I have a limited understanding of Maths. I do not even know what a 'Real' is! Is there any chance you can explain some of your points in more simple terms please?

Oh, sorry. I assumed because you posted in the Calculus sub-forum that you had an understanding of higher-level maths. I can try and explain a bit better. Basically, there's four main kinds of numbers - integers, rational numbers, irrational numbers, and real numbers. In higher math classes, you'll encounter some more, but most of them aren't widely used. The integers are commonly called "whole numbers". There is a subset of the integers that's called either "natural numbers" or "counting numbers". This is only the positive integers. There's a huge debate as to whether those sets actually mean the same thing, or if zero is included in either set. Some say zero is in the natural numbers, but not the counting numbers; some say zero is in the counting numbers, but not the natural numbers; some say zero is in both; and still others say zero is in neither.

Then there's "rational numbers." These are what are typically called "fractions." If a number can be written in the form a / b, where both a and b are integers, then the number is rational. Note that the integers are a subset of the rational numbers because any integer can be written as n / 1. The irrational numbers are any numbers that

*cannot* be written as a ratio of two integers. Examples of these include \(\displaystyle \sqrt{2}\) and \(\displaystyle \pi\).

Finally, we have the real numbers. In some sense, this can be considered the set of all numbers. It includes the integers, the rationals, and the irrationals. As I mentioned, there are other sets of numbers that aren't included in the reals, but you don't really need to worry about them for now.

Specifically:

You say that it is true that __it is ever increasing__ but it is __not strictly a requirement __- is that a contradiction? ...or is this the 'side-effect' you refer to...but if it is always the case then it will always happen won't it?.

I think the best explanation I can offer here is an analogy. All airplanes fly in the sky, but that's not why they're airplanes. There are many things that fly in the sky that are not airplanes, such as birds. In much the same way, all exponential functions have an increasing growth rate, but that's not why they're exponential functions. This

**page from Purple Math** might be able to provide a better explanation of what an exponential curve is.

You mention that an exponential curve must have an exponent - will that be a single number? Does this relate to the 1.618 number - the golden ratio? If it does - then how does this relate to the lower numbers that do not seem to relate to this number?

In order for a function to be considered exponential, it must have an exponent, and this exponent must contain the input variable. But it need not be a single term. For instance, \(\displaystyle f(t) = 1.5 \cdot 2^{t-5}\) is an exponential function, as is \(\displaystyle f(x) = e^{x^2}\). The base of the exponent is typically some arbitrary constant, but it also need not be. If you'd defined

*x* in a previous formula, you might have an exponential curve that looks like \(\displaystyle f(t) = 6 \cdot (x^2 + 7)^t\).

Can you explain why the second question is 'semantics'? How should I phrase the question? The situation I have (in common parlance if you like) is that I am being told that **'the Fibonacci sequence grows exponentially'**. Is this true, partly true or not true at all?

Basically, I say it's a matter of semantics because in a lot of mathematics, people tend to be sticklers for precision and accuracy. A lot of the time this matters greatly. Like, say, you're making a rocket ship. If you get the math even a little bit wrong or accidentally use the wrong units, the rocket might explode.

**Here is a CNN article** about a time an orbiter was lost because one of the engineers working on it was using inches and feet while everyone else was using centimeters and meters. That being said, there are also sometimes when it doesn't really matter. This is a sort of gray area, where it's not entirely clear if it matters or not.

As Tkhunny pointed out, the numbers of the Fibonacci sequence don't grow at a constant ratio, so for that reason it can't really be considered an exponential function. Another reason why it might not be considered an exponential function is because it's only defined for the integers, so it's not even really a function at all, let alone an exponential one.

However, when just "shooting the breeze," as it were, I don't think these distinctions particularly matter. People often say "the Fibonacci sequence" in place of "the function which generates the Fibonacci numbers," and everyone seems to understand what was meant. The bottom line is, other people may disagree, but I say that yes, the Fibonacci sequence

**is** an exponential function.