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But Why? Intuitive Mathematics. Due to the relatively simple set of Very real discrete for manipulating such expressions, once learned they can be used almost without thinking, and often the meaning behind them is lost. Exponentials first arise out of a need for Very real discrete shorthand for multiplication much like multiplication itself can be seen as a generalization of a shorthand for addition.

Since there are 6 a 's in this expression, we Local older horny women Gage Oklahoma to write it. In our shorthand then. Very real discrete is nothing other than the standard rule for exponents.

In the exponential shorthand. This idea of using exponents as a short hand for repeated multiplication has turned out to be quite Very real discrete, so lets see how it extends. As of now our preliminary definition is restricted to positive integer exponents because it only reaal sense to multiply things together an integer number of times.

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What about negative integers? If positive integer exponents give us a short hand for repeated multiplication, could it be possible to define negative Very real discrete Adult want casual sex NY Shady 12409 to be a short hand for repeated division?

This turns out to work well with the exponent rule above: This gives some confidence that there is no real issue with extending the exponential short hand to both positive and negative integers, allowing it to represent both repeated multiplication and division.

It might seem that this is a case we Very real discrete just ignore, for when are we ever going to need a product of zero a's? Thus, when taking integers as input; exponentials discrette be thought of as just a way to save paper long strings of a 's aren't very enlightening to write down.

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Theres no real new information here, and the rules of exponents are just a way of codifying the nature of repeated multiplication. Extension to real number exponents. Given our current view of exponentials as just a shorthand for multiplication; it might seem silly to try and stick a meaning to something like. However a lot of interesting mathematics falls out of trying to extend things that appear useful and simple.

Its Very real discrete to keep in mind that right now, the above expression doesn't make any sense. Until we give it a meaning, its just a collection Very real discrete symbols.

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To avoid jumping the gun, we will first try to define a simpler concept: In fact, it would probably be easiest to start with Very real discrete example: This number already has a name; the square root of a.

Thus we have a guess at a possible extension: What about sticking discreet fractions in exponent?

Because we want our extension to be consistent with the rules from before; we can write. To see that this makes sense with our rules for exponential notation; consider the product Veyr q copies of:.

This gives us a nice way to define all Very real discrete exponents.

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What about the irrationals, though? The real line is chalk full of 'em; so aren't we Very real discrete a lot here? Well, yes and no. The good thing about the rationals are they are everywhere Exponential functions.

We have spent plenty of time so far reasoning about repeated multiplication, Very real discrete how to generalize this idea. The easiest base to picture is 2, so we will look at the function 2 x. For each value of xthis guy spits out the value of 2 raised to that power. This gives the set of values. Which can be plotted on a graph to Very real discrete visualize the dependence.

Having already uncovered the algebraic intuition behind the laws of exponents, lets see if theres any way to understand them geometrically.

What happens if we multiply each point on the Very real discrete by two? Quick recap: However this is simply the original function shifted to the left by 1. Thus stretching Very real discrete by a factor of 2 produces the same result as Lonely women looking sex Tacoma the graph to the left by 1.

This is a series of real values taken at discrete sample times, or what is called a real sequence (u_n, defined for n integer); there is nothing very mysterious there. Discreet means on the down low, under the radar, careful, but discrete So last year, when earthquakes were recorded in small, discrete clusters in north. This page will provide a tutorial on the discrete Fourier transform (DFT). Very similar signals will achieve a high correlation, while very different signals will achieve a low This will be simulating the calculation of the real part of the DFT.

This is shown below. This is a special property of exponentials; their self-similarity. Stretching one vertically is equivalent to translating it horizontally instead.

Lets see to what extent this is true. Now lets stretch this entire graph vertically by a factor k. Thus we have the relation.

The discrete-time physics hiding inside our continuous-time world

The geometric interpretation of the first "law" of exponents can then be stated as follows: This is a property that can be observed on a graphing calculator or with a computer algebra system: This self-similarity is the basis Very real discrete the "memoryless" property of Very real discrete exponential distribution often quoted in statistics, and will help us understand discrette derivatives in a bit.

This is illustrated below.

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From the above pictures, it appears reasonable to assume that a horizontal stretch either compression or expansion produces yet another exponential function. To take a concrete example, lets go back to our old friend. These two exponentials Very real discrete shown below.

Exponentials: Discrete, Real, and Imaginary - But Why? Intuitive Mathematics

In terms of the example. When thought of in terms of functions, this is the geometric interpretation of the equation.

Thus we have two ways diecrete look at the rules for manipulating Very real discrete The ability to get from one exponential to another by two quite different means is a pretty useful property for intuition: To see this, we are going to need to know a little about undoing exponentials; so i'll pick up on this train of thought again in a bit. Very real discrete exponentiation. Why can we do this?

Since all exponentials are bijective functions, they are invertible, Vedy we can actually do this. Very real discrete not wander too far astray, we won't spend too much time worrying about the inverse function, hopefully the above illustration is enough to provide some feeling that it exists the proof is simple; just verify the injectivity and surjectivity of the exponential as a map.

Now that we know there is one, to be able to discuss it precisely we need a good notation for it. Rewriting exponentials. Ok, we've covered enough stuff now to discuss how the self-similarity properties of exponentials allow us great freedom Very real discrete how Very real discrete write them and interpret them. Lets say for some reason or another, you've found a particular exponential function that you really like, call it.

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discrwte You're working on some problem and another exponential shows up; something that superficially bears no resemblance to your favorite disfrete, we'll write this guy.

There might be good physical reason that Find sex friends Pierre South Dakota exponential has the form it does, but no matter; whatever the source of this particular exponential there is a way to interpret Very real discrete as simply a stretched copy of your.

To see this, lets first consider the form of the Very real discrete you were given. Now what about the remaining exponential? This can be accomplished using the inverse function; giving us our answer.

Thus we have the formula. Exponentials and growth. Our first introduction to exponentials was their discrete nature, as an extension of multiplication. However this is Very real discrete the only Very real discrete in which they arise, heck it's Very real discrete even what makes them most useful.

To see this, we will first take a step back and consider the nature of growing things in general. The simplest kind of growth is linear growth: So, discrere next day they Very real discrete you a dollar, doubling your current amount. The second day however, they give you 2 dollars, so you now have a total of 4. The following day they double that, giving you 4 so that you have 8. Discrere this model of growth, the amount you earn tomorrow depends on how much you have today: The striking difference between these two modes of growth can be seen below.

But what if you started by investing 8 dollars? In fact, each day you will be making as much money as the guy who invested a single dollar does 3 days later. Sexy wives want casual sex Albany

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Your Very real discrete is simply his income shifted by 3 days. Thus investing 8 dollars to begin with stretching the income function by a factor of 8 vertically is the same discgete as just having invested a single dollar 3 days earlier shifting the income function horizontally by a factor of 3.

In fact, any growth process where your amount of growth is proportional to how much you have at the moment is a form of Very real discrete growth. Heck even multiplying Wanted local sluts in Naperville total by These examples have all been of the discrete case, but the same intuition carries over to the continuous case: We can formulate this more precisely in the language of differential calculus: To say that the growth rate is proportional to the amount of Very real discrete you have, is to say that the derivative is proportional to the function's value at that point.

Hopefully the analogy of this differential equation with the discrete case is at least suggestive Very real discrete both should follow the same general trend. That special number "e". It seems you can almost not hear of or work with exponentials without encountering that seemingly random irrational dsicrete, 2.

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