# If the Fibonacci sequence is plotted as a curve - is it an exponential curve?

#### KeithMath

##### New member
In general I am guessing that an exponential curve has to 'grow at an ever increasing rate' - is this true?

Specifically, is the Fibonacci sequence of numbers 'exponential'? My question is based on the fact that it starts of with 1, 2, and 3 ...this is linear and the increase from 2 to 3 (50%), is less than the increase from 1 to 2 (100%)

I can see that later on in the sequence that it is exponential as it increases at a regular amount. But what about the lower numbers?

Any views? Would love to hear them as I really need to settle an argument!

Thanks in advance #### ksdhart2

##### Senior Member
In general I am guessing that an exponential curve has to 'grow at an ever increasing rate' - is this true?

Well, sort of. It's true that the growth rate of an exponential curve is always increasing, but that's not strictly a requirement for the curve to be exponential - it's just a side effect, a "bonus" if you will. An exponential curve is called exponential because it contains an exponent at its heart. The general form of the exponential curve is $$\displaystyle f(t) = k \cdot r^t$$, where k is some positive constant (usually specified to be the initial value at time t = 0), and r is the amount each term grows when compared to the previous one.

Because the derivative is defined as the instantaneous rate of change at some point, we can investigate whether the growth rate is increasing, decreasing, or constant, by examining the derivative of the derivative (aka the second derivative). That happens to be: $$\displaystyle \dfrac{\partial^2}{\partial t^2} (k \cdot r^t) = k \ cdot r^t \cdot ln^2(r)$$. The only way this could ever be 0 is if r = 1, but then it wouldn't be an exponential curve. Therefore, it's always positive, and thus the growth rate of the exponential curve is always increasing.

Specifically, is the Fibonacci sequence of numbers 'exponential'? My question is based on the fact that it starts of with 1, 2, and 3 ...this is linear and the increase from 2 to 3 (50%), is less than the increase from 1 to 2 (100%)

I can see that later on in the sequence that it is exponential as it increases at a regular amount. But what about the lower numbers?

Any views? Would love to hear them as I really need to settle an argument!

Thanks in advance This, too, is a matter of semantics. The Fibonacci sequence itself isn't an exponential curve because it's only defined over the integers. However, there are extensions which are defined over the reals. If you use such an extension, you'll find that the Fibonacci numbers are generated by an exponential curve, although you'd be hard pressed to find that result purely by looking at the Fibonacci numbers themselves. You can look at the growth rate by taking the limit of: $$\displaystyle \displaystyle \lim_{n \to \infty} \dfrac{F_n}{F_{n-1}}$$ and the result of that will be $$\displaystyle \dfrac{1 + \sqrt{5}}{2} \approx 1.618$$. This is a special mathematical constant typically denoted by the symbol $$\displaystyle \phi$$ (the Greek letter phi).

If you're familiar with linear algebra, you might find this page and this other page helpful in showing how to derive a closed form for the Fibonacci numbers. But, the bottom line is that the Fibonacci numbers can be defined by the following function:

$$\displaystyle f(n) = \dfrac{(1+\sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \cdot \sqrt{5}}$$

This formula only works for the integers, so to extend it to the reals, we need to do a bit of manipulation. The end result is:

$$\displaystyle f(n) = \dfrac{\phi^n}{\sqrt{5}}$$

Here, the Fibonacci numbers are defined by plugging in any integer argument for n, then rounding down to the nearest integer. This, then, is an exponential curve, where $$\displaystyle k = \dfrac{1}{\sqrt{5}}$$ and $$\displaystyle r = \phi$$. That tells us that the growth rate is always increasing. Indeed, the second derivative of the function is: $$\displaystyle \dfrac{\partial^2}{\partial n^2} \left( \dfrac{\phi^n}{\sqrt{5}} \right) = \dfrac{(n-1) \cdot n \cdot \phi^{n-2}}{\sqrt{5}}$$. Again, this is always positive except for when n = 0 and n = 1.

#### KeithMath

##### New member
Any chance you can exaplin that in more simple terms please?

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?

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?.
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?
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?
It is a long time since i have done algebra so the more the basic the explanation, the better for me Thanks all the same for the help so far...I have a lot of people trying to get to the bottom of this - it does seem a tricky one.

PS: you may have answered it but I cant tell #### tkhunny

##### Moderator
Staff member
If it is "Exponential", the same change in sequential terms must show the same RATIO of change. This is not asymptotic behavior. It must happen everywhere.

1
1
2
3
5
8
etc

1/1 =1
2/1 = 2 -- Not Exponential
3/2 = 1.5
5/3 = 1.66666666....
8/5 = 1.6

The following is an example of an exponential growth pattern

1
2
4
8
16
etc...

2/1 = 2
4/2 = 2
8/4 = 2
16/8 = 2

Same ratio EVERY TIME.

#### KeithMath

##### New member
So the Fibonacci sequence is NOT exponential?

If i understand tkhunny correctly - you are saying that it is NOT exponential. You use the word asymptotic - does this mean 'kind of'. So the Fibonacci sequence looks 'a bit' exponential but in order to be exponential - it has to be 100% so all of the time. Do I understand you correctly?

#### ksdhart2

##### Senior Member
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.

#### JeffM

##### Elite Member
I fear I must respectfully disagree with the previous post. Definitions are very important in math.

Some people find it convenient to define the natural numbers so as to have zero as the least member; others find it convenient to define the natural numbers so as to have one as the least member. There is a debate about which definition is more useful, but it is basically a debate about convenience.

The primary reason that different kinds of numbers have been developed is to ensure closure. If you try to add any two natural numbers, you get a natural number. In other words, the natural numbers are "closed" with respect to addition (and multiplication). But you cannot subtract all pairs of natural numbers and get a natural number. The natural numbers are not closed with respect to subtraction.

If you try to subtract one integer from another, you do get another integer. So the integers are closed under addition, subtraction, and multiplication. All natural numbers are integers, but not all integers are natural numbers. Similar processes lead to rational numbers, real numbers, and complex numbers.

Exponential functions are carefully defined. All airplanes fly, but not all things that fly are airplanes. It does not lead to precise thought to call bats airplanes. The function that generates the Fibonacci sequence is not exponential.

#### KeithMath

##### New member
Surely this is either yes or no...or is it a matter of opinion?

Firstly, I would like to thank all 3 of you for trying to help me out here.

I have asked several friends and colleagues and no-one has given me a DEFINITIVE answer yet. Everyone seems perplexed! Well maybe JeffM has given me a 100% answer, but this isn't quite the same as the answer from tkhunny and it is the opposite to ksdhart2. I have very simple maths knowledge but I cannot see why this is hard to answer. Would it help if I limited the question to whole integers only? ...and they would be positive numbers too. In fact I am specifically looking at the numbers 1, 2, 3, 5, 8, 13, 21, 32, 53. Are they an exponential sequence (or is the function that creates them 'an exponential function')...not that I fully understand the bit in brackets!

Would be neat to sort this #### mmm4444bot

##### Super Moderator
Staff member
I've always thought if the points from a discrete function lie on an exponential curve, then the function is a discrete exponential function. Conversely, if a continuous exponential function cannot be found to pass through a set of discrete points, then the set of discrete points do not belong to a discrete exponential function.

#### KeithMath

##### New member
so what is the answer then?

mmm4444bot are you saying that 1,2,3,5,8,13,21,... is, or is not, points on an exponential curve?

NB: the numbers 1 2 and 3 are not on a curve - they are in a line.

#### mmm4444bot

##### Super Moderator
Staff member
mmm4444bot are you saying that 1,2,3,5,8,13,21,... [are] or [are not] points on an exponential curve?
They do not lie on an exponential curve (using the standard definition, posted earlier). :cool:

#### KeithMath

##### New member
So, is any of the sequence exponetial?

mmm4444bot - thanks for the quick reply. When you get higher up in the sequence (144,233,277,...), the numbers do grow at an ever increasing rate around 'the golden ratio' (1.618) - at this point could you say they are growing exponentially?

#### Subhotosh Khan

##### Super Moderator
Staff member
mmm4444bot - thanks for the quick reply. When you get higher up in the sequence (144,233,277,...), the numbers do grow at an ever increasing rate around 'the golden ratio' (1.618) - at this point could you say they are growing exponentially?
In the limit, as n → ∞, the successive terms of Fibonacci's (Fn) sequence grows exponentially.

#### KeithMath

##### New member
is that a definitive answer then?

Do I have this right? mmm4444bot and Subhotosh Khan are saying that at the lower end (i.e. 1, 2, 3, 5, 8, 13) the Fibonacci sequence is NOT an exponential curve and does not grow exponentially...but at the higher end (and all the way up to infinity) it does grow exponentially. As you are both the highest level of contributor and therefore maths gurus in my opinion - can I take this as a definitive answer to my original question? (i.e. 1,2,3,5,8,13,21 does not grow exponentially)

#### tkhunny

##### Moderator
Staff member
I don't see any confusion at all. Not sure were you are reading it. Let's agree with just about everyone.

The sequence, in its entirety, is NOT exponential. This is easily shown by simply comparing two ratios of values the same distance apart in the sequence.

"In the limit" - This is the "asymptotic" behavior I suggested. It begins to behave as an exponential the farther it goes. This does NOT mean that somewhere, if you go far enough, it WILL BE an eponential. It is a limiting process, not the arrival of a freight train.

Having said that, what is it that you want? If your application or process is more interested in transient behavior, then this is definitely NOT an exponential. If your application or process is more interested in stable, long-term behavior, then maybe it is exponential in that application or process. It doesn't have to be one or the other. There is some dependence on what you want.

People say, "...grows exponentially..." ALL THE TIME. Most of the time, they are absolutely wrong when their data are compared to actual well-defined, mathematically-constructed exponentials. Does that mean the growth is NOT exponential? If there is only one meaning, then yes. However, we know that dictionaries are histories, not unalterable stone tablets. The common usage of a term can be very different over many years.