Second highest and lowest numbers

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Harte

Senior Member
Messages
4,562
Re: Second highest and lowest numbers

An irrational number, like any other number, when expressed as a decimal has an infinite number of digits to the right of the decimal point. What makes the number irrational is that the digits to the right of the decimal point never repeat.

For example, the number three can be expressed as 3.000... with the zeroes repeating forever. The ratio one third (1/3) can be expressed .33333 with the threes repeating forever. But the number pi, as an irrational number, never has an infinitely repeating digit, or set of digits.

Now, we know pi is something ike 3.1415... on and on forever. We can ignore the 3 to the left of the decimal, which gives us another irrational number0.1415... This is a number between zero and one.

Now, I can't remember pi to 10 or twelve places, so I'm going to make these digits up, mainly just for the purposes of illustration. Lets call our irrational number;
0.141527658926...

Realizing that the digits in this number go on forever, it is easy to see that you could change one digit in this string and still have an irrational number, say change the 4 to a 3:
0.131527658926.

We now have a new irrational number, different from our first one.

We could take this new number and change one digit in it to generate another irrational number, and so on, and then repeat with our original number, only changing a different digit in the first step.

Also, let's take three consecutive digits and change them in an orderly fashion, say the digits 152:

0.131537658926
0.131637658926
0.132637658926.

Just continuing in this fashion gives us a thousand new irrational numbers. Anyone need any further evidence that the irrationals outnumber the rationals? Consider we are doing this by starting with just one irrational number. You could do it with the natural base e or the square root of two, or with any other rational number.

Also, start with a rational number that repeats. say 0.33333.
beginning with one of the 3's, add one to each three in a geometric progression (the second digit 3, the fourth digit 3, the eighth digit 3, and on forever) this turns 1/3 into an irrational number. Now do it again, only add 2 to each three in geometric progression. Now do it adding 3, now 4, ad infinitum. Now repeat the whole thing but use an exponential progression instead (add one to the 2nd, 4th, 16th, 32nd digits, etc.)

I think it's pretty plain now that there are more irrational numbers between zero and one than rational numbers on the entire number line.

Harte
 

thenumbersix

Member
Messages
290
Re: Second highest and lowest numbers

<div class='quotetop'>QUOTE(\"Keroscene\")</div>
This reminds me of something I heard once and was wondering if an awnser was actually possible.

If the temperature is zero degrees out today, and it will be twice as cold tommorrow, what will the temperature be?[/b]

:huh: my head hurts....

My favourite number thingy :

111,111,111 * 111,111,111 = 12345678987654321

I'm sure there something deep and meaningful there, damned if I know what though :D
 

fanavans

Junior Member
Messages
71
Re: Second highest and lowest numbers

Harte, Thanks for the clarification!

Thenumbersix, its kinda like the 11:11 thing.... Which I understand in the least.


What amazes me though is that there are really the same amount of even numbers, odd numbers and numbers... That seems bizzare...
 

Harte

Senior Member
Messages
4,562
Re: Second highest and lowest numbers

<div class='quotetop'>QUOTE(\"fanavans\")</div>
Harte, Thanks for the clarification!

Thenumbersix, its kinda like the 11:11 thing.... Which I understand in the least.


What amazes me though is that there are really the same amount of even numbers, odd numbers and numbers... That seems bizzare...[/b]

To understand the fun little game with elevens, remember what multiplication by 11, or 111 or1,111 is saying:

11*12=
12
*11
12
+120
132
Notice the sum in the central area of this. It is only 12+12 with the second 12 offset, of course because 11 is made up of two "1's", the first being 1 times 10, which is the cause of the offset in the sum.
Because of this, it is easy to remember 11 times anything.

Given a 2 digit number Dd, 11*Dd=D(D+d)d (the parentheses do not indicate multiplication here.)

Given a 3 digit number abc, 11*abc = abc+abco, or

abc
+abco
a(a+B)(b+c)c (again,parentheses for clarity, not to denote multiplication)

Eleven times anything can be remembered by the fact that the result is just the original number added to it's own string of digits offset by one to the left. Using actual numbers, we can say:
11*253=2(2+5)(5+3)3=2783. Note the first and last digits remain unchanged after the multiplication.

A similar rule can be observed for 111, only the offset will be one, then two spaces to the left and gets a bit more complicated. In the extreme example provided above by thenumbersix (a good name for a number theory junkie if there ever was one), the fact that all the digits involved are ones makes the rule of the offset in multiplying transparently obvious. The "ones" in the multiplicand make for an easier vision of exactly what multiplying by a string of one's results in.

There is another rule that can help you remember some of your multiplication table. Any multiple of three will, when reduced by the addition of the constituent digits, reduce to a multiple of the number 3. For example, the number 213: 2+1+3=6=multiple of 3 so 213 is a multiple of three. Also, 315765: 3+1+5+7+6+5=27 (multiple of 3) and 2+7=9 (multiple of 3). This can help you check if a number is prime. If the digits add to a multiple of three in this manner, the number cannot be prime.

As a corollary to this "3" business, note that any multiple of the number 9 will reduce to a multiple of 9, then to 9 itself:
81: 8+1=9
99: 9+9=18, 1+8=9
3573: 3+5+7+3=18, 1+8=9
5895: 5+8+9+5=27, 2+7=9

This is what they called fun in the Mathematics Department.

Harte
Ps. Edited several times to try to get the vertical products to line up right. H
 

fanavans

Junior Member
Messages
71
Re: Second highest and lowest numbers

<div class='quotetop'>QUOTE(\"Harte\")</div>
I know a fairly straightforward way to envision this fact, I'll post it later if anyone is interested.[/b]

I'm interested!

Oh, and the fact was:
There are more irrational numbers between zero and one than there are rational numbers on the number line. A rational number is a number that be expressed as a ratio, you probably know. So the number \"2\" can be represented as 2/1, this set also contains all the fractions you can write (3/4, 88/89, etc.) They are still outnumbered by the irrational numbers, even the ones between zero and one. ?
 

Harte

Senior Member
Messages
4,562
Re: Second highest and lowest numbers

<div class='quotetop'>QUOTE(\"fanavans\")</div>
I'm interested!

Oh, and the fact was:[/b]

Fanavans,
I posted it above.

Harte
 

fanavans

Junior Member
Messages
71
Re: Second highest and lowest numbers

Ok,

I was hoping for something like the infinity hotel story, where the hotel has an infinite number of rooms and an infinite number of guests turns up...

Alas, I cannot remember the rest...

Fanavans
 

StarLord

Senior Member
Messages
3,186
Re: Second highest and lowest numbers

That would be the infinite number of elevators suffer due to the infinite number of buttons being pushed at the infinite number of the same times by the infinite number of room service deliveries bringing an infinite number of Hamburgers and Fries which causes the infinite number of deepfat fryers that made those infinite number of orders of fries to over load which in turn causes the infinite number of circut breakers to malfunction which leads to an infinite number of elevator doors opening at the same time which overloads an infinite number of power stations to go down which causes an infinite number of brownouts just after those infinite number of deepfat fryers catch fire which causes an infinite number of sprinklers to go off which empties a not so infinite lake in about a quantum New York Second.

So, they were right, Hamburgers & Fries are bad for all of yous.
 

fanavans

Junior Member
Messages
71
Re: Second highest and lowest numbers

I'd prefer an infinite number of julia roberts' in ontop of an infinite number of pianos...
 

gl100

Member
Messages
281
Re: Second highest and lowest numbers

While I?m in complete agreement with your Julia Roberts preference, a piano would be infinitely low on my list of places for her to be on top of. ;)
 

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