Let's start a thread for Useful Mathematical Formulae

Miscellaneous and Personal Stuff I Must Share
21

Nobody’s mentioned Pythagorean’s? a^2 + b^2 = c^2.

Ideal Gas Law, PV=nRT.

Pressure = Force / Area.

Moment = Force * Distance

Velocity is the first derivative of position.
Acceleration is the first derivative of position and the second derivative of position.
Jerk is the first derivative of acceleration…

Similarly, the shear force in a member is the derivative of it’s curve.
The moment force is the second derivative…
What’s the third derivative? I know it’s something important, it’s been awhile.

22

An antenna F feet high can transmit/receive approximately 1.2*(Square root of F) miles. (Example: 100-foot antenna works for 1.2*10 = 12 miles). Or, if you’re F feet tall, you can see 1.2 F miles standing on the beach (Cecil did a column on this – it’s a derivation using the Pythagorean equation).

One nautical mile equals exactly 1.852 kilometers. This is not an approximation like km to feet. It’s EXACT, like 5280 feet to one statute mile, or 12 inches to one foot. This may seem like a trivial distinction, but it’s important to software geeks and navigators.

23

[QUOTE=Santo Rugger]

Velocity is the first derivative of position.
Acceleration is the first derivative of position and the second derivative of position.
Jerk is the first derivative of acceleration…
[/QUOTE]

Who are you calling a jerk? Seriously though, I don’t understand. Can you explain or link to the wiki? I mean regarding the jerk part.

24

The speed of light, c, is 3 x 10⁸ m/s. No muss, no fuss.

A new cent=2.5 g.

A nickel=5.0 g.

Those last two came in handy when divvying up a…never mind.

25

[QUOTE=Autolycus]
Who are you calling a jerk? Seriously though, I don’t understand. Can you explain or link to the wiki? I mean regarding the jerk part.
[/QUOTE]
Not a problem. It’s used in situations where the acceleration happens quickly enough that the motion does its thing before the materials start reacting.
[QUOTE=Wiki]
Jerk is often used in engineering, especially when building roller coasters. Some precision or fragile objects — such as passengers, who need time to sense stress changes and adjust their muscle tension, or suffer, e.g., whiplash — can be safely subjected not only to a maximum acceleration, but also to a maximum jerk. Jerk may be considered when the excitation of vibrations is a concern. A device which measures jerk is called a “jerkmeter.”
[/QUOTE]
I’ve never heard of a “jerkmeter”, I suspect that’s a bit of gratuitous editing. The only time I’ve measured it, we used accelerometers that were tapped into a blast wall and simply took the derivative of the curve.
[QUOTE=Wiki]
Jerk is also important to consider in manufacturing processes. Rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in uneven lines of a cut. This is why modern motion controllers include jerk limitation features.

[/QUOTE]
I forget the specifics, but I’ve also used jerk when analyzing high velocity impact dynamics. Metals are very cool, in that they can withstand pressures high above their yield strengths when said pressures are initiated very fast (impulse) and for a very short time period. It’s almost as if the metal becomes an incompressible liquid for a short period of time, and, in fact, that’s how explosive welds are analyzed.

26

[QUOTE=Santo Rugger]

Velocity is the first derivative of position.
Acceleration is the first derivative of position and the second derivative of position.
Jerk is the first derivative of acceleration…

Similarly, the shear force in a member is the derivative of it’s curve.
The moment force is the second derivative…
What’s the third derivative? I know it’s something important, it’s been awhile.
[/QUOTE]
And the derivative of a real number is xn^x-1. So the derivative of x^4 is 4x^3

Not a formula but a help for math classes, especially geometry. If you have a graphing calculator, I’m assuming a TI-83 cuz that’s what I have but I bet other brands do this as well, then you can program formulas into it so that you hit a a few buttons, input your numbers and it applies the correct formula. Check your manual for instructions on how to program.

27

The volume of a sphere is 4/3 pi r^3*.
Its area is the derivative of that, or 4 pi r^2.

The circumference of a circle is the derivative of its area, pi r^2 goes to 2 pi r.

*Note that the rank (size of power) is the same as number of dimensions in each case.

There are 124 pints in a keg (15.5 gallons times 128 ounces per gallon divided by 16 ounces per pint).

An acre is how much land one man and two oxen can plow in one day.

The sun travels approximately one degree around the sun each day.

If we have a three variable system, we can solve for any variable by covering up the other two. For example


            distance
rate =       time

If we cover up distance, we get distance = rate * time
If we cover up time, we get
          distance
time =     rate

Appoligies to Ignaz for not seeing their posting of Pythagoras.

[QUOTE=chacoguy420]
The true speed limit can be calculated by adding 4mph to the posted speed for speeds posted less than 50mph or by adding 8mph to speeds posted greater than 50mph.

So, where the posted speed limit is 45, the real speed limit is 49mph.

I’ve been driving 83 on the interstates for year and never even turned a cops head.
[/QUOTE]
In this vein, the true “safe” speed for corners, as advised by the yellow cautionary signs that come before them, can be found by doubling said speed and subtracting 10%. So, using your example, the maximum “safe” speed of a cautionary 45mph curve would be 90-9, or 81 mph.
I’m mostly kidding.

28

If you take all of a man’s veins out of him, and line them up end to end…
…That man would die.

29

Maybe not exactly useful, but interesting if you like Pythagorean triangles; here’s a formula that generates every right triangle with integer sides. Take any two numbers M and N such that M>N>0. If you want only right triangles with the sides reduced to the lowest terms, further specify that M-N is odd, and that M and N have no common factors. Your two numbers then generate the three sides of a right triangle A, B, and C through the following formula:
A=M²-N² B=2MN C=M²+N². For example, M=2 N=1 gives A=3 B=4 C=5

30

Lets go back to basics, shall we?

Reflexive Property of Equality:
a = a

Symmetric Property of Equality:
If a = b then b = a.

Transitive property of Equality:
a = b and b = c, then a = c

What?

31

[QUOTE=Santo Rugger]
The sun travels approximately one degree around the sun each day.
[/QUOTE]

:stuck_out_tongue:

A block of land 208 ft x 208 ft is approximately an acre.

A “section” of land (as in Sixteenth Section) is one square mile.

32

Remember this sequence:

1
1.25
1.6
2
2.5
3.2
4
5
6.4
8

These are 10^0, 10^0.1, 10^0.2,…,10^0.9.

So, for instance, the base ten log of 6.4 is 0.8. Remembering that powers of ten add one to the base ten logs, the log of 640 is 2.8. The log of the number of days in a year must be about 2.55 (the Windows calculator confirms it’s 2.562…).

This sequence is three threads interleaved. One of them is 1, 2, 4, 8. Another is 1.25, 2.5, 5, and 10 would be next (starting the next cycle). The third is 16, 32, 64, and in the next cycle 128, familiar powers of two, though they need a decimal point shift, and the 128 becomes 125, as this isn’t perfect but good to a percent or so. These patterns make it easier to remember the sequence.

With base ten logs you can do all sorts of things. Remembering this sequence as the question rather than the answer means you have powers of ten too. And multiplying these by 2.3 gives you natural logs, which are even more versatile.

33

[QUOTE=Sampiro]

I’ve seen this called "The “Number of Handshakes” Formula but the first word problem I heard that used it involved pictures, so I’ll go with that.

There are eight siblings. They’ve had group shots and individual shots made, but now they want to have a series of pictures made in which each sibling will be pictured with exactly one sibling until ultimately each sibling will have a picture of himself/herself one-on-one with each other sibling (i.e. A will have 7 pictures, one showing him standing with B, then with C, with D, with E, with F, with G, with H, and with I, and B will have a picture made with A, C, D, E, F, G, H, and I, etc.). Each sibling will have 7 pictures of themself in all. How many pictures will need to be made (not including copies for each sibling to have a picture)?
The handshake version is “8 people are in a room. If every person shakes hands with every other person one time, how many handshakes will occur?” A surprising number of people will instantly answer “64”, incidentally.

Since I’m not sure how to write it the way I would in a word processor it may seem clumsy, but the formula for this one is:

In the below X is the number of pictures needed, N = the number of siblings.

X = half the sum of N (N - 1) (I know that’s clumsy looking)

X = half the sum of 8 (7), or half of 56. There will need to be 28 pictures in order to have a shot of each sibling with each sibling.

[/QUOTE]

It seems to me that this would be the summation of 1 to 7 (or 7 choose 2) = 21.

34

[QUOTE=Pseudocode]
It seems to me that this would be the summation of 1 to 7 (or 7 choose 2) = 21.
[/QUOTE]
The summation of 1 to 7 is 28. What’s 7 choose 2?

35

[QUOTE=Santo Rugger]
The summation of 1 to 7 is 28. What’s 7 choose 2?
[/QUOTE]

Sorry, I saw what he did and misremembered summation as 7(6)/2, or n(n-1)/2, instead of n(n+1)/2.

36

if you take someone’s hourly rate of pay and multiply it by 2000 you have a decent approximation of their annual wage. (this obviously works in the other direction as well)

37

Divide centimeters by 5. Multiply result by 2 to get inches.

1"=2.54cm, but 2.5 is close enough for, say, buying clothes from overseas off eBay.

38

[QUOTE=NinetyWt]
A block of land 208 ft x 208 ft is approximately an acre.
[/QUOTE]
Or, for the metrically minded, a block of land 100m x 100m is exactly one hectare.

39

[QUOTE=Santo Rugger]
The summation of 1 to 7 is 28. What’s 7 choose 2?
[/QUOTE]
The number of possible combinations of any two objects in a set of seven, I believe.

40

[QUOTE=Ignatz]
And to be more exact: C = 5/9 (F-32) and conversely, F = 9/5 C + 32
[/QUOTE]

I know that. I was giving the useful approximation for those who can’t work out five-ninths in their head. :stuck_out_tongue: And “Minus forty” is the temperature for which you don’t need to give the scale (use either formula above and substitute F=C).