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Let's continue our discussion of
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common sets of numbers
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we just discussed a moment ago.
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The set of all integers, which is notated
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with a capital Z,
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and the set of all rational numbers.
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In other words the set of all fractions,
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which is notated with a capital Q.
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So, is there anything beyond the domain
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of the rational numbers?
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Anything that's not a fraction
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that we can think of?
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Well, one of the most famous numbers of
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them all is actually not a rational number
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The number "pi".
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Pi has some sort of quirky properties,
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right?
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We know the decimal expansion
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begins 3.1415... and so forth.
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One of the interesting aspects of pi is
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that decimal expansion goes on forever,
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so it's infinite
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and it's also non-repeating,
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or you could say non-recurring.
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So, the distribution of the numbers
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in the decimal expansion of pi
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are, effectively, random.
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I can't write pi as the ratio of
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one whole number (one integer)
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over another.
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So if a number can't be written
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in this fashion,
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if it can't be expressed
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in rational form
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then it's called irrational.
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So pi is an irrational number.
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There's a number that's not in Q,
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let's label that
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in our Venn diagram.
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Somewhere out here is the number pi.
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You might know that the number "e",
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also sort of a special number in math
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and science, is 2.7... etc, etc,
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has also this nice property
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that its decimal expansion
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is non-repeating and infinite,
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so it's likewise an irrational number.
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So e is also out here.
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In fact, there is an infinite set of
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irrational numbers.
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How can I just argue that there
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is an infinite set of irrationals?
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One fact is that if you take
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the square root of a prime number,
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you get an irrational result.
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So it's decimal expansion is
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non-repeating and infinite.
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Number two, is that the number
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of primes is infinite.
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Now, prime, just to remind you
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is a whole number
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greater than or equal to 2
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whose only divisors are 1 and itself.
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So the square root of a prime
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is irrational
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and the number of primes is infinite.
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Now, this is not necessarily obvious,
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right?
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We don't know that sort of a-priori.
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In fact, I'm including a little
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bonus lecture where I proof a la Euclid
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that there's an infinite number of primes.
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If you put these two together,
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then you can show that there is
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an infinite set of irrationals.
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Let's check that out.
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If there's an infinite number of primes
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and the square root of any prime
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is irrational
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then there's the reason again that
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there's an infinite number
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of irrational numbers.
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A slightly different justification.
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Let's call this justification number one,
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that there's an infinite set
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of irrationals.
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Justification number two,
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from sort of a "complexity" perspective,
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that there's an infinite number
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of irrationals.
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What I've said before is that
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the decimal expansion of
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an irrational number is non-repeating
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and infinite.
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So we can go in reverse here,
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because if I am just constructing
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infinite non-repeating decimal expansions
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I could ask, how many are possible?
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Now, that's a big counting problem,
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but as it turns out
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there's an infinite number of such
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decimal sequences that I can construct.
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For instance, let's say I just start by
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flipping a coin or asking a computer
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to compute a super-random number
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and according to the outcomes,
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these coin flips, or whatever,
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I fill in some number.
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How many different such
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decimal expansions are possible?
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There is an infinite number of ways
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I can express a non-repeating
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infinite decimal expansion.
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So there are an infinite number of
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irrational numbers, because there's
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an infinite number of such
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decimal expansions I could construct.
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This leads us to our next set of numbers,
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and the most important
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set of numbers of them all,
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which is the reals.
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The reals consist of the rational numbers
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let's just say "plus", kind of informally,
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the irrational numbers.
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Take note here.
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The reals, without a doubt,
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are the most important and
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commonly encountered number system
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in any applied science field.
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So, when you do calculus,
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you're working with the reals.
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When you do linear algebra,
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differential equations, you're basically
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working with the reals.
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Sometimes other sets,
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but generally speaking
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it's the reals or a subset of the reals.
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In proper notation, we would say
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it's really the union of those two sets.
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So, now, I can add another
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concentric circle to my Venn diagram,
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which is showing the sequence of
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common sets of numbers.
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Now I've got the reals as the
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outermost shell here.
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If I graph the real line, as it's called,
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I get a straight line, importantly though,
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with no gaps.
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On the other hand, if I plot all the
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rational numbers on the line,
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I get lots of gaps.
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In fact, it's a well-known
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mathematical result
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that between any two rational numbers
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lies an irrational number.
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The reals then give us kind of this
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full-fledged closed domain,
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in the sense there're no gaps.
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In other words, when I plot the reals,
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I get, importantly,
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a continuum of numbers.
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So, if we plot the reals,
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we get a continuum.
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Now, what happens if I sort of track
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the motion of an object
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or a particle in space?
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Let us imagine we see a particle moving
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and we track it,
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and it goes along this path here.
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It's starting in the lower left
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moving to the right
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and as it goes, say, time elapses.
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So, the way I experience the motion of
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that particle is a little bit debatable,
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but generally the accepted interpretation
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is that I experience that motion
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as a continuous process.
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So, therefore, it would be of use
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as mathematicians and scientists
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to use a model that replicates
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that continuous behavior.
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So, in other words, we would probably
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like to use, generally speaking,
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a number system that is a continuum.
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Now, you've heard of the
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space-time continuum, right?
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I won't get too new agey here, but we
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generally experience the real world
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as a space-time continuum,
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and the number system we want to use
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to model that is a continuum.
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Now, similarly,
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just to add to that notion,
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for those of us that have maybe had
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a little bit of calculus or seen functions
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and, why not?
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We can think of a function...
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Usually in calculus we'd like to study
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functions that are continuous, right?
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Those have kind of nice
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mathematical properties.
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So, if we take a function
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that's continuous,
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maybe as a model for the motion
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of the particle, and we use it
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in conjunction with the domain
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for that function that's also continuous,
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together, we get a continuous model.
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And that, generally speaking,
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is gonna be a good description
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of real world phenomena.