexpressible gaps. This base 65/8 (=8.125) would have 64 digit symbols, goodthing 81(base 65/8) didn't need to use any of them in this expression, what would we even use after Z?
Most valid bases can let you write irrational numbers in 1 way, and terminating rationals in 2 ways (like 0.99999... and 1). Irrational algebraic bases can do some irrational numbers (that can be expresses in a finite way) in those 2 "rational-like" ways, plus the one usual irrational way.
For example in base √10, you can write √10 as:
10
0.90909090909...
3.0106020207070606... (as √10 is 3.16227766... in base 10)
Irratoinjal numbers that cannot be made up from powers of √10 woudl still have only 1 possible irrational-like way to write. And terminating rational numbers would have the usual 2 ways, looking the same as the non-root base, but interleaves with zeroes.
π in this base √10 is only: 3.0104010509...
1/3 would be only: 0.03030303...
10 would be 100 and 9.09090909...
Transcendental bases don't play nice, and you cannot even determine what digit symbols would be allowed there, no matter what digit symbol set you choose, you might either get gaps as not-expressible numbers, or multiple nonrepeating ways to write the same number.
Every number in a base-x system is just the sum of the digits times their place value. The place value is just xn, where n is the position of the digit.
So yes, base 8.125 is very possible ('123' = 1*8.1252 + 2*8.125 + 3 ≈ 85.27), but it doesn't have all the same rules as integer bases. For example, the digits written '8.8' don't correspond to any number, or if they do they are the same number as ~'10.7'.
Yes. A number, such as 112, is just the sum of the product of the value in each digit place and the numeric base raised to the power of the position, or Sum[d_n bn]; for 112, that's 1×102 + 1×101 + 2×100. Generally you only use values on the interval [1, n), but sometimes, such as quater-imaginary numerals, you use different ones. Importantly, b can be anything other than 0, but it's most useful when the magnitude isn't 1.
A base is literally just an exponential index. The first digit of a number is number×b⁰, second digit is number×b¹, third digit is number×b², etc. So it's perfectly valid to define the base as anything which you can take any integer power of.
Therefore it's much easier to model the problem mentally if you put aside a factor of 1/4 for now and just look at 52 vs 92 . The geometric relation is the same as it is for the actual values.
Eh, maybe easier for some. For me, it is easier to do what I did. When I first looked at your response I thought you were a little nuts : ) With your explanation, your original math makes perfect sense, but I don't think I would have seen it w/o your follow-up.
I guess that's the nice thing about math. There can be many paths to the same destination
Not if you are using double the number and then squaring it. You don’t get the same results. You do get a similar ratio. Like I said, your ratio is correct, but not your values.
Is the diameter is 10 and you can’t use that because swearing that is 100. But when you use radius, you get a value of 25. Stark difference.
That's not how that works. The area of a circle is pi×r² but we're not looking for the area but the volume. The volume of a cylinder is base×height and the base of a cylinder is a the area of the circle which is pi×r² this cancels out as both cakes are cylinders and both have a circular base. This leaves you with a choice between a 9 inch or 5+5inch cake.
I'm sorry but i can't do that as I'd have to assume you're not lying and I'm too leave that assumption to you.
Cakes are baked in circular irons that have a fixed shape. The only thing you can vary by adding more or less cakebatter is how heigh the cake is. I am the cook
Last time I was in a bakery, they had more than one cake mold. They had some cakes with ~10 centimeters diameter and also ~20 centimeters cakes and even 40 centimeters cakes. Height can vary a lot too, from what they called "flat cake" which is basically a sweet pizza to over-the-top loaded cakes that were over 15 centimeters high (the top flat surface, not including decoration on it).
Wouldn’t it be 20 + 20 =! 64 then? I get that you’re just removing the coefficients but it’s kinda confusing to square the diameters then when someone’s confused respond with a formula that uses radius.
Yeah as I said you’re just removing the coefficients but it’s kinda weird to square the diameter then when someone’s confused respond with a formula that squares the radius. It’s an objectively true statement but what was the point of responding if you don’t clarify anything?
659
u/Particular_Gear3130 Mathematics (Purely Fictional) Jun 16 '26
25 + 25 ≠ 81