# Difference between revisions of "Math"

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==Miscellaneous== | ==Miscellaneous== | ||

+ | *See also [[Statistics]]. | ||

+ | |||

*[https://twitter.com/wtgowers/status/1456558355225485317 Gower Twitter thread] on thinking you've found a great result, but you made a mistake (2021). | *[https://twitter.com/wtgowers/status/1456558355225485317 Gower Twitter thread] on thinking you've found a great result, but you made a mistake (2021). | ||

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*[https://eml.berkeley.edu/~cshannon/e204_21.html Shannon's math camp notes] on analysis, etc. | *[https://eml.berkeley.edu/~cshannon/e204_21.html Shannon's math camp notes] on analysis, etc. | ||

+ | |||

+ | ---- | ||

+ | |||

+ | ==Erdos Numbers== | ||

+ | *My Erdos Number is 5, from | ||

+ | |||

+ | :*Rasmusen-Connell-Farb-Lubotzky-Alon-Erdos, | ||

+ | |||

+ | :*Rasmusen-Janssen-Sierksma-Doignon-Fishburn-Erdos | ||

+ | |||

+ | :*Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos | ||

+ | |||

+ | ---- | ||

+ | ==Geometry== | ||

+ | [https://mathcs.clarku.edu/~djoyce/java/elements/elements.html Joyce] has an amazingly good website on Euclid's Elements. | ||

---- | ---- | ||

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---- | ---- | ||

+ | |||

==Irrational Numbers== | ==Irrational Numbers== | ||

*I read that Kronecker opposed the use of irrational numbers because they can't be constructed from integers by a finite number of steps. But you can construct pi as the ratio not only as an infinite series, but as the ratio of circumference to diameter. What's wrong with that? | *I read that Kronecker opposed the use of irrational numbers because they can't be constructed from integers by a finite number of steps. But you can construct pi as the ratio not only as an infinite series, but as the ratio of circumference to diameter. What's wrong with that? | ||

+ | |||

+ | *If you don't use infinity, you can't derive the area of a circle, can you? Or the length of a curve? | ||

+ | |||

+ | *Not using proof by contradiction is separate from not using infinity in proofs, right? | ||

+ | |||

+ | *'''Theorem: The square root of 2 is irrational.''' | ||

+ | |||

+ | :'''Proof:''' Suppose not. If the square root of 2 is rational, Sqrt(2) = a/b for some integers a and b. First, note that this implies that Sqrt(2) = x/y for some integers x and y that are not both even. That's because if Sqrt(2) = a/b and both a and b are even, then it is also true that a and b are divisible by 2, so Sqrt(2) = .5a/.5b with both .5a and .5b being integers. If .5a and .5b are both even, we can repeat the division by 2, and we can keep doing this till either the numerator or denominator is odd (or both). | ||

+ | |||

+ | :So suppose Sqrt(2) = x/y, where x and y are not both even. We will show that this implies that both x and y are even, a contradiction. | ||

+ | |||

+ | :First, square both sides so 2 = x^2/y^2. Then 2y^2 = x^2. This implies that x^2 is even. But then x must be even too, since if x is odd, x(x-1) is even because it multiplies an odd number (x) times an even one (x-1), and so x(x-1)+x is odd because it adds an even number (x(x-1)) to an odd number (x). | ||

+ | |||

+ | :If x is even, we can write it as x = 2m for some number m. But then 2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED. | ||

+ | |||

+ | |||

+ | |||

---- | ---- | ||

+ | |||

==Line Integrals== | ==Line Integrals== | ||

I should set up a latex file in a rasmapedia directory to explain line integrals. | I should set up a latex file in a rasmapedia directory to explain line integrals. |

## Latest revision as of 19:48, 17 April 2022

## Contents

## Miscellaneous

- See also Statistics.

- Gower Twitter thread on thinking you've found a great result, but you made a mistake (2021).

- Shannon's math camp notes on analysis, etc.

## Erdos Numbers

- My Erdos Number is 5, from

- Rasmusen-Connell-Farb-Lubotzky-Alon-Erdos,

- Rasmusen-Janssen-Sierksma-Doignon-Fishburn-Erdos

- Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos

## Geometry

Joyce has an amazingly good website on Euclid's Elements.

## Math Education

- My StackExchange question on synthetic division and whether it is every useful for anything.

## Irrational Numbers

- I read that Kronecker opposed the use of irrational numbers because they can't be constructed from integers by a finite number of steps. But you can construct pi as the ratio not only as an infinite series, but as the ratio of circumference to diameter. What's wrong with that?

- If you don't use infinity, you can't derive the area of a circle, can you? Or the length of a curve?

- Not using proof by contradiction is separate from not using infinity in proofs, right?

**Theorem: The square root of 2 is irrational.**

**Proof:**Suppose not. If the square root of 2 is rational, Sqrt(2) = a/b for some integers a and b. First, note that this implies that Sqrt(2) = x/y for some integers x and y that are not both even. That's because if Sqrt(2) = a/b and both a and b are even, then it is also true that a and b are divisible by 2, so Sqrt(2) = .5a/.5b with both .5a and .5b being integers. If .5a and .5b are both even, we can repeat the division by 2, and we can keep doing this till either the numerator or denominator is odd (or both).

- So suppose Sqrt(2) = x/y, where x and y are not both even. We will show that this implies that both x and y are even, a contradiction.

- First, square both sides so 2 = x^2/y^2. Then 2y^2 = x^2. This implies that x^2 is even. But then x must be even too, since if x is odd, x(x-1) is even because it multiplies an odd number (x) times an even one (x-1), and so x(x-1)+x is odd because it adds an even number (x(x-1)) to an odd number (x).

- If x is even, we can write it as x = 2m for some number m. But then 2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED.

## Line Integrals

I should set up a latex file in a rasmapedia directory to explain line integrals.

They are really "curve integrals".

1. You want to integrate x^2 over all points between 2 and 6.

2. You want to integrate x^2 + y^2 over all points between y=0, x in [2,6].

3. You want to integrate x^2 + y^2 over all points on the straight line between (2,2) and (6,6) (not intervals now--- vectors).

4. You want to integrate x^2 + y^2 over all points on the curve y = x^3 between (2,8) and 3,27). This is the real stuff.

5. You want to integrate x^2 + y^2 over all points in the area bounded by (0,0) and (0,2) and (2,0) and (2,2).

6. You want to integrate x^2 + y^2 over all points in the area bounded by y = x^3 and y = log x (or some two curves that cross).

## Slide Rules

December 6, 2021: In late 2021, Russian video appeared that showed a heavy bomber in flight with one of the crew using s slide rule, apparently to calculate course and/or fuel consumption rate. Most pilot training, especially for the crews of long-range aircraft, includes instruction on how to use special slide rules for such calculations in the event of problems with the electronic navigation and flight management instruments that do this automatically. For generations, ever since long-range flight became possible, manual tools were used for these calculations, along with a special bubble sextant to obtain the location of an aircraft. On the surface the original sextant is used for this but in the air, there is no fixed horizon to base these calculations. The bubble-sextant creates an artificial horizon that enables aerial navigation that shows position within ten kilometers or less. Surface sextant navigation is even more accurate and electronic navigation, especially using GPS, is accurate enough for landing aircraft.

Media tends to refer to EMP (Electromagnetic Pulse) created by a distant nuclear weapon detonation as the reason for these manual backups.