WHAT IS THE IMPORTANCE AND SIGNIFICANCE OF "E" (EULER'S NUMBER)?
Euler's number is an irrational number like π. The number goes like this e =2.71828...and there is no repeat pattern.
You can get the information on the web, but these are some of my thoughts.
Very roughly, the derivative of a function is the slope of the graph of the function.
If a function happens to be linear, then its derivative is constant. For the function y =x, the "derivative of y with respect to x" is equal to 1, because the slope of the graph y =x is 1.
But when a function is not linear, then its derivative is itself a function, because the slope of the tangent will be greater or less at different points of the function's graph. For the function y = x2, the "derivative of y with respect to x" is 2x, because at any point (x,x2) the tangent to the graph has slope 2x.
Now, since the derivative of a function can itself be a function, it's natural to ask: is there any function which is its own derivative? It turns out that the answer is yes, that there is exactly one such function on the real numbers, and that function is y = ex the exponential function whose base is Euler's number.
So any time you have a phenomenon that can be modeled with an exponential (or logarithmic) function, and you want to do some calculus on the function, you'd best use e. This includes cases of exponential growth, compound interest, and radioactive decay. There are also applications in probability and statistics.
You can get the information on the web, but these are some of my thoughts.
- "e" is the base of Natural Logarithm (ln). Given that x = ln y, then y = e^x
- "e" is also used in complex numbers, e^ix = cos x + i sin x
- "e" has a special place in Calculus, d/dx (e^x) = e^x, d/dx (ln x) = 1/x
- "e" is also used in the definitions of hyperbolic functions, sinh x , cosh x and tanh x
Very roughly, the derivative of a function is the slope of the graph of the function.
If a function happens to be linear, then its derivative is constant. For the function y =x, the "derivative of y with respect to x" is equal to 1, because the slope of the graph y =x is 1.
But when a function is not linear, then its derivative is itself a function, because the slope of the tangent will be greater or less at different points of the function's graph. For the function y = x2, the "derivative of y with respect to x" is 2x, because at any point (x,x2) the tangent to the graph has slope 2x.
Now, since the derivative of a function can itself be a function, it's natural to ask: is there any function which is its own derivative? It turns out that the answer is yes, that there is exactly one such function on the real numbers, and that function is y = ex the exponential function whose base is Euler's number.
So any time you have a phenomenon that can be modeled with an exponential (or logarithmic) function, and you want to do some calculus on the function, you'd best use e. This includes cases of exponential growth, compound interest, and radioactive decay. There are also applications in probability and statistics.
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