Why does it have such significance?
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If you do the basic calculus operations on the equation e^x it doesn’t change. This makes it very important.
It’s not quiet like, but almost like the +0 or multiply by 1 of calculus.
If you do the basic calculus operations on the equation e^x it doesn’t change. This makes it very important.
It’s not quiet like, but almost like the +0 or multiply by 1 of calculus.
If you do the basic calculus operations on the equation e^x it doesn’t change. This makes it very important.
It’s not quiet like, but almost like the +0 or multiply by 1 of calculus.
The answer involves calculus. Without calculus you can’t really appreciate the importance of e. Note the ancient Greeks knew about pi but not e.
The most basic important differential equation is dy/dx = cy for a constant c: something changes at a rate proportional to itself. The proportionality constant is c. Many ordinary differential equations that can be solved explicitly turn out to be related to equations like that.
If dy/dx = cy and y(0) = y*_0_* (the initial value), then the solution to that differential equation is y(x) = y*_0_*e^(cx). All of these are *exponential* curves (increasing if c > 0, decreasing if c < 0, and constant curves if c = 0).
In particular, to take the simplest case, if dy/dx = y and y(0) = 1 (so the proportionality rate is 1 and the initial value is 1) then y(x) = e^(x).
There are connections between e and trigonometric functions, but that involves further calculus plus complex numbers, so it is not as basic a reason as what I wrote above.
The answer involves calculus. Without calculus you can’t really appreciate the importance of e. Note the ancient Greeks knew about pi but not e.
The most basic important differential equation is dy/dx = cy for a constant c: something changes at a rate proportional to itself. The proportionality constant is c. Many ordinary differential equations that can be solved explicitly turn out to be related to equations like that.
If dy/dx = cy and y(0) = y*_0_* (the initial value), then the solution to that differential equation is y(x) = y*_0_*e^(cx). All of these are *exponential* curves (increasing if c > 0, decreasing if c < 0, and constant curves if c = 0).
In particular, to take the simplest case, if dy/dx = y and y(0) = 1 (so the proportionality rate is 1 and the initial value is 1) then y(x) = e^(x).
There are connections between e and trigonometric functions, but that involves further calculus plus complex numbers, so it is not as basic a reason as what I wrote above.
The answer involves calculus. Without calculus you can’t really appreciate the importance of e. Note the ancient Greeks knew about pi but not e.
The most basic important differential equation is dy/dx = cy for a constant c: something changes at a rate proportional to itself. The proportionality constant is c. Many ordinary differential equations that can be solved explicitly turn out to be related to equations like that.
If dy/dx = cy and y(0) = y*_0_* (the initial value), then the solution to that differential equation is y(x) = y*_0_*e^(cx). All of these are *exponential* curves (increasing if c > 0, decreasing if c < 0, and constant curves if c = 0).
In particular, to take the simplest case, if dy/dx = y and y(0) = 1 (so the proportionality rate is 1 and the initial value is 1) then y(x) = e^(x).
There are connections between e and trigonometric functions, but that involves further calculus plus complex numbers, so it is not as basic a reason as what I wrote above.
Suppose you have a 100 gram plant, and it grows by 100% in one month. At the end, you have a 200 gram plant.
But suppose it first grows 50% in half a month, and then 50% in half a month. If you compute it like this, it ends up bigger, because the first 50% of extra plant also grows in the second half: the growth compounds on itself. You ultimately end up with a 225 gram plant.
What if you split that 100% instead into “first 25%, then 25%, then 25%, and finally one last 25%”? Now the compounding growth gives you a slightly over 244.14 gram plant. What if you split into ten 10% one after another. A slightly over 259.37 gram plant. How about a hundred consecutive 1% increases? A slightly over 270.48 gram plant.
A million 0.0001% increases? Slightly over 271.828 gram plant. Is this number starting to look familiar? Yup, it’s getting closer and closer to e times our starting 100 grams.
This is one way to think about what e means. It’s a conversion coefficient between single-step growth, and continuous growth that builds upon itself: doubling turns into multiplying by e when split evenly across infinitely tiny steps that compound. Thus it is a very fundamental number when dealing with any sort of continuous compounding growth (or in other words, continuous growth that is proportional to the size of the original thing).
Suppose you have a 100 gram plant, and it grows by 100% in one month. At the end, you have a 200 gram plant.
But suppose it first grows 50% in half a month, and then 50% in half a month. If you compute it like this, it ends up bigger, because the first 50% of extra plant also grows in the second half: the growth compounds on itself. You ultimately end up with a 225 gram plant.
What if you split that 100% instead into “first 25%, then 25%, then 25%, and finally one last 25%”? Now the compounding growth gives you a slightly over 244.14 gram plant. What if you split into ten 10% one after another. A slightly over 259.37 gram plant. How about a hundred consecutive 1% increases? A slightly over 270.48 gram plant.
A million 0.0001% increases? Slightly over 271.828 gram plant. Is this number starting to look familiar? Yup, it’s getting closer and closer to e times our starting 100 grams.
This is one way to think about what e means. It’s a conversion coefficient between single-step growth, and continuous growth that builds upon itself: doubling turns into multiplying by e when split evenly across infinitely tiny steps that compound. Thus it is a very fundamental number when dealing with any sort of continuous compounding growth (or in other words, continuous growth that is proportional to the size of the original thing).
Suppose you have a 100 gram plant, and it grows by 100% in one month. At the end, you have a 200 gram plant.
But suppose it first grows 50% in half a month, and then 50% in half a month. If you compute it like this, it ends up bigger, because the first 50% of extra plant also grows in the second half: the growth compounds on itself. You ultimately end up with a 225 gram plant.
What if you split that 100% instead into “first 25%, then 25%, then 25%, and finally one last 25%”? Now the compounding growth gives you a slightly over 244.14 gram plant. What if you split into ten 10% one after another. A slightly over 259.37 gram plant. How about a hundred consecutive 1% increases? A slightly over 270.48 gram plant.
A million 0.0001% increases? Slightly over 271.828 gram plant. Is this number starting to look familiar? Yup, it’s getting closer and closer to e times our starting 100 grams.
This is one way to think about what e means. It’s a conversion coefficient between single-step growth, and continuous growth that builds upon itself: doubling turns into multiplying by e when split evenly across infinitely tiny steps that compound. Thus it is a very fundamental number when dealing with any sort of continuous compounding growth (or in other words, continuous growth that is proportional to the size of the original thing).
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