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Homer
05-12-2005, 04:18 PM
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

istewart
05-12-2005, 04:22 PM
That's fairly cool, actually.

mostsmooth
05-12-2005, 04:24 PM
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

chaas4747
05-12-2005, 04:29 PM
[ QUOTE ]
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

Was waiting for this.

Homer
05-12-2005, 04:30 PM
[ QUOTE ]
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

40*34 + 9 equals I think 1369. I guess your point is that it doesn't always make things much easier and you're right. It just got me thinking about possible shortcuts of which I am unaware.

Porcupine
05-12-2005, 04:30 PM
Unless I'm missing something>

I'd make a = 3 to square 37, so 34*40 + 9 = 1369.

Thanks Homer!

edit: Homer responded the same time I did, but I think that 34*40 + 9 (basically 34*4 + 9) is easier to do quickly in your head than 37*37.

GuyOnTilt
05-12-2005, 04:31 PM
[ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

a = 3. 40 x 34 = 340 x 4 = 1360. 1360 + 9 = 1369.

GoT

GuyOnTilt
05-12-2005, 04:34 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

Was waiting for this.

[/ QUOTE ]

It still works and serves its purpose of making it such easier to do in your head. Normally if I had to do this in my head I'd do 900 + 210 + 259, but Homer's way is much faster and simpler.

GoT

swolfe
05-12-2005, 04:41 PM
[ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

use a = 13

50*24 + 169 = 1369

EDIT: i picked 13 because 50*24 is instantaneous whereas i'd have to think a little about 40*34

Stuey
05-12-2005, 04:53 PM
[ QUOTE ]
Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]

I want a book also or a website that has examples of math systems. Years ago I knew a cool system using visual place holders ect. it was simple but had it's uses. I will try to find it when I have more time.

Morgan O'Mally
05-12-2005, 04:53 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

Was waiting for this.

[/ QUOTE ]

It still works and serves its purpose of making it such easier to do in your head. Normally if I had to do this in my head I'd do 900 + 210 + 259, but Homer's way is much faster and simpler.

GoT

[/ QUOTE ]

My boy is Wicked Smart!

BradyC
05-12-2005, 05:02 PM
The only one I remember you probably already know... Quick way to multiply 11 x any two-digit number... just add the two digits together and put the remainder in the middle. Ex: 11 x 23... 2 + 3 = 5 put the five in the middle 253... and if the middle number is two digits just carry it over to the left number like 11 x 58... 5 + 8 = 13 put the 3 in the middle and carry the one over the 5 which gives you 638. I know it's basic but I am noobish.

Josh W
05-12-2005, 05:07 PM
I had a math prof. in college who was faster at arithmetic than I was. He's the only person I've ever met who was. He's published books, had infomercials, and calls himself the "Mathemagician". Seeing him mulitply 5-digit number is a 20 seconds is wicked.

And humbling.

Anyways, Homer, your trick was one I figured out for myself in 7th grade. In 6th grade, I was really dumb. See, I sorta figured out then that:

1^3 + 2^3 + 3^3 + ....N^3 = (1+2+3+...n)^2

But I never tried to prove it in 6th grade. I just kept making n one increment larger, and seeing if it still held true. I got up to like 73, then decided it must be true for all numbers. I had a buddy of mine, Nick Harris, do one half of the equation, and I'd do the other half.

The 'trick' you mention is kinda funny, because I've always had n^2 memorized up to n = 50. So it was always just the opposite for me. If I wanted to multiply 41*53, I'd square 47 then subtract 36 for 2173. (although, given those two numbers now, I'd go (53*40) + 53....).

I thought if I could ever memorize the squares up to 100^2, I'd be able to multiply any two digit numbers within maybe 4 seconds.

But then, I made an even greater discovery.

Women.

(they are even tougher to figure out...if you want a real challenge...)

Josh

Jeff W
05-12-2005, 05:08 PM
Simple one, but you can use the thinking to devise other tricks.

Divide any number by 9:

1/9=.1111_
2/9=.2222_
3/9=.3333_
...
9/9=.9999_=1(They are equivalent)
10/9=9/9+1/9=1.1111_
11/9=9/9+2/9...
...

Not sure how useful it is. My physics advisor is the U.S. physics olympiad coach and knows a ton of these tricks. I will ask him to recommend a book or something on this.

JaBlue
05-12-2005, 05:10 PM
kind of interesting but i utilize the distributive property much faster in my head when I'm doing these kind of calculations.

gumpzilla
05-12-2005, 05:20 PM
There's a segment in one of the Feynman books - I think it's the Los Alamos from Below section of Surely You're Joking - that talks about Feynman trying to compete with Hans Bethe at speedy mental arithmetic. A technique very similar to this is mentioned, I believe for squaring numbers near 50.

One thing that strikes me is that fast approximate multiplication might have been somewhat easier in the past when people actually used logarithm tables.

DrPublo
05-12-2005, 05:39 PM
[ QUOTE ]
[ QUOTE ]
While teaching some boring Algebra stuff today (FOIL and factoring) I came up with a math shortcut that I thought was kind of cool. Anyway, one problem was to expand (x-2)(x+2) (it is x^2 - 2x + 2x - 4, or x^2 - 4). My brain started to wander and I was thinking that means x^2 = (x-2)(x+2) + 4. So, if you wanted to square a number like 98, you could do 100*96, then add 4 to get 9604. In fact, you could generalize the formula:

x^2 = (x-a)(x+a) + a^2

So, if you wanted to square 94, you could make a = 6 and do 100*88 + 36 = 8836.

Anyway, does anyone know of any other shortcuts like this or of a book I could get to learn how to solve problems more quicky in my head. It doesn't really serve any purpose other than to entertain myself, but I'm curious if there is something like this out there.

[/ QUOTE ]
square 37 for me using your shortcut

[/ QUOTE ]

37^2 = 40*34 + 9
= 1360+9 = 1369

Right?

The Doc

incognito
05-12-2005, 05:49 PM
Funny, I was just going through my old Netscape bookmarks file yesterday and found this:

http://www.math.hmc.edu/funfacts/allfacts.shtml

There are a few tricks of the type you're looking for in there.

Rotating Rabbit
05-12-2005, 05:49 PM
Interesting, I'd do it 900+210+210+49. I'm curious that you sum the last two already.

partygirluk
05-12-2005, 05:54 PM
Click on me (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=exchange&Number=1690563&Fo rum=f20&Words="71%20%2A%2068"&Searchpage=0&Limit=25&Main=1686344&Search=true&wh ere=bodysub&Name=&daterange=1&newerval=1&newertype =y&olderval=&oldertype=&bodyprev=1#Post1690563)

Homer
05-12-2005, 06:07 PM
Heh, that's pretty cool. I never read that post since it was the last in the thread.

<font color="white">Alright, that's a lie. The minute I saw your post I decided to wait 3 months and claim the idea as my own to ascend to my rightful throne as lord of the OOT math nerds.</font>

tbach24
05-12-2005, 06:13 PM
That triangle with all the numbers is pretty cool. Pretty useful too

lu_hawk
05-12-2005, 06:14 PM
you can take a common way to solve squares and go one step further and make it easy to multiply numbers that aren't the same but are close to each other using an anchor. for example, to square 47 you square 50 to get 2500, subtract 3*100, and then add 9. so you get 2209. and you used the formula:

x^2 = a^2 + (x-a)(2a) - (x-a)^2

'a' can be anything but you pick a multiple of 10 to make it easy to do in your head. someone else referred to the way feynman described how to do squares near 50 in your head and this is feynman's way, but describing it in this formula shows that the number doesn't need to be near 50. so for example, to get the square of 38 in your head:

40^2=1600
subtract 2*80=160 and you get 1440
add 2^2=4 and you get 1444

but the formula above is specific to squares you can go back a step and make it easy to multiply numbers that are close but not the same. so if you want to multiply x times y, and 'a' is your anchor number:

x*y = a^2 + (x-a)*a + (y-a)*a + (x-a)(y-a)

yes this seems complicated but it's the same as feyman's way of squaring but you just add 2 steps. so to multiply 48*45 in your head:

50^2=2500
subtract 2*50=100 and get 2400
subtract 5*50=250 and get 2150
add 2*5=10 and get 2160.

it might take some practice but this is the best way i have found, and after awhile you don't have to think about it too much.

mmbt0ne
05-12-2005, 06:49 PM
Mulitply any 2-digit number ab by 11 and it's a(b+a)b

Example:

12*11 = 1(1+2)2 = 132
17*11 = 1(1+7)7 = 187
39*11 = 3(3+9)9 = 429

Luzion
05-12-2005, 07:09 PM
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

Jazza
05-12-2005, 07:11 PM
most of you know the shortcuts for figuring out if a whole number is divisible by:

2 - if it's even
3 - if the sum of the digits is divisble by 3
4 - if the last two digits are divisible by 4
5 - if the last digit is 0 or 5
6 - if it's divisible by 2 and 3 (duh)
8 - if the last 3 digits are divisible by 8
9 - if the sum of the digits is divisible by 9

but no one ever told me one for 7, and i thought that was bullshit, so i found it myself:

write the number in question backwards

under this number write the number 132645132645132645...

now multiply each digit with the number below it, then add them all together

if this number is divisible by 7, then the original is

so, lets say you wanted to know if 27351223 was divisible by 7, well...

2*1+7*3+3*2+5*6+1*4+2*5+2*1+3*3=84, and 84 is divisible by 7, so 27351223 is divisible by 7

also not that you can't figure out if your new number is divisible by 7, repeat the procss:

4*1+8*3=28 which is divisible by 7, so 84 is too

this methods kinda blows though, it's not as easy as the other ones, which is probably why they don't teach it

jason_t
05-12-2005, 07:25 PM
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

Luzion
05-12-2005, 07:30 PM
[ QUOTE ]
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

[/ QUOTE ]

well hey, no one ever said it was an original shortcut... /images/graemlins/grin.gif

jason_t
05-12-2005, 07:32 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

[/ QUOTE ]

well hey, no one ever said it was an original shortcut... /images/graemlins/grin.gif

[/ QUOTE ]

Gauss came up with it when he was five.

gumpzilla
05-12-2005, 07:37 PM
[ QUOTE ]

Gauss came up with it when he was five.

[/ QUOTE ]

I wonder if this story is apocryphal or not. But at any rate, if this is the old Gauss learns his teacher good, he was ten, I believe.

tbach24
05-12-2005, 07:37 PM
[ QUOTE ]
Gauss came up with it when he was five.

[/ QUOTE ]

I love this story. Is it really true?

Luzion
05-12-2005, 07:37 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

[/ QUOTE ]

well hey, no one ever said it was an original shortcut... /images/graemlins/grin.gif

[/ QUOTE ]

Gauss came up with it when he was five.

[/ QUOTE ]

I guess you are familar with the story of how one of his teachers gave the same problem to his class to occupy them for awhile, and he solved it right away using the method I just explained. Blah... /images/graemlins/tongue.gif

jason_t
05-12-2005, 07:42 PM
[ QUOTE ]
[ QUOTE ]
Gauss came up with it when he was five.

[/ QUOTE ]

I love this story. Is it really true?

[/ QUOTE ]

Well, ever since E.T. Bell's book Men of Mathematics it is a major part of mathematical folklore.

tbach24
05-12-2005, 07:46 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Gauss came up with it when he was five.

[/ QUOTE ]

I love this story. Is it really true?

[/ QUOTE ]

Well, ever since E.T. Bell's book Men of Mathematics it is a major part of mathematical folklore.

[/ QUOTE ]

I think I read it in some book about all these physics dudes. Hmm, En was the name or something. Lousy book. That's when I told my teacher I wasn't doing any more physics work and that I would be occupying all my time working on fantasy baseball. I rule.

jason_t
05-12-2005, 07:47 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

[/ QUOTE ]

well hey, no one ever said it was an original shortcut... /images/graemlins/grin.gif

[/ QUOTE ]

Gauss came up with it when he was five.

[/ QUOTE ]

I guess you are familar with the story of how one of his teachers gave the same problem to his class to occupy them for awhile, and he solved it right away using the method I just explained. Blah... /images/graemlins/tongue.gif

[/ QUOTE ]

Yeah, there are some other pretty remarkable stories about Gauss. Another story goes like this: his father owned a business. When he was three, Gauss was watching his father compute taxes and was just standing there constantly correcting his father. That's wrong, nope that's wrong, etc.

When he was 15, he discovered the statement of the prime number theorem which explains the distribution of the primes (although failed to prove it) and when he was 17 he proved the fundamental theorem of algebra (every polynomial has a root) which had been unproven for hundreds of years.

TStoneMBD
05-12-2005, 07:53 PM
while we are on the subject, there is an easy way to do square root. for instance, if you wanted to find the square root of 9 you could do the following formula in your head:

a= your number^2
b= a(2)
c= b(2)
d= c(7)/b^2

ab^2+cd/(b/3^2)(cd/abc)^2=x

x = 3

Luzion
05-12-2005, 07:54 PM
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Heres an easy way to add up a sequence of numbers. For example, lets say from 1 to 100.
1 + 2 + 3 + 4 + 5 + 6 ..... + 100 = ?

The sum of the first and last number, which is 1 + 100 = 101.
The sum of the second and second to last number, which is 2 + 99 = 101.
The sum of the third and third to last number, which is 3 + 98 = 101.

Get the pattern? Summation of two numbers that equal 101 occurs 50x in this sequence. 1+100, or 2+99, or 3+98, or 4+97, or 5+96...

1 + 2 + 3 + 4 + 5 + 6 + 7 .... + 100 = 101 x 50 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 x 5 = 55

Summation of

1 to 1000 = 1001 x 500 = 50050
17 to 9862 = 9879 x 4923 = 48,634,317

Its a neat idea, and could be fairly useful...

[/ QUOTE ]

Good job Gauss.

[/ QUOTE ]

well hey, no one ever said it was an original shortcut... /images/graemlins/grin.gif

[/ QUOTE ]

Gauss came up with it when he was five.

[/ QUOTE ]

I guess you are familar with the story of how one of his teachers gave the same problem to his class to occupy them for awhile, and he solved it right away using the method I just explained. Blah... /images/graemlins/tongue.gif

[/ QUOTE ]

Yeah, there are some other pretty remarkable stories about Gauss. Another story goes like this: his father owned a business. When he was three, Gauss was watching his father compute taxes and was just standing there constantly correcting his father. That's wrong, nope that's wrong, etc.

When he was 15, he discovered the statement of the prime number theorem which explains the distribution of the primes (although failed to prove it) and when he was 17 he proved the fundamental theorem of algebra (every polynomial has a root) which had been unproven for hundreds of years.

[/ QUOTE ]

Lots of mathematicians seem to have interesting stories. I remember in high school when I got bored in math class, I would flip through the book and read the short biographies on mathematicians that they would have every couple of sections. I think I recall reading the same stories of Gauss correcting his dads computations and the story of adding up 1 to 100 in class.

jason_t
05-12-2005, 08:17 PM
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Yeah, there are some other pretty remarkable stories about Gauss. Another story goes like this: his father owned a business. When he was three, Gauss was watching his father compute taxes and was just standing there constantly correcting his father. That's wrong, nope that's wrong, etc.

When he was 15, he discovered the statement of the prime number theorem which explains the distribution of the primes (although failed to prove it) and when he was 17 he proved the fundamental theorem of algebra (every polynomial has a root) which had been unproven for hundreds of years.

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Lots of mathematicians seem to have interesting stories. I remember in high school when I got bored in math class, I would flip through the book and read the short biographies on mathematicians that they would have every couple of sections. I think I recall reading the same stories of Gauss correcting his dads computations and the story of adding up 1 to 100 in class.

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My personal favorite is about G.H. Hardy. Hardy was an English mathematician who worked in classical analysis and analytic number theory. One of the problems he devoted tremendous effort to is the Riemann Zeta problem. This problem is still unsolved today, as it has been for ~150 years. It's almost surely the most famous unsolved math problem. You may have heard of it as it's been covered in the NY Times a few times in the past two years, PBS has done a series on it and there are two popular math books published in the past two years on it. The Clay Mathematics Institute is offering one million dollars to the first mathematician to solve it. Hardy liked to travel. He took a trip to Denmark once to see Harold Bohr (mathematician brother of the physicist Niels Bohr). He was to travel back to England by sea but the North Sea is notoriously dangerous and the forecast was particularly bad. Hardy was a devout athiest but believed that God had a personal grudge against Hardy(!). Hardy believed this so firmly that he believed God wanted Hardy to have no glory. So Hardy wrote a postcard to his friend J. Littlewood saying "Riemann Zeta hypothesis. I have proven it. Details when I return. Regards, G.H." The idea being that the mathematical community would believe Hardy but God wouldn't cause Hardy's ship to sink because Hardy would die with the glory of having the world believe he solved the Riemann Zeta problem.

Jeff W
05-12-2005, 08:34 PM
Hahahaha, that is among the greatest anecdotes I've ever heard.

Luzion
05-12-2005, 08:47 PM
I agree. Great story. /images/graemlins/grin.gif

kyro
05-13-2005, 02:26 AM
I do this kinda crap when I'm bored. The one Homer came up with is something I figured out when I was 13 or so. There's a bunch that I read in some book, like squaring numbers where the first two digits are the same and the second two digits add up to ten.

39 * 31
(3*(3+1))(9*1)
(12)(09)
1209

RocketManJames
05-13-2005, 03:10 AM
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Unless I'm missing something&gt;

I'd make a = 3 to square 37, so 34*40 + 9 = 1369.

Thanks Homer!

edit: Homer responded the same time I did, but I think that 34*40 + 9 (basically 34*4 + 9) is easier to do quickly in your head than 37*37.

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I think that overcounting then subtracting then re-adding might be simpler, but I could be wrong. Or, undercounting then adding then re-adding.

Examples:

37*37 = 40*40 - 2*(3*40) + 3*3 (uses 3 as the "difference" number)

42*42 = 40*40 + 2*(2*40) + 2*2 (uses 2 as the "difference" number)

In Edit: I understand that this is basically the same FOIL method as Homer's, but I think that breaking it down like this, while it requires 3 steps, is simpler due to the simpler numbers.

-RMJ