# What are the best books/resources for Rapid Math /Arithmetic Hacks?

November 12, 2020 6:36 PM Subscribe

I've always been interested in quick mental math/arithmetic. e.g. Sometimes I find myself checking a bill by adding it up the way we were taught in school. I'm sure there are better ways. What are the best books/resources? Looking for math techniques that can be used often, are easy to remember and easy to use!

I had tons of fun with the Trachtenberg system. I know there are other collections of mental math, but that was my starting place.

posted by Anonymous Function at 10:04 PM on November 12, 2020

posted by Anonymous Function at 10:04 PM on November 12, 2020

The Secrets of Mental Math by Art Benjamin and Michael Shermer.

You can catch Art's mathmagic show from the TED stage from back in 2007.

posted by muddgirl at 10:05 PM on November 12, 2020 [1 favorite]

You can catch Art's mathmagic show from the TED stage from back in 2007.

posted by muddgirl at 10:05 PM on November 12, 2020 [1 favorite]

I grew up with 算得快 "Count it fast", maybe, would be a fair translation?

Let's see what I remember:

1. 37 * 3 is 111, so your 37s times table should be doable with 111, 222, 333, etc. as landmarks.

2. 5 is just 10 divided by 2, so multiplying by 5 is the same as halving the number and adding a 0.

3. Sevenths are a rotating repeating decimal -- 0.142857, 0.285714, 0.428571, 0.571428..., etc.

I'd like to try the Trachtenberg method! I never learned to multiply by abacus but I suspect it's a similar skill.

posted by batter_my_heart at 11:23 PM on November 12, 2020

Let's see what I remember:

1. 37 * 3 is 111, so your 37s times table should be doable with 111, 222, 333, etc. as landmarks.

2. 5 is just 10 divided by 2, so multiplying by 5 is the same as halving the number and adding a 0.

3. Sevenths are a rotating repeating decimal -- 0.142857, 0.285714, 0.428571, 0.571428..., etc.

I'd like to try the Trachtenberg method! I never learned to multiply by abacus but I suspect it's a similar skill.

posted by batter_my_heart at 11:23 PM on November 12, 2020

Oh, hmm, none of that is useful for checking bills. For that, I think you just want old fashioned estimation.

I actually remember the state of CT's "teach to the test" math curriculum trying to teach this: if you round several things down (e.g. 2.25, 2.15, 2.10, 2.10, 2.25 all could be rounded to 2), then you need to add a little extra and expect the sum to be closer to 11 than 10, as if they were all heaping spoonfuls. But if some were rounded up and some down, then you could expect the heaps and divots to cancel one another out.

tbh at the grocery store, I get suspicious if the "% saved" is below 20%. I don't usually buy things at full price, so usually that percentage is 25%-35%; if it isn't, I've either bought something unusual or have mis-scanned myself.

posted by batter_my_heart at 11:33 PM on November 12, 2020 [1 favorite]

I actually remember the state of CT's "teach to the test" math curriculum trying to teach this: if you round several things down (e.g. 2.25, 2.15, 2.10, 2.10, 2.25 all could be rounded to 2), then you need to add a little extra and expect the sum to be closer to 11 than 10, as if they were all heaping spoonfuls. But if some were rounded up and some down, then you could expect the heaps and divots to cancel one another out.

tbh at the grocery store, I get suspicious if the "% saved" is below 20%. I don't usually buy things at full price, so usually that percentage is 25%-35%; if it isn't, I've either bought something unusual or have mis-scanned myself.

posted by batter_my_heart at 11:33 PM on November 12, 2020 [1 favorite]

Cut the knot has a nice collection of tips and tricks with references. As it turns out, squares are surprisingly useful.

posted by kmt at 12:25 AM on November 13, 2020

posted by kmt at 12:25 AM on November 13, 2020

It may be in one of the links above, but I didn't see "casting away 9's" mentioned. For instance:

11.27 + 9.88 + 141.16 = 162.31

To check that you can add 1 + 1 + 2 + 7 + 9 + 8 + 8 + 1 + 4 + 1 + 1 + 6 = 49 = 4

But you can speed that up (instead of going to 49) by collapsing 2 digit numbers every time you get them (like 1 + 1 + 2 + 7 = 11 = 2). So you can do it very quickly!

Also, it's called casting away 9's because if you get 9 you start again (and/or you ignore 9's). Like above, 4 + 7 = 2, and 2 + 9 = 2, and 2 + 8 = 1 and so on. This is because 9 + 3 = 12 = 3, and 9 + 7 = 16 = 7, so 9's become like zeros, and zeros stay zeroes and 8's (for the advanced folks) can be treated like -1 since 8 + 5 = 13 = 4.

It verifies the answer of 162.31 above since 1 + 6 + 2 + 3 + 1 = 9 + 3 + 1 = 3 + 1 = 4 just like all the digits above added up to.

Now, it's not a mathematical proof, it's just used to show *IF THERE IS* a mistake, but if the digits add up there still could be a mistake.

I typically use this to check a bill quickly, such as at a restaurant.

posted by forthright at 5:12 PM on November 13, 2020

11.27 + 9.88 + 141.16 = 162.31

To check that you can add 1 + 1 + 2 + 7 + 9 + 8 + 8 + 1 + 4 + 1 + 1 + 6 = 49 = 4

But you can speed that up (instead of going to 49) by collapsing 2 digit numbers every time you get them (like 1 + 1 + 2 + 7 = 11 = 2). So you can do it very quickly!

Also, it's called casting away 9's because if you get 9 you start again (and/or you ignore 9's). Like above, 4 + 7 = 2, and 2 + 9 = 2, and 2 + 8 = 1 and so on. This is because 9 + 3 = 12 = 3, and 9 + 7 = 16 = 7, so 9's become like zeros, and zeros stay zeroes and 8's (for the advanced folks) can be treated like -1 since 8 + 5 = 13 = 4.

It verifies the answer of 162.31 above since 1 + 6 + 2 + 3 + 1 = 9 + 3 + 1 = 3 + 1 = 4 just like all the digits above added up to.

Now, it's not a mathematical proof, it's just used to show *IF THERE IS* a mistake, but if the digits add up there still could be a mistake.

I typically use this to check a bill quickly, such as at a restaurant.

posted by forthright at 5:12 PM on November 13, 2020

Here is how I actually subtract since I always found borrowing confusing.

Suppose the problem is 232 - 199 (write it the normal vertical way. I just don't know how to make it appear that way in this comment.)

I ask myself 9 + what equals 12? The answer is 3, so I write down 3 and carry the 1 (from 12).

Next, what do I add to 9 + the carried 1 = 10 to get 13? The answer is 3, so I write down 3 and carry the 1 (from 13).

Finally, what do I add to 1 + the carried 1= 2 to get 2? The answer is 0.

So, 033 = 33 is the answer.

Let's do one with lots of 0's where the borrowing would be really confusing.

1000 - 456 = ?

6 + what = 10? 4 and carry the 1.

5 + carried 1 = 6 + what = 10? 4 and carry the 1.

4 + carried 1 = 5 + what = 10? 5. ( notice here there actually is 10 above the 4, so I don't have to think about carrying here).

Answer: 544

Hope this makes sense.

posted by wittgenstein at 6:32 AM on November 14, 2020

Suppose the problem is 232 - 199 (write it the normal vertical way. I just don't know how to make it appear that way in this comment.)

I ask myself 9 + what equals 12? The answer is 3, so I write down 3 and carry the 1 (from 12).

Next, what do I add to 9 + the carried 1 = 10 to get 13? The answer is 3, so I write down 3 and carry the 1 (from 13).

Finally, what do I add to 1 + the carried 1= 2 to get 2? The answer is 0.

So, 033 = 33 is the answer.

Let's do one with lots of 0's where the borrowing would be really confusing.

1000 - 456 = ?

6 + what = 10? 4 and carry the 1.

5 + carried 1 = 6 + what = 10? 4 and carry the 1.

4 + carried 1 = 5 + what = 10? 5. ( notice here there actually is 10 above the 4, so I don't have to think about carrying here).

Answer: 544

Hope this makes sense.

posted by wittgenstein at 6:32 AM on November 14, 2020

I just came across The Universe of Discourse : Testing for divisibility by 19 and an older post The Universe of Discourse : Testing for divisibility by 7. Both are the mathy maths proofs for the divisibility by n tests.

posted by zengargoyle at 12:36 PM on November 22, 2020

posted by zengargoyle at 12:36 PM on November 22, 2020

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posted by exogenous at 8:34 PM on November 12, 2020