Speed of Thinking: Adding Large Numbers Instantly

[Monte_Carlo]

Since I don't have have a formal math background, I'm always under impression that academically trained mathematicians/physicists/computer scientists have superb arithmetic abilities. For example, they can add large numbers on a fly.

Say a physics Ph.D. just got job in a bank. He is a quant. There is a presentation going on. Suddenly, there is a need to add five numbers very fast. And he is new on the job. His boss is in the room. The numbers are below.

$211,043
$229,951
$174,606
$164,342
$164,760


Is it reasonable to expect this fresh math whiz to be able to instantaneously add these numbers? I mean in his head?

 

[proof]

Since I don't have have a formal math background, I'm always under impression that academically trained mathematicians/physicists/computer scientists have superb arithmetic abilities. For example, they can add large numbers on a fly.

Say a physics Ph.D. just got job in a bank. He is a quant. There is a presentation going on. Suddenly, there is a need to add five numbers very fast. And he is new on the job. His boss is in the room. The numbers are below.

$211,043
$229,951
$174,606
$164,342
$164,760


Is it reasonable to expect this fresh math whiz to be able to instantaneously add these numbers? I mean in his head?

not at all. many(most?) mathematicians are horrible at basic arithmetic.

 

[Doc Al]

Since I don't have have a formal math background, I'm always under impression that academically trained mathematicians/physicists/computer scientists have superb arithmetic abilities. For example, they can add large numbers on a fly.

You've been misinformed.

 

[Dr Lots-o'watts]

I was told by two math professors of mine that neither the ability to calculate quickly, nor the ability to solve high school "written problems" were indicative to successful research in math.

"Intuition" was the best quality they came up with at the time. Personally, I believe the ability to maintain interest, motivation and focus on a (math research) project, including extensive study in the relevant field, is the most crucial quality. It has to be stimulating. There is quite a world of difference between being an accountant and being a topologist.

 

[Monte_Carlo]

I acknowledge your responses, and I find it contradictory that some in the field of mathematics say basic abilities are not important.

Perhaps for mathematical research, an ability to count fast is not a virtue. Let's say though, a mathematician is employed in industry. How are they distinguished from the rest of the non-mathematical peers? What is being mathematical then, in common understanding?

 

[Mu naught]

I acknowledge your responses, and I find it contradictory that some in the field of mathematics say basic abilities are not important.

Perhaps for mathematical research, an ability to count fast is not a virtue. Let's say though, a mathematician is employed in industry. How are they distinguished from the rest of the non-mathematical peers? What is being mathematical then, in common understanding?

You think mathematicians spend their time doing things computers can do many millions of times faster?

Mathematicians who are employed in industry spend their time looking for ways to model real-world problems and come up with optimizations and relations. None of this involves crunching numbers - that's what computers are for. In fact, many mathematicians don't even ever use numbers at all, they manipulate symbols. Values mean nothing to most mathematicians.

 

[russ_watters]

I think fast computation has more to do with memory. I can't keep a new 10 digit phone number in my head for more than a few seconds without repeating it over and over and take forever to permanently remember. I'm doing above average as an engineer though. My sister (in finance) knows all of her credit card numbers as well as a host of obscure phone numbers by heart. Her friends call her Rain Woman.

 

[leroyjenkens]

I think fast computation has more to do with memory. I can't keep a new 10 digit phone number in my head for more than a few seconds without repeating it over and over and take forever to permanently remember. I'm doing above average as an engineer though. My sister (in finance) knows all of her credit card numbers as well as a host of obscure phone numbers by heart. Her friends call her Rain Woman.

Heck, even if I constantly repeat a phone number out loud until I can get to some paper to write it down, this is what happens: I'll use a fake phone number as an example.

555-2948
555-2948
555-2948
555-2948
555-2498
555-2498
555-2498 and that's what I write down.

 

[JaredJames]

I'm terrible at mental arithmatic. But with pen an paper I can do pretty well at anything put in front of me from basics to rather complex problems within a reasonable amount of time.

This reminds me of a maths lecturer I had from two years ago. We were doing differentiation and he was running through some of the more complex examples. He put a series of 10 on the board and gave us time to try them. When it came to checking them he could run off answers virtually instantly from his head, which had taken us 5 minutes a piece to do.

However, I find people like this are either really gifted and few and far between or work with them so often it just becomes natural (as with any other task you complete).

 

[chaoseverlasting]

I think it would be reasonable to expect an approximate answer. Roughly around 900,000 would be a decent answer. Depends on person to person though. There are some people who don't use calculators (or use them minimally), they would probably be pretty good at such stuff.

 

[Monte_Carlo]

Ok, so does GRE quantitative score demonstrate math ability? It's not all number crunching of course, but similar? And GRE quantitative doesn't mean much, then why people boast about it?

 

[chaoseverlasting]

Ok, so does GRE quantitative score demonstrate math ability? It's not all number crunching of course, but similar? And GRE quantitative doesn't mean much, then why people boast about it?

From what I've heard, the GRE quant section doesn't really mean much. I haven't given the GRE so I can't really say.

 

[D H]

Since I don't have have a formal math background, I'm always under impression that academically trained mathematicians/physicists/computer scientists have superb arithmetic abilities. For example, they can add large numbers on a fly.

 

[G037H3]

@ OP

a savant is someone who could do that (I also recall some 'mathemagician' with the surname of Young) because savants work in an algorithmic manner, note that being able to do that doesn't mean you're good at math...a savant wouldn't be able to come up with a mathematical proof, for instance

mathematicians think in an abstract/lateral thinking manner

also, with large sums like that, i start from left to right, my thinking being that i get the majority of the answer out of the way off the bat, but you just have to be flexible in changing a previous number when you go to a new row

 

[Doc Al]

One of the rare exceptions that I am personally aware of is Arthur Benjamin, a math professor who is also a lightning mental calculator. (http://www.math.hmc.edu/~benjamin/mathemagics/index.html" ) He's quite amazing! He tells you how he does it, but that doesn't make it any less incredible.

 

[G037H3]

One of the rare exceptions that I am personally aware of is Arthur Benjamin, a math professor who is also a lightning mental calculator. (http://www.math.hmc.edu/~benjamin/mathemagics/index.html" ) He's quite amazing! He tells you how he does it, but that doesn't make it any less incredible.

Ah, that's his name. I thought it was Young for some reason.

 

[G037H3]

Ok, so does GRE quantitative score demonstrate math ability? It's not all number crunching of course, but similar? And GRE quantitative doesn't mean much, then why people boast about it?

the math/quantitative GRE is to filter out students who haven't actually learned much math as undergraduates

 

[Newai]

Heck, even if I constantly repeat a phone number out loud until I can get to some paper to write it down, this is what happens: I'll use a fake phone number as an example.

555-2948
555-2948
555-2948
555-2948
555-2498
555-2498
555-2498 and that's what I write down.

That's just about everyone.

 

[mugaliens]

I was told by two math professors of mine that neither the ability to calculate quickly, nor the ability to solve high school "written problems" were indicative to successful research in math.

"Intuition" was the best quality they came up with at the time. Personally, I believe the ability to maintain interest, motivation and focus on a (math research) project, including extensive study in the relevant field, is the most crucial quality. It has to be stimulating. There is quite a world of difference between being an accountant and being a topologist.

I'd have to say this axiom holds true for excellence in lot of disciplines, from mineral to managerial.

 

[Jamma]

A lot of people have this impression about mathematicians. In fact, whenever there is a sum to be done, like working out how much we have to pay each for a taxi, even my friends always tell me to do it, since I'm a mathematician and should be able to do the sum much more easily because I have a mathematics degree!

In fact, I'm really not very good at mental arithmatic, pretty average really. A lot of people have said the same thing here, so it seems that it is a completely irrelevant factor of whether or not someone is a good mathematician (in fact, it would appear if you just took these posts that it is a detriment, but I'd say in general, mathematicians are probably a little better at metal arithmatic than the average person).

People often think of mathematicians as calculators; sort of a cold machine of logic without much imagination or life. But in fact, the MOST important thing for being a good mathematician is a great imagination, the ability to abstract simple ideas, unravel them and to try to visualise things in a novel and creative way. If you have this, then you will be good at solving a range of problems, all of which need some spark of imagination to solve, even though the way of solving it is purely logical. This imagination is even more important for doing maths research, where you HAVE to think of things in different ways since the problems become more and more difficult, and if you don't do this, then someone will have beaten you to the result anyway.

 

[D H]

IIRC, the speed of thinking is between 5 to 20 mentations per second (a person's Stroud number).

John M. Stroud (1967), "The Fine Structure of Psychological Time," Annals of the New York Academy of Sciences, 138:2, 623-631
http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1967.tb55012.x/abstract

 

[zomgwtf]

I remember in grade 12 calculus we had to do an exercise every morning to improve our mental math. We had to multiple numbers in this chart.

It would be a large chart say 10x10 and along the colmns there were different 3 digit numbers and along the rows there were different 3 digit numbers and you had to multiply them, all 100 in your head. You were given like 10 minutes and yeah...
Not that any of this matters I just thought it was interesting and the discussion was about mental math :tongue:
I do not think that all mathematicians are mental math wizards although they would probably be able to give a very accurate guess. They also would probably be able to work out seemingly difficult problems in their head that is relevant to their research just because the familiarity with the process.

 

[dkotschessaa]

(delurking)

Interesting topic. As part of my preparation for entering back into school for a Math degree one of the things I've actually been working on (besides Pre-calculus, calculus and physics) has been basic arithmetic. I've actually gone and reviewed basic multiplication and division tables and have been learning various arithmetic tricks.

I'm well aware that mathematicians aren't calculators. I have several reasons for doing this:

- the mental exercise
- it's actually very rewarding
- it helps you understand what numbers are and how they work
- Saves time, believe it or not. Calculators are fast, but a person operating a calculator is not necessarily so. Even many problems will at some point require an answer that will require some basic arithmetic, and I don't want to be reaching for a calculators in order to figure out 6 * 8.
- it DOES get bothersome telling people you are into math and then having to explain why you can't figure out the tip at dinner!

If you start learning these little tricks for multiplying, adding, dividing, squaring etc., you really do start to understand numbers in a new way, which I'm very excited about. You also learn to come up with new ways of solving problems - the imagination factor that was mentioned before.

-DaveKA

 

[mugaliens]

I've actually gone and reviewed basic multiplication and division tables and have been learning various arithmetic tricks.

Cool! It's fun, isn't it?

It DOES get bothersome telling people you are into math and then having to explain why you can't figure out the tip at dinner!

I move the bill's decimal place to the left, note the value, then double it. Tip is somewhere between the first and second value, depending on quality of service. At any rate, I always tip $2 as a minimum for a meal.

 

[JaredJames]

At any rate, I always tip $2 as a minimum for a meal.

Those lucky McDonalds staff!

 

[Jamma]

(delurking)

Interesting topic. As part of my preparation for entering back into school for a Math degree one of the things I've actually been working on (besides Pre-calculus, calculus and physics) has been basic arithmetic. I've actually gone and reviewed basic multiplication and division tables and have been learning various arithmetic tricks.

I'm well aware that mathematicians aren't calculators. I have several reasons for doing this:

- the mental exercise
- it's actually very rewarding
- it helps you understand what numbers are and how they work
- Saves time, believe it or not. Calculators are fast, but a person operating a calculator is not necessarily so. Even many problems will at some point require an answer that will require some basic arithmetic, and I don't want to be reaching for a calculators in order to figure out 6 * 8.
- it DOES get bothersome telling people you are into math and then having to explain why you can't figure out the tip at dinner!

If you start learning these little tricks for multiplying, adding, dividing, squaring etc., you really do start to understand numbers in a new way, which I'm very excited about. You also learn to come up with new ways of solving problems - the imagination factor that was mentioned before.

-DaveKA

This probably is a good exercise, but don't spend too much time doing it. In a university degree, there will be very little adding, dividing, subtracting and multiplying to do, especially if you go for pure maths, and there are many other very important things that you can do in preparation.

These little tricks can all be explained algebraically, if you do a course on number theory, some of these things that you've noticed might be brought up.

However, it's probably good if you notice these things earlier, it could give you a (small) headstart. For example, you may have noticed that a number is divisible by 3 when the sum of its digits are also- it's a nice opening exercise to prove this.

These little tricks you notice may be quite helpful, but the best thing you can do with them is this: whenever you think that you notice some nice trick or pattern, try to prove or disprove your general hypothesis formally.

Good luck with the degree!

 

[dkotschessaa]

Great idea, Jamma. I'm spending some time on it now - but that's because I have the time (I'm unemployed) and I'm enjoying it. I've noticed that it's actually helped me with my pre-calc study quite a bit though. Perhaps the usefulness of these things will fade as I get (back) into higher math. Typically I've always done better at math the more advanced I got, so lower level arithmetic was something I wanted to master on a more personal level. But at least on the level of algebra, somewhere, at some point, you're having to add/subtract/multiply/divide to figure out x or something, and it's nice to know the basic math facts to complete the problem rather than reach for a calculator.

Your advice about proofs is great. Another thing, that I forgot to add, is that I think some of these methods might be adapted in some way to higher mathematics. I don't know and I'd like some feedback on that. I don't necessarily mean they have to do with "mental differential equations" (though that'd be cool) but that learning how to manipulate numbers/symbols on a low level could help one devise methods for doing so on a higher level.

There's interesting stuff out there I haven't looked at yet, including http://en.wikipedia.org/wiki/Vedic_mathematics" , and the abacus system (which basically involves practicing with an abacus until you don't need one - (standard learning for children in certain countries, I believe).-Dave KA

 

[Jamma]

Great idea, Jamma. I'm spending some time on it now - but that's because I have the time (I'm unemployed) and I'm enjoying it. I've noticed that it's actually helped me with my pre-calc study quite a bit though. Perhaps the usefulness of these things will fade as I get (back) into higher math. Typically I've always done better at math the more advanced I got, so lower level arithmetic was something I wanted to master on a more personal level. But at least on the level of algebra, somewhere, at some point, you're having to add/subtract/multiply/divide to figure out x or something, and it's nice to know the basic math facts to complete the problem rather than reach for a calculator.

Your advice about proofs is great. Another thing, that I forgot to add, is that I think some of these methods might be adapted in some way to higher mathematics. I don't know and I'd like some feedback on that. I don't necessarily mean they have to do with "mental differential equations" (though that'd be cool) but that learning how to manipulate numbers/symbols on a low level could help one devise methods for doing so on a higher level.

There's interesting stuff out there I haven't looked at yet, including http://en.wikipedia.org/wiki/Vedic_mathematics" , and the abacus system (which basically involves practicing with an abacus until you don't need one - (standard learning for children in certain countries, I believe).


-Dave KA

Yeah, so these are the sort of things that I am talking about. It probably isn't that useful to learn these sorts of things (although the Trachtenberg system may be helpful), but it's a nice exercise to prove why they are true.

For example, why is the general multiplication rule for the Trachtenberg system true? Well, just write down your numbers a and b as a=a_0.10^0+a_1.10^1+...+a_n.10^n and b similarly and write out a few of the first terms of the multiplication of these two. So you can then easily see that in the i'th digit, you will get the sum of numbers
a_0.b_i+a_1.b_(i-1)+...+a_i.b_0
i.e. the sum of all multiples of a_x and b_y where x and y sum to i. You can do this for i=0 up to i={the highest digit of a and b}. So after each computation you just need to add this new digit to your old number (of course, if the last one "spilled over" into the i'th digit you need to take this into account).

A lot of these other proofs work in the same way. Hope this makes sense.

 

[Acuben]

Glad to know I am not the only one