Math Facts
September 1, 2016 11:22 AM   Subscribe

Beyond multiplication tables, what sort of mathematical knowledge is useful to have memorized? Also, what alternative methods have been of help for you?

I've been working my way through Khan Academy sans calculator, so tips and tricks for solving without getting into ridiculous numbers would be helpful. For instance, factoring numbers when solving division problems and canceling out.

I know that understanding is more important than just rote memorization, but surely there are some things worth knowing. Not pi, I assume, but maybe conversions, squares, primes?

'Foreign' practices are of considerable interest to me as well, learning lattice multiplication and Austrian subtraction has done me a lot of good. I tend towards the messier side, so anything that makes it harder to make foolish errors is appreciated.
posted by Trifling to Education (31 answers total) 31 users marked this as a favorite
I would say for "real-world" applications, it's good to know geometric formulas (area/perimeter/circumference calculations of various 2- and 3-dimensional shapes), as well as how to calculate angles in triangles using trigonometry (sine/cosine/tangent).
posted by brainmouse at 11:26 AM on September 1, 2016 [7 favorites]

My son does these math competitions called "number sense." It's basically a test of math calculations where you have to do them all in order and you can't use any kind of scratch paper or marks - you do it in your head and write down the answer. So he's ridiculously good at this kind of thing, it's really kind of neat to watch.

Anyway - I think a lot of the calculations that are involved in the test are useful (some are not, like doing roman numerals math in your head). He had to learn it all of course - there are tons of tricks and methods and stuff.

This is a kind of long-winded way of saying - there are lots of websites and printed material to help your kids (or yourself) learn number sense stuff. I'd recommend checking that out.
posted by RustyBrooks at 11:33 AM on September 1, 2016 [1 favorite]

Magic Number Nine?
posted by PuppetMcSockerson at 11:37 AM on September 1, 2016 [1 favorite]

Cross multiplying. As in 4 is to 5 as 6 is to x
(4/5=6/x -> 4x=30 -> x=30/4 -> x=7.5) I use this all the time to scale things up/down with the same proportions.
posted by sexyrobot at 11:41 AM on September 1, 2016 [8 favorites]

It's often handy to remember that 210 ≈ 103.
posted by kickingtheground at 11:53 AM on September 1, 2016 [1 favorite]

The most practical math tool I learned in engineering school was how to keep track of units. It's a simple technique (write down everything and then cancel stuff out with appropriate conversion factors) but it has built-in error-detection, making conversion mistakes all but impossible regardless how complicated the calculation may be.
posted by cardboard at 11:59 AM on September 1, 2016 [13 favorites]

If you're doing anything close to code, powers of 2.
posted by PMdixon at 12:05 PM on September 1, 2016 [4 favorites]

Squares are good to have memorized because they open up a lot of other basic math, because:
(x-y) * (x+y) = x^2-y^2

Therefore, since you've memorized that 17^2 = 289, you also know:
18*16 = (17+1) * (17-1) = (17^2) - (1^2) = 289 - 1 = 288
21*13 = (17+4) * (17-4) = (17^2) - (4^2) = 289 - 16 = 273

Any two even or two odd numbers are x+y and x-y, so knowing those squares can help you do that math more quickly than trying to do "21 times 13 is 21 times 10 plus 13 times 1..."
posted by Etrigan at 12:10 PM on September 1, 2016 [6 favorites]

The decimal value of common fractions, especially those with single digit denominators (e.g. 1/8 is 0.125, 3/8 is 0.375 -- these facts are surprisingly useful!).

The value of 2^X where X is whatever number you feel like. It's helpful to hear a number and know immediately whether it can be reached only by doubling the number two repeatedly.
posted by Mrs. Pterodactyl at 12:21 PM on September 1, 2016 [5 favorites]

It's good to know what the square root of two and three are, to a few decimal places.
It's good to know this in the same way it's good to know several of the special right triangles, along with the basic trig table that goes with them.

These can help you out in all kinds of real-world situations, where taking a cosine is how you project to get the length of something in terms of the length of something else.

These are areas where many construction workers (carpenters, masons, etc) are better and faster with mental math than many accountants and other number folk.
posted by SaltySalticid at 12:23 PM on September 1, 2016 [1 favorite]

Not pi, I assume

Actually it is pretty useful to know that pi is approximately 3.14

This will allow you to do things like calculate the area of a circle.

You don't need to memorize the other digits though.

As to what else is useful to know, much of that depends on what you are wanting to do with math. As SaltySalticid points out, some people are faster at certain types of mental math than others -- that's because accountants don't generally have a need to figure out the length of a diagonal. So, whatever you tend to use more is probably something you want to memorize.
posted by yohko at 12:30 PM on September 1, 2016 [2 favorites]

Learn to factor with primes and stop at the middle.

So, divide by two until you can't divide by two anymore. Then divide by three, five, seven, whatever.

You stop at the middle because if you divide 30 by 2 and get 15, there will be no factors larger than 15. You are wasting your time to go higher. Your largest and smallest factors pair up.

So, 30 /2 =15 /3= 5

Your prime factors for 30 are 2, 3 and 5. You can build all your other factors from cross multiplying the prime factors, such as 10, 6 and 15.
posted by Michele in California at 12:35 PM on September 1, 2016

Percentages. Both for tipping and figuring out store discounts. The quickest trick for any round percentage is to take 10% of the number and then multiply. So a 20% tip on a $14.86 check would be ~ $3. 60% off a 38.99 pair of shoes is ~ $24.

In fact, that reminds me. Another great math tip for practical adult life is that rounding is fine. You will never need to leave a $2.87 tip or be positive that the discount on those shoes works out to $22.90. It's never that dire. Just don't round when doing DIY projects.

It shocks me how many people can't calculate a tip when it's literally "figure out what 10% is and then double that".
posted by Sara C. at 12:54 PM on September 1, 2016 [3 favorites]

I didn't expect to have to remember the order of operations ever again, but now I'm doing a tiny bit of computer programming, and there it is. And lo and behold, I did not forget Please Excuse My Dear Aunt Sally, which I think I memorized in fourth grade.
posted by ArbitraryAndCapricious at 1:34 PM on September 1, 2016 [3 favorites]

Not pi, I assume

Actually it is pretty useful to know that pi is approximately 3.14

This will allow you to do things like calculate the area of a circle.

posted by yohko at 2:30 PM on September 1

In my years working in construction I used 3.14 all. the. damn. time. when cutting holes in sheet metal, in drywall, in wood, in acoustical ceilings.

Also, the 3 4 5 method to make sure something is square (ie when building decks or foundations or just whatever). It can be 6 8 10 or 12 16 20, whatever, but multiples of 3 4 5. (simple vid demonstrating it). A piece of string -- braided is better by far than twisted -- in the toolbox is pretty much a must for any carpenter. (A box of dental floss is good, too, for various applications; super-light string but strong enough to be pulled tight.)

These aren't "math facts" probably, more like arithmetic facts. But you'll want them in your pocket if you find yourself making a living with a hammer in your hand.
posted by dancestoblue at 2:09 PM on September 1, 2016 [1 favorite]

it's not directly mathematical but really handy in learning/communicating math to know what common greek letters are called on sight, I definitely cannot think in terms of "this squiggly thing" and "the other thing" but have often tried.
posted by Nomiconic at 2:14 PM on September 1, 2016

Trig functions defined on a right triangle: soh cah toa
sine = opposite over hypotenuse
cosine = adjacent over hypotenuse
tangent = opposite over adjacent

Powers of 2 up to 2^16.
posted by Bruce H. at 2:17 PM on September 1, 2016

If you're learning calculus it is useful to have a few basic trigonometric identities down, so that you don't have to referring when simplifying a problem.
posted by peacheater at 2:21 PM on September 1, 2016

One trick I use in my head for estimating (when I'm not near a suitable calculator) is, when finding odd fractions of something, to multiply numerator and denominator by something to bring it closer to an intuitive fraction.

I might have no idea what 7/25 is, but 28/100 is trivial.

Or go the other way: 15/24 is just shy of 2/3 - divide the denominator until you get to one you know (like Michele's factoring), then roughly divide the numerator the same way (in my head, here, I pushed the numerator up one to 16, divided by 2 thrice, got 2/3-1/24, and 1/24 is pretty close to 1/25, which is 4%, or 0.04, so we're somewhere around 0.62. That path is a better choice for a fraction you can't approximate by multiplying.)

I also find myself using SOHCAHTOA when I'm trying to estimate height at a distance, but then I usually need a calculator for the trigonometric step.
posted by WasabiFlux at 2:54 PM on September 1, 2016 [1 favorite]

How about "X percent of y is the same as y percent of x". Example: ask someone for 12% of 25 and they may have to think about it, but 25% of 12 is easy.
posted by klausman at 2:55 PM on September 1, 2016 [20 favorites]

Somewhere on Metafilter in the last six months, it was pointed out that NASA used 3.1416 for pi. It's more accurate than you think - the error is around .000007.
posted by wittgenstein at 3:03 PM on September 1, 2016 [1 favorite]

I memorized the factorials, and sums of strings of numbers. (The product and sum of 1 to n). It also astounds people if you know the decimal expansion of 1/7, 2/7, ect.
posted by Valancy Rachel at 3:59 PM on September 1, 2016

My life is sort of unusual, but every couple years I'm glad I know the quadratic formula
posted by the marble index at 4:28 PM on September 1, 2016

I use the binomial series to do quick multiplications, divisions, powers, and roots in my head that are good to two or three significant figures.

The binomial series lets you raise a number close to 1 to any power by doing
(1+x)n = 1 + nx + ½ n (n-1) x² + ... =~ 1 + nx
Each term in the series has another power of x. If x is small (i.e. your number 1+x is "close to one"), the higher powers of x are really small, and you can ignore them. Especially useful are the negative powers:
(1+x)-n =~ 1 - nx
This means, for example, that (y/0.95) is approximately 5% larger than y: "divide by 0.95"= multiply by (1-0.05)-1 = multiply by 1.05. Dividing by 0.95 sounds hard, but adding five percent is not so bad. (You'd add ten percent by moving the decimal over one place; five percent is half as much.)

With practice you can see more patterns. For example in my class today I had on a whiteboard the ratio
4.2 / 1.8
This is approximately 4/2 = 2. The numerator is 5% bigger than 4, and the denominator is 10% smaller than 2, so the ratio is 5% + 10% = 15% bigger than 2: that gives 4.2/1.8 =~ 2.30. Actual answer is 2.333, so the error starts in the third significant figure. Figuring this out is usually quicker than reaching for a calculator.
posted by fantabulous timewaster at 4:44 PM on September 1, 2016

MIT football cheer:

Hold that line,
Hold that line,
posted by charlesminus at 4:50 PM on September 1, 2016

If you are talking test taking, be very familiar with the common right triangles: e.g. 2-3-5 and 1,2, sqrt(3).

Trig and first year calculus have a bunch of common substitutions.
posted by SemiSalt at 4:52 PM on September 1, 2016

Trig functions defined on a right triangle: soh cah toa
sine = opposite over hypotenuse
cosine = adjacent over hypotenuse
tangent = opposite over adjacent

Better than "soh cah toa" is "Some one's herding cows and horses through our attic". I heard that once, many years ago, never forgot it.

The precedence in arithmetic for evaluating expressions, BODMAS (brackets order division multiplication addition subtraction), comes up a lot but everyone knows that one.

I've found it very useful to know sqrt 2 is 1.4142135. Comes up all t.he time (A4 etc paper sizes, aperture numbers, length of the diagonal of a unit square)
posted by w0mbat at 11:54 PM on September 1, 2016

The math trick I return to again and again is comparing percentages.

12/23 = x/100

(12 x 100)/23 = x

I use it to compare prices per ounce in the store, I use it to convert from one unit to another, etc.
Cross multiply and divide.
posted by Foam Pants at 4:24 PM on September 2, 2016

OK, I'm going to post my shortcuts for averages.

To find the average of some numbers clumped in a narrow range, you can pick a "benchmark", average the deviations from the benchmark, and add the result to the benchmark. Example: Say you want the average of 66, 62, 71, 58, and 63. These are all around 60, so make that your benchmark. The deviations are 6, 2, 11, -2, and 3. Average deviation is 4, so the average of the original numbers is 64.

Relatedly, to check an average, use the average itself as a benchmark; the deviations (positive and negative) from the average must cancel out. If they don't, you know how far you need to adjust your average -- see previous trick!

To keep a running average of several numbers in your head, it helps to know that the nth number "pulls" the average by 1/n of its distance from the previous average. For example, if you have an average of 80 on four exams, then a score of 95 on the fifth exam will raise your average by one-fifth of (95–80), which is 3. New average: 83.
posted by aws17576 at 8:04 PM on September 2, 2016 [1 favorite]

Speaking of checking your work, casting out nines is handy (but not foolproof -- it generally won't catch errors resulting from transposing two digits, for example).
posted by aws17576 at 8:09 PM on September 2, 2016

Math but not arithmetic: Geometric constructions. I sometimes find it handy to draw a parallel or perpendicular line (or an angle bisector, or midpoint, or whatever) using just a string-and-pencil "compass."
posted by sibilatorix at 4:43 PM on September 3, 2016

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