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Problem-Solving Trick No One Taught You: RMS-AM-GM-HM Inequality - YouTube
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hey this is presh talwalkar
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if someone asks
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hey can you tell me the average of the
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numbers
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a mathematician might jokingly say which
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mean do you mean
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most people are referring to the
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arithmetic mean which is the most common
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simple average
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but in some applications
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other means may be more appropriate
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for example the geometric mean is useful
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to calculate the average annual growth
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rate
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and it's used in economics for example
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then there's the harmonic mean which is
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useful to calculate things like the
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average speed
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there's also something called the root
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mean square which is useful in
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electrical engineering and it's also
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called the quadratic mean
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interestingly enough there is a
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relationship between all of these means
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it's abbreviated by their acronym
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the rms am gm and hm inequality
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this is useful in math competitions and
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it's also useful in theoretical proofs
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in this video i'm going to present the
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version for two variables because
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there's a nice geometric visualization
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of it
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so first let's state it
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for positive numbers a and b
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the largest quantity will be the root
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mean square this is equal to the square
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root of the quantity a squared plus b
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squared all over 2.
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this will always be greater than or
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equal to a plus b over 2 which is called
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the arithmetic mean
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this will always be greater than or
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equal to the square root of a times b
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which is the geometric mean
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and this will always be greater than or
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equal to
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two divided by the quantity one over a
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plus one over b which is the harmonic
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mean
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furthermore equality holds if and only
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if a is equal to b
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so this seems like an extremely
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complicated statement
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but we'll be able to demonstrate why
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this is true geometrically
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so let's get started with the geometric
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construction
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we start out with one length of a and
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another length of b
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now we construct a semicircle with
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diameter a plus b
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next we construct a radius of this
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semi-circle
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notice it has length that's half of the
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diameter
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therefore it will be a plus b over 2
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which is the arithmetic mean
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now let's construct the root mean square
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this segment right here is equal to the
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length of the radius minus the length of
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a
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this will be b minus a all over 2. now
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suppose the vertical radius is
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perpendicular to the diameter
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we now construct the hypotenuse of this
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right triangle
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using the pythagorean theorem we can
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figure out its length
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it will be equal to the root mean square
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which is the square root of the quantity
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a squared plus b squared all over 2.
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from this diagram we can see the root
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mean square will always be greater than
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or equal to the arithmetic mean because
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the hypotenuse is never smaller than one
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of its legs
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now let's construct the geometric mean
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we'll construct a perpendicular where
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the two line segments meet
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now a triangle inscribed in a
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semi-circle is always a right triangle
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so we've constructed an altitude on a
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hypotenuse of a right triangle
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using those properties we can deduce
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that the length of the red segment is
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equal to the square root of a times b
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and this is the geometric mean
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now let's construct the harmonic mean
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but before we do that let's relate the
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geometric mean to the arithmetic mean
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we'll construct a hypotenuse from the
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top of this red line to the center of
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the semicircle
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the hypotenuse is a radius of the circle
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so its length is equal to the arithmetic
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mean which is a plus b over two
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and from this diagram we can deduce that
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the arithmetic mean is always greater
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than or equal to the geometric mean
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so now let's construct the harmonic mean
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we'll draw an altitude to this
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hypotenuse
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using the properties of an altitude on
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hypotenuse
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we'll try and figure out the length of
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this purple segment which we'll call x
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so using these similar triangles we get
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that the geometric mean squared is equal
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to x times the arithmetic mean
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we substitute in those values and then
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solve for x
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another way of writing this is 2 divided
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by 1 over a plus 1 over b which is
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exactly the harmonic mean
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so now from this diagram we can deduce
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the geometric mean is always greater
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than or equal to the harmonic mean
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so now let's put all of these lengths
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together in the same diagram
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we can compare the lengths geometrically
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we have that the root mean square is
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greater than or equal to the arithmetic
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mean we also know the arithmetic mean is
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greater than or equal to the geometric
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mean and then we have that the geometric
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mean is greater than or equal to the
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harmonic mean this is all because the
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hypotenuse of a right triangle is
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greater than or equal to either of its
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legs
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we can combine all of these inequalities
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together
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and we have the root mean square
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arithmetic mean geometric mean and
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harmonic mean inequality
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voila
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but we want to see what happens if a is
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equal to b
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obviously if a is not equal to b then we
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have strict inequalities because the
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hypotenuse of a right triangle is
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strictly larger than the leg of its
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triangle but now what happens if a is
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equal to b
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do we get that all of these lengths are
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equal to each other
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well let's once again visualize this
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geometrically
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here's an animation of b approaching a
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that i made with the desmos calculator
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i provided a link to desmos.com in the
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video description
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let's see what happens as we make the
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length of b approach the length of a
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notice how all the lengths are
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converging to the same value and when b
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is equal to a they all are exactly equal
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so this illustrates that the
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inequalities become equal if a is equal
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to b
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so to put this all together we've
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demonstrated the root mean square
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arithmetic mean geometric mean and
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harmonic mean inequality for two
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variables
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an equality holds if and only if a is
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equal to b
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this is the inequality for two variables
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but it actually generalizes to n
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variables in the following form
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this is not easy to visualize
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geometrically so i provided a link to
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this in the video description
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but these inequalities are very useful
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in problem solving and pay attention
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because they're going to appear in some
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of my upcoming videos
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thanks for watching this video
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