That’s a good question, and is the very foundation of math.

So, I’ll give it to you by relating it to Sine and Cosine. On a right triangle, if the hypotenuse is 1, the other two legs will equal something. And if you make a circle—hang on I’m going somewhere—with the hypotenuse, the legs will be equal to sine and cosine, based on the angle made by the hypotenuse and the cosine leg. So every single degree has a measurement for those numbers—which there’s no algebraic way to deduce, except through maybe pythagorean theorem or analytic geometry, but you also just have to measure it; which may also be what they mean by you can’t do a proof through algebra, except by trial and error

So, that number, which for sine of 60 degrees, the number is √3/2. So, in Roman Numerals or Binary, there will be a way to represent that. But the thing itself, is what the number actually represents.

So, in all things, a number represents a physical object. If Gravity, it’s Gravitational Force; if Mass or Weight; if Quanta; if atomic force or charge in a Chemistry Equation. So, those will represent the same value, based on the context of the equation.

Now a formula, is the next thing to learn. A formula is a shape, that relates to the numbers we measure, and it derives measurements for unknowns through logic, by figuring from what is known, to the unknown. As that’s what a formula is meant to do. A formula is the logic of a shape, so in Trig Functions, the formula is the shape itself, and the sine or cosine or tangent are the numbers we use, to derive information about unknowns. Such as to find area, or other side lengths. Which is highly useful. Because that’s what math is doing, is using known information, and logic, to come to conclusions about unknowns, because the logic of the shapes agree. And through ratios of the object’s similarity, we can augment this, or decrease it, to find new numbers.

1/17/26

That is like a pythagorean theorem, but I don't think a+b=c is the same as a^n+b^n=c^n. Basic algebra shows that you can't transpose exponents through addition. You can through multiplication, but they need to retain the same coefficient. You did create something like a pythagorean theorem, but in things like Heron's Formula, you see how when you multiply the numbers, it becomes different, and stops working. So, I think it's basic algebra to say that those two equations are different, and can't have the same number theory applied to them. Just because of how exponents work.

I also think to derive formulas, you have to do them through observing structures. A math formula is like a sentence. It describes a shape of some kind--which is why we use them in science--so I think the logic of a^n+b^n=c^n where n equals 2 is describing a specific pattern, where we discovered it through looking at a square. What a^3+b^3=c^3 would be, we don't know, but the logic is still valid, but it describes a specific shape. That's all our formulas are doing.

1/18/26

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I mean, if it's just saying arithmetic works, we just know it does. It's actually slightly different than a^n+b^n=c^n, if you want to get into the nitty gritty. But, if it's just validating the logic of algebra, we already knew this. It's just something we already knew. It's actually counterproductive to say we don't. You'd invalidate all of math doing that. And I think math works well enough, that we know it's valid.

But generally, exponentiation is a bit different than addition. Is what I'm saying. When you exponentiate, you do something a little bit different. Just like all four operations are different from one another. But, we know they work with absolute certainty, because they produce testable results. It's just something like division, where we see math work in all places, we know it doesn't need to be changed and take the leap into trusting it.

1/20/26

a + b = c has something to do with prime numbers. It's not about validating addition or arithmetic. I have no idea what it means. And I'm okay with that.