Mental Math Tricks: Worth It or Parlor Trick?

[ajgrebel]

Are they actually worth looking into, to develop a quicker and deeper understanding of math or are they just a step above parlor tricks?

 

[S_Happens]

I don't know about a deeper understanding, but there are certainly cases where certain shortcuts are useful. Squaring 2 digit numbers or breaking down multiplication of 3+ digit numbers into several operations is very useful when time is a concern, such as during an exam.

I always like to find the shorter way. When you do, it will be pretty obvious which one(s) will be useful by how often you use them and how easy they become. Like stated, squaring 2 digit numbers or multiplication of 2 large numbers is what I encounter the most at school and work, so I certainly think those are useful.

Personally I have no use for squaring 4 or 5 digit numbers.

Were you talking about something else?

 

[Steff196]

Any "trick" would have a mathematical underpinning for why it works and therefore I would consider it useful. The rule of 3 is a good example- if a number's digits add up to something divisible by 3 then the number itself is divisible by three e.g. 141 1+4+1=6 so it is divisible by 3. This is no "trick," there is a mathematical reason for it (I don't feel like writing the proof but I'm sure you could find it on wiki). Also, being able to multiply two and three digit numbers quickly in your head comes in handy on tests (when not allowed to use a calculator).

 

[ajgrebel]

Thanks for all the info. I just wanted to make sure that I wasn't wasting my time on something that would not be benificial beyond the mere parlor tricks.

 

[QuarkCharmer]

That's a good one steff, I will certainly remember that trick. Is there some sort of resource for information like this?

 

[configure]

That's a good one steff, I will certainly remember that trick. Is there some sort of resource for information like this?

If you mean for divisibility, you can refer to http://en.wikipedia.org/wiki/Divisibility_rule

 

[Mark44]

Any "trick" would have a mathematical underpinning for why it works and therefore I would consider it useful. The rule of 3 is a good example- if a number's digits add up to something divisible by 3 then the number itself is divisible by three e.g. 141 1+4+1=6 so it is divisible by 3. This is no "trick," there is a mathematical reason for it (I don't feel like writing the proof but I'm sure you could find it on wiki). Also, being able to multiply two and three digit numbers quickly in your head comes in handy on tests (when not allowed to use a calculator).

There is a similar rule involving 9s. If a number's digits add up to 9 or a multiple of 9, then the number itself is divisible by 9. In times past, people were taught to check their addition by the technique of "casting out 9s."

Here's an example where I have intentionally made an error.

 34
 45
 51
+28
---
168

To check I add up the digits of each of the terms to be added. If the digits of any number form a two-digit number, I can add those digits to produce a single digit. If the digits add up to 9, I can throw it away (that's the casting out part).
34 --> 7
45 --> 9 --> 0
51 --> 6
28 --> 10 --> 1

The four numbers at the left boil down to 7 + 0 + 6 + 1 --> 14 --> 5
The digits in the answer I showed, 168, add to 15 --> 6
Since 5 and 6 aren't equal, I know I have made a mistake. This prompts me to check my addition, where I notice that I carried 2 into the 10's column but I should have carried only 1. After adding the numbers again, I see that the sum is 158, the digits of which add to 5.

The mathematics underlying this is part of number theory, and modular arithmetic. Rewriting a number as the sum of its digits is in effect working in modulo 9, in which all numbers fall into one of nine equivalence classes. Each of these classes represents the remainder when the number is divided by 9. It can be proved that the sum of the digits of any integer is in the same equivalence class as the integer itself.

 

[Steff196]

The mathematics underlying this is part of number theory, and modular arithmetic. Rewriting a number as the sum of its digits is in effect working in modulo 9, in which all numbers fall into one of nine equivalence classes. Each of these classes represents the remainder when the number is divided by 9. It can be proved that the sum of the digits of any integer is in the same equivalence class as the integer itself.

Exactly what I was going to say. Modular arithmetic and number theory give you the ability to use your own tricks whenever you need them (don't try to memorize that wikipedia list). The wikipedia link given doesn't do the theory behind the techniques justice. If you want to learn some similar techniques a good place to start would be to learn the euclidean algorithm and working with modular arithmetic.