Recent content by logicgate

Thanks for the answer. I have a question : Do inverse functions relate to horizontal transformations of graphs ? Like for example the function |x-1| is treated as |x| shifted one unit to the right. Is it because the inverse of x-1 is x + 1 ?

So assuming I have a graph of a parent function f(x) and I want to graph for example the function f(2x+1). I need to find a way to manipulate the function f(x) to make it look like the function f(2x+1). For the parent function f(x) I have coordinates of (x , y). And for the function f(2x+1) I...

So using the multiplicative inverse axiom we have : 1) x . x^-1 = 1 2) x . (1/x) = 1 I have no idea why do mathematicians define the multiplicative inverse of a number x to be the "fraction" 1/x. But I know for sure that multiplying any number a for example by the multiplicative inverse of x is...

Thank you for your answer.

As I understand, a ratio is a comparison between two or more quantities. Ratios involve two or more numbers. Whereas a fraction is a single real number. Why are ratios and fractions the same when ratios involve two or more different numbers whereas fractions represent only ONE real number like...

It's very easy to prove the area formula for a rectangle when both length and width are positive integers, but I cannot prove it when length or width or both are rational or irrational numbers. I need an intuitive proof that is as simple as possible without using very advanced math like calculus.

Euclid defines equivalent ratios as the following : "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former...

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e...

This might sound like a stupid question but I am just wondering why is it that x times yz equals xyz and not xyxz ? Why don't we distribute multiplication in this case ?

I tried searching on the internet for hours to find an answer, but I didn't find any.

How did you arrive to the conclusion that ln(1/a)^x = xln(1/a) ? This is exactly what I want to prove.

use power series

In the book "Calculus by Michael Spivak" it says that a^x = e^xln(a) is a definition. And I am not convinced to accept this as true without a proof.