genuinelydense
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what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts
tthnaks
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TSR Learn Together
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alow
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(Original post by genuinelydense)
what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts
tthnaks
e is the number, which raised to the power x (where f\left( x \right)=e^{x}), f\left( x \right)=f'\left( x \right)
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Doctor_Einstein
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(Original post by genuinelydense)
what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts
tthnaks
Anything that can be modelled with "e" can be modelled with any other base as well.

Just as 2^20 is the same as 4^10, the base of any power function can be converted to a new base.

E.g. 2^20 = e^(ln(2)*20).

So any function like y = e^(kx), can be written in a form with a different base instead of e, like y = 10^(kx), but the values of k will differ.

So the question is, why is e chosen to be the base of so many equations, rather than 10, or 2, or any other number. The answer is because the gradient of the function y = e^x is also given by the function y = e^x. e is unique in this way, and therefore it allows calculations of gradients much easier using the chain rule, than what would be possible if the function was written in a different form like y = 10^(kx). The second form is still valid, but it is harder to take the derivative of this second base 10 form.

I don't know if you have covered derivatives and the chain rule yet, but when you learn it you will understand why e is chosen.

It may be of interest for you that e^x can also be expressed by this infinite function:

Image

where the expression 3! simply means 3x2x1. Similarly, 4! means 4x3x2x1 and so on.
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Old_Simon
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(Original post by Doctor_Einstein)
Anything that can be modelled with "e" can be modelled with any other base as well.

Just as 2^20 is the same as 4^10, the base of any power function can be converted to a new base.

E.g. 2^20 = e^(ln(2)*20).

So any function like y = e^(kx), can be written in a form with a different base instead of e, like y = 10^(kx), but the values of k will differ.

So the question is, why is e chosen to be the base of so many equations, rather than 10, or 2, or any other number. The answer is because the gradient of the function y = e^x is also given by the function y = e^x. e is unique in this way, and therefore it allows calculations of gradients much easier using the chain rule, than what would be possible if the function was written in a different form like y = 10^(kx). The second form is still valid, but it is harder to take the derivative of this second base 10 form.

I don't know if you have covered derivatives and the chain rule yet, but when you learn it you will understand why e is chosen.
Once you reach Euler it all becomes a lot clearer.(to the OP)
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steve44
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(Original post by genuinelydense)
what is so special about the number e and why can so many things be modeled by it? As in, I know why that the number π exists because it's the ratio of diameter to circumference of a circle, but e... just seems to pop out of nowhere!

just finished learning about capacitors, nuclear decay and chemical half lives, so this is quite important to my understanding of these concepts
tthnaks
Using the property a^{x + \delta} = a^x a^\delta where a is any positive number we get


{d a^x \over dx} = \lim_{\delta \rightarrow 0} {a^{x+\delta} - a^x \over \delta} = a^x \lim_{\delta \rightarrow 0} ({a^\delta - 1 \over \delta}).

So

\dfrac{d}{dx} a^x = Const. \; a^x

where Const. depends on the numerical value of a. The fundamental definition of the number e is then the value of a such that Const. = 1 in this equation.

That is, the number e is defined by the condition that \lim_{\delta \rightarrow 0} ({e^\delta - 1 \over \delta}) = 1

which rearanged becomes

e = lim_{\delta \rightarrow 0} (1 + \delta)^{1/\delta}.

Try on your calculator (1+0.000001)^{1000000} to get an approximation of e. You should get 2.718.
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genuinelydense
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(Original post by steve44)
Using the property a^{x + \delta} = a^x a^\delta where a is any positive number we get


{d a^x \over dx} = \lim_{\delta \rightarrow 0} {a^{x+\delta} - a^x \over \delta} = a^x \lim_{\delta \rightarrow 0} ({a^\delta - 1 \over \delta}).

So

\dfrac{d}{dx} a^x = Const. \; a^x

where Const. depends on the numerical value of a. The fundamental definition of the number e is then the value of a such that Const. = 1 in this equation.

That is, the number e is defined by the condition that \lim_{\delta \rightarrow 0} ({e^\delta - 1 \over \delta}) = 1

which rearanged becomes

e = lim_{\delta \rightarrow 0} (1 + \delta)^{1/\delta}.

Try on your calculator (1+0.000001)^{1000000} to get an approximation of e. You should get 2.718.
thanks, this is a pretty clear explanation of what e is. How do you rearrange that first expression to get the one in bold though?
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steve44
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Which expression do you mean?
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alow
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(Original post by genuinelydense)
thanks, this is a pretty clear explanation of what e is. How do you rearrange that first expression to get the one in bold though?
You can't put LaTeX in bold by using normal [b][/b] tags so we can't see what expression you're talking about.
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