The value of Pie is 22/7 . You can not equate this to an exact decimal.
π was proven to be transcendental (not the root of any polynomial) and hence non-rational (not of the form a/b with a and b integers) by Von Lindemann in 1882. The most accessible proof today of this fact today is through the Lindemann–Weierstrass Theorem (see en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem). It takes quite a bit of mathematical maturity to understand the proof, but it is not ridiculously hard.
You can copy and paste from wikipedia but unless you know the basics of what you are talking about it is not much help. Pie = 22/7.
Of course Pie is an irrational number which basically means the numbers after the decimal point never end. Pie and 22/7 are approximately equal.
@ANNONYMOU5 said in #1:
> We know pi is about 3big, and will never end, meaning no matter how precise we try to define this using numbers only, it will never be the same as pi.
> But how do we know that? How and why do professors know that pi will never end?
TL;DR: Because π is irrational and irrational numbers all have infinite, never repeating decimal expansions.
Because π is irrational (and this can be mathematically proved: en.wikipedia.org/wiki/Proof_that_π_is_irrational). It's not part of the rational numbers. Rational numbers (like 1, 3, 1/2, -4, -21/7, 17/[-1000], etc.) are all numbers that are simply quotients of two integers: r = q/p, where q, p are integers and r is a rational number.
All rational numbers have either a finite decimal representation (like 1/8 = 0.125) or they have an infinite decimal representation with only finitely many decimal places that are not infinitely repeating, like 1/3 = 0.3 (3 repeating) or 1/7 = 0.142857 (142857 repeating) or 1/12 = 0.083 (3 repeating).
Irrational numbers however all have infinite, non repeating decimal representations. That means that you cannot write their decimal form down in full.
Why is that? Well, let's start by thinking about what a decimal representation really is. It's just a sum of different fractions (i.e. rational numbers). Example: 1/8 = 0.125 = 0/1 + 1/10 + 2/100 + 5/1000
We have three (and only three) cases:
1) Finite decimal representations
2) Infinite decimal representations with only finitely many non repeating digits before they infinitely repeat
3) Infinite decimal representations which never repeat (i.e. which have infinitely many non repeating digits)
Caveat: Cases 1) and 2) are really the same, because finite decimal representations are just the special case of infinite decimal representations that end in 9 repeating (in a general base N number system the same is true for N-1 repeating, but we usually work in base 10). For example: 1/8 = 0.125 = 0.1249 (9 repeating).
For the same reason 0.9 (9 repeating) = 1.
The first two belong to rational numbers, the last one to irrational numbers. Why?
Let's start with something easy:
In case 1) we simply have an addition of finitely many rational numbers. Rational numbers are closed under addition, meaning that any two rational numbers can be added together and the result will again be a rational number:
Let x and y be rational numbers. Then:
x + y = q/p + u/v = (q*v)/(p*v) + (u*p)/(v*p) = (q*v + p*u)/(p*v) = z, where q, p, u and v are all integers.
Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
So that means that in case 1) we will of course obtain a rational number. All numbers with finite decimal representation are rational.
In case 2) it is a bit harder to prove that it can only correspond to rational numbers. At first everything is just like in case 1). We simply add finitely many non repeating (at least not infinitely repeating) rational numbers. For example:
1/12 = 0.083 (3 repeating) = 0.08 + 0.003 (3 repeating) = 0/1 + 0/10 + 8/100 + residual term
What's a bit harder about case 2) is getting to grips with this residual term, as it's an infinite series rather than a finite sum of rational numbers. In the case of 1/12 the residual term is:
0.003 (3 repeating) = 3/1000 + 3/10000 + 3/100000 + ... = 3/(10^3) + 3/(10^4) + 3/(10^5) + ... = ∑_(k=3)^∞ 3/(10^k),
where ∑ is the greek upper case letter sigma which denotes a summation.
Luckily every term in this infinite series is strictly less than 1 (3/1000 is less than one and all other terms are even less than that) and therefore we can evaluate the series. It converges to a finite value. It's simply part of a geometric series: en.wikipedia.org/wiki/Geometric_series#Closed-form_formula
We can thus replace it with its closed form:
∑_(k=3)^∞ 3/(10^k) = ∑_(k=0)^∞ 3/(10^k) - ∑_(k=2)^∞ 3/(10^k)
= 3/(1 - 1/10) - (3 + 3/10 + 3/100)
= 3/(9/10) - 3.33
= 1/(3/10) - 3.33
= 10/3 - 333/100
= 1000/300 - 999/300
= 1/300
Let's (for the sake of our sanity) check whether or not that residual term plus 0.08 still returns 1/12:
8/100 + 1/300 = 24/300 + 1/300 = 25/300 = 5/60 = 1/12
So it all works out nicely! Notice that our residual term is also a rational number! That's not a coincidence. The closed form of the geometric series is simply the quotient of two rational numbers. And rational numbers are also closed under division (well, you cannot divide by zero, but the closed form of the geometric series never allows division by zero because |r| is strictly less than 1)! So this always gives a rational number. This proofs that case 2) also always corresponds to a rational number.
That Case 3) can only correspond to irrational numbers can be shown by process of elimination. Here's the proof sketch Wikipedia gives:
"To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, there can never be a remainder greater than or equal to m. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats."
This shows that any rational number produces a decimal expansion corresponding to cases 1) or 2). "If r is rational, its decimal expansion corresponds to either case 1) or case 2)."
I've shown above that any decimal expansion corresponding to cases 1) or 2) belongs to a rational number. "If a decimal expansion corresponds to either case 1) or case 2), the number r represented by the decimal expansion is rational."
Both of these statements combined give:
A number's decimal representation corresponds to case 1) or 2) if and only if the number is rational.
Conversely, irrational numbers cannot have decimal representations corresponding to case 1) or 2), which leaves only case 3) for irrational numbers.
And that's how mathematicians know that the decimal expansion of π never terminates and never repeats.
> And there must be some kind of pattern, but I was told there isn't.
Why would a pattern be necessary? A string of truly random numbers also doesn't have a pattern. Why should an irrational and transcendental number like π have a (repeating) pattern?
I'm not a mathematician though, so take anything I say with a grain of salt.
> We know pi is about 3big, and will never end, meaning no matter how precise we try to define this using numbers only, it will never be the same as pi.
> But how do we know that? How and why do professors know that pi will never end?
TL;DR: Because π is irrational and irrational numbers all have infinite, never repeating decimal expansions.
Because π is irrational (and this can be mathematically proved: en.wikipedia.org/wiki/Proof_that_π_is_irrational). It's not part of the rational numbers. Rational numbers (like 1, 3, 1/2, -4, -21/7, 17/[-1000], etc.) are all numbers that are simply quotients of two integers: r = q/p, where q, p are integers and r is a rational number.
All rational numbers have either a finite decimal representation (like 1/8 = 0.125) or they have an infinite decimal representation with only finitely many decimal places that are not infinitely repeating, like 1/3 = 0.3 (3 repeating) or 1/7 = 0.142857 (142857 repeating) or 1/12 = 0.083 (3 repeating).
Irrational numbers however all have infinite, non repeating decimal representations. That means that you cannot write their decimal form down in full.
Why is that? Well, let's start by thinking about what a decimal representation really is. It's just a sum of different fractions (i.e. rational numbers). Example: 1/8 = 0.125 = 0/1 + 1/10 + 2/100 + 5/1000
We have three (and only three) cases:
1) Finite decimal representations
2) Infinite decimal representations with only finitely many non repeating digits before they infinitely repeat
3) Infinite decimal representations which never repeat (i.e. which have infinitely many non repeating digits)
Caveat: Cases 1) and 2) are really the same, because finite decimal representations are just the special case of infinite decimal representations that end in 9 repeating (in a general base N number system the same is true for N-1 repeating, but we usually work in base 10). For example: 1/8 = 0.125 = 0.1249 (9 repeating).
For the same reason 0.9 (9 repeating) = 1.
The first two belong to rational numbers, the last one to irrational numbers. Why?
Let's start with something easy:
In case 1) we simply have an addition of finitely many rational numbers. Rational numbers are closed under addition, meaning that any two rational numbers can be added together and the result will again be a rational number:
Let x and y be rational numbers. Then:
x + y = q/p + u/v = (q*v)/(p*v) + (u*p)/(v*p) = (q*v + p*u)/(p*v) = z, where q, p, u and v are all integers.
Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
So that means that in case 1) we will of course obtain a rational number. All numbers with finite decimal representation are rational.
In case 2) it is a bit harder to prove that it can only correspond to rational numbers. At first everything is just like in case 1). We simply add finitely many non repeating (at least not infinitely repeating) rational numbers. For example:
1/12 = 0.083 (3 repeating) = 0.08 + 0.003 (3 repeating) = 0/1 + 0/10 + 8/100 + residual term
What's a bit harder about case 2) is getting to grips with this residual term, as it's an infinite series rather than a finite sum of rational numbers. In the case of 1/12 the residual term is:
0.003 (3 repeating) = 3/1000 + 3/10000 + 3/100000 + ... = 3/(10^3) + 3/(10^4) + 3/(10^5) + ... = ∑_(k=3)^∞ 3/(10^k),
where ∑ is the greek upper case letter sigma which denotes a summation.
Luckily every term in this infinite series is strictly less than 1 (3/1000 is less than one and all other terms are even less than that) and therefore we can evaluate the series. It converges to a finite value. It's simply part of a geometric series: en.wikipedia.org/wiki/Geometric_series#Closed-form_formula
We can thus replace it with its closed form:
∑_(k=3)^∞ 3/(10^k) = ∑_(k=0)^∞ 3/(10^k) - ∑_(k=2)^∞ 3/(10^k)
= 3/(1 - 1/10) - (3 + 3/10 + 3/100)
= 3/(9/10) - 3.33
= 1/(3/10) - 3.33
= 10/3 - 333/100
= 1000/300 - 999/300
= 1/300
Let's (for the sake of our sanity) check whether or not that residual term plus 0.08 still returns 1/12:
8/100 + 1/300 = 24/300 + 1/300 = 25/300 = 5/60 = 1/12
So it all works out nicely! Notice that our residual term is also a rational number! That's not a coincidence. The closed form of the geometric series is simply the quotient of two rational numbers. And rational numbers are also closed under division (well, you cannot divide by zero, but the closed form of the geometric series never allows division by zero because |r| is strictly less than 1)! So this always gives a rational number. This proofs that case 2) also always corresponds to a rational number.
That Case 3) can only correspond to irrational numbers can be shown by process of elimination. Here's the proof sketch Wikipedia gives:
"To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, there can never be a remainder greater than or equal to m. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats."
This shows that any rational number produces a decimal expansion corresponding to cases 1) or 2). "If r is rational, its decimal expansion corresponds to either case 1) or case 2)."
I've shown above that any decimal expansion corresponding to cases 1) or 2) belongs to a rational number. "If a decimal expansion corresponds to either case 1) or case 2), the number r represented by the decimal expansion is rational."
Both of these statements combined give:
A number's decimal representation corresponds to case 1) or 2) if and only if the number is rational.
Conversely, irrational numbers cannot have decimal representations corresponding to case 1) or 2), which leaves only case 3) for irrational numbers.
And that's how mathematicians know that the decimal expansion of π never terminates and never repeats.
> And there must be some kind of pattern, but I was told there isn't.
Why would a pattern be necessary? A string of truly random numbers also doesn't have a pattern. Why should an irrational and transcendental number like π have a (repeating) pattern?
I'm not a mathematician though, so take anything I say with a grain of salt.
Oh my goodness, my head hurts. @Thalassokrator ,where did you learn mathematics? What is pie even for? I only know so how to calculate stuff using it, but never really knew how to calculate pie or what pie means.
I checked with the wikjpedja page and I can now proudly share all my knowledge about this: "_". I didn't get anything.
What are integers?
Does ∑ basically mean the same as x+y=z?
I did understand what the three cases are, tho
>x + y = q/p + u/v = (q*v)/(p*v) + (u*p)/(v*p) = (q*v + p*u)/(p*v) = z, where q, p, u and v are all integers.
Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
I could understand the rule with rational+rational = rational, but in what way does this prove it?...
>Why would a pattern be necessary? A string of truly random numbers also doesn't have a pattern. Why should an irrational and transcendental number like π have a (repeating) pattern?
My thought was that if you just randomly throw numbers , you'd somewhen get a repeating pattern. And because we use the decimal system, there are only 10 letters the number could be made of...
God you wrote so much and I understood so little
I checked with the wikjpedja page and I can now proudly share all my knowledge about this: "_". I didn't get anything.
What are integers?
Does ∑ basically mean the same as x+y=z?
I did understand what the three cases are, tho
>x + y = q/p + u/v = (q*v)/(p*v) + (u*p)/(v*p) = (q*v + p*u)/(p*v) = z, where q, p, u and v are all integers.
Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
I could understand the rule with rational+rational = rational, but in what way does this prove it?...
>Why would a pattern be necessary? A string of truly random numbers also doesn't have a pattern. Why should an irrational and transcendental number like π have a (repeating) pattern?
My thought was that if you just randomly throw numbers , you'd somewhen get a repeating pattern. And because we use the decimal system, there are only 10 letters the number could be made of...
God you wrote so much and I understood so little
The reason is because we cant measure the area of curved surfaces in an exact manner. We dont have a tool for that. We actually only can get the precise area of squared or rectangle shapes.
What we do instead is to draw squares or rectangles in the rounded surface, then draw more tinier square rectangles in the blank spaces and sum the area of all.
On a circle, what happens is that once all the area is "covered", if you zoom in, you will still find blank places where you have to continue doing so. Then zoom in again, etc.
Doesnt matter how many times you repeat the process, you cant get all the area of a curved surface with squares or rectangles. You can keep zooming in for eternity and you still wont be able to do so.
Pi is just a way to avoid doing all the squares and rectangle drawing, but it can only give you an approximation, a very good one, but never exact.
This doesnt just happens with circles, but any curved surface as well. Thats like the whole point and the reason for calculus being invented.
What we do instead is to draw squares or rectangles in the rounded surface, then draw more tinier square rectangles in the blank spaces and sum the area of all.
On a circle, what happens is that once all the area is "covered", if you zoom in, you will still find blank places where you have to continue doing so. Then zoom in again, etc.
Doesnt matter how many times you repeat the process, you cant get all the area of a curved surface with squares or rectangles. You can keep zooming in for eternity and you still wont be able to do so.
Pi is just a way to avoid doing all the squares and rectangle drawing, but it can only give you an approximation, a very good one, but never exact.
This doesnt just happens with circles, but any curved surface as well. Thats like the whole point and the reason for calculus being invented.
The first proof I learnt on proving the irrationality of pi was Niven's proof and its still my personal favourite.
I am too lazy to type but a bit of single variable calculus and trigonometry and here you go -- en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Niven's_proof
The proof has many things not well derived and expects the reader to fill the gaps.
@Alientcp
In other coordinate systems, we use different infinitesimal units for calculating area.
Say in polar coordinates, we use sectors to calculate integrals.
I am too lazy to type but a bit of single variable calculus and trigonometry and here you go -- en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Niven's_proof
The proof has many things not well derived and expects the reader to fill the gaps.
@Alientcp
In other coordinate systems, we use different infinitesimal units for calculating area.
Say in polar coordinates, we use sectors to calculate integrals.
@ANNONYMOU5 said in #16:
> Oh my goodness, my head hurts. Thalassokrator ,where did you learn mathematics? What is pie even for? I only know so how to calculate stuff using it, but never really knew how to calculate pie or what pie means.
Pi is simply the constant ratio of any circle's circumference and its diameter. It's the same for every circle, no matter the size (because the circumference is determined by the diameter, the ratio of the two doesn't change).
If you have a wheel of diameter 1 (for example 1 metre across), then rolling the wheel along a straight line exactly once (one 360º revolution) will cover a distance of exactly Pi (metres).
One revolution of a wheel that's 1 metre across moves the wheel by about 3.1415 metres: en.wikipedia.org/wiki/File:Pi-unrolled-720.gif
> I checked with the wikjpedja page and I can now proudly share all my knowledge about this: "_". I didn't get anything.
Don't feel bad. It's my bad. Wikipedia pages about maths are often very technical, sorry for not using something more accessible as a source. Then again, the proofs for pi's irrationality are all fairly technical (from the standpoint of the kind of mathematics that's usually taught in schools). You need at least differential and integral calculus for some of them (and other prerequisites for other proofs). Mathematical proofs are also often very dense, economical (they use as little space as possible) and therefore hard to read anyone who isn't a mathematician.
> What are integers?
Numbers like 0, 1, 2, -4, -7, 2000, -39875243, 19, 55, -3 are integers.
To review: Natural numbers are all counting numbers, the numbers we all first learn as a child: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... all the way to infinity (although infinity is a concept, not a number)
Integers include all natural numbers. But they also include the number 0. And the negative versions of all natural numbers. So they also include -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, ... all the way to negative infinity (which also is a concept, not an actual number).
Does that answer your question? :-)
> Does ∑ basically mean the same as x+y=z?
A bit more than just that. It's a notation for summation: en.wikipedia.org/wiki/Summation#Capital-sigma_notation
Example: ∑_(k=0)^10 k = 0 + 1 + 2 + 3 + 4 + 5 +6 + 7 + 8 + 9 + 10 = 55
Some people also write sum_(k=0)^10 k = 55 and that's the same.
k is the "Index of summation". It increases by one with each step, until it reaches the upper bound. The bounds are written below and above the ∑ symbol. The (k=0) means that you start your index of summation at 0. The 10 above the ∑ symbol is the upper bound. That means you stop at index 10. The term behind the ∑ symbol is what you're actually summing up. In this case it's very easy, you just sum up the indices of summation k themselves. So you go from k=0 to 1 in the next step, 2 in the next and so on until you arrive at the upper bound 10.
That's how this weird notation works.
Another example:
∑_(k=0)^6 k^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 0 + 1 + 4 + 9 + 16 + 25 + 36 = 91
Here you sum up the square numbers k^2: en.wikipedia.org/wiki/Square_number
Another example:
∑_(k=0)^4 (1/2)^k = (1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 + (1/2)^4 = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 31/16
Here you sum up the powers of one half.
> I did understand what the three cases are, tho
>
> Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
>
> I could understand the rule with rational+rational = rational, but in what way does this prove it?...
All it shows is that finite decimal expansions (like 1/8 = 0.125) can only come from rational numbers. And that infinitely long rational expansions with finitely many digits that don't infinitely recur, followed by a sequence of periodically recurring digits that extend out to infinity (like 1/12 = 0.08333333333333333333333333333333333333333 ... and 3s all the way to infinity) can also only come from a rational number.
The finite decimal expansion (case 1) can be understood as a special case of the infinitely repeating decimal expansion in case 2). It's just that the sequence of periodically recurring digits is all zeros. 1/8 = 0.125 = 0.1250000000000000000000000000000000000 ...
Makes sense?
There's an "If and only if" (en.wikipedia.org/wiki/If_and_only_if) relation between a number's rationality and the case 2) nature of its decimal expansion. If it's rational, it has to have a case 2) decimal expansion. And vice versa, if it has a case 2) decimal expansion, it HAS TO be rational.
Therefore irrational numbers (numbers that are not rational) cannot have periodic decimal expansions (case 2)) and must have case 3). Irrational numbers must have decimal expansions that extend out to infinity and never repeat periodically.
> My thought was that if you just randomly throw numbers , you'd somewhen get a repeating pattern. And because we use the decimal system, there are only 10 letters the number could be made of...
Good thinking! Of course you get finite sequences of numbers in the decimal expansion of Pi that come up more than once (in fact, they come up an infinite amount of times, if I recall correctly).
So you might have something like π = 3.1415.....123456789........123456789....123456789......
The finite sequence "123456789" turns up several times (in fact infinitely many times, if I recall correctly) in the decimal expansion of π. But the spacing between these occurrences is not regular. Maybe the first occurrence is from the 100th decimal place to the 109th decimal place. But then the second occurrence is between the 510th decimal place and the 519th decimal place. But the third occurrence happens much faster, from the 531th to the 540th decimal place. Then you have to wait more than 10000 decimal places until you see that particular sequence of numbers again, let's say from the 10875th decimal place to the 10884th decimal place.
So you are right, you can find finite sequences of numbers that show up over and over again in the decimal expansion of π. But they don't show up periodically. They show up randomly. There are arbitrarily long sequences of one number repeated over and over again in the decimal expansion of π. You might find π = 3.1415................82437500000000000000000000000000000.....
And get real exited because you thing π might be rational after all.
But at some point (after finitely many zeros) the string of zeros always ends, crushing your dreams and aspirations and you're back to seemingly random numbers. It might continue:
π = 3.1415................82437500000000000000000000000000000.....000000013489327598.....
There's no regular periodic pattern in there. If there were, pi would be a rational number. But it's been proved to be irrational.
> God you wrote so much and I understood so little
Don't beat yourself up, my explanation might not have been that great. An actual mathematician would probably be able to give you a better explanation. Also keep in mind that understanding the irrationality of pi is not particularly easy. Maths is hard.
Sorry that this post wasn't actually any shorter than my first post!
> Oh my goodness, my head hurts. Thalassokrator ,where did you learn mathematics? What is pie even for? I only know so how to calculate stuff using it, but never really knew how to calculate pie or what pie means.
Pi is simply the constant ratio of any circle's circumference and its diameter. It's the same for every circle, no matter the size (because the circumference is determined by the diameter, the ratio of the two doesn't change).
If you have a wheel of diameter 1 (for example 1 metre across), then rolling the wheel along a straight line exactly once (one 360º revolution) will cover a distance of exactly Pi (metres).
One revolution of a wheel that's 1 metre across moves the wheel by about 3.1415 metres: en.wikipedia.org/wiki/File:Pi-unrolled-720.gif
> I checked with the wikjpedja page and I can now proudly share all my knowledge about this: "_". I didn't get anything.
Don't feel bad. It's my bad. Wikipedia pages about maths are often very technical, sorry for not using something more accessible as a source. Then again, the proofs for pi's irrationality are all fairly technical (from the standpoint of the kind of mathematics that's usually taught in schools). You need at least differential and integral calculus for some of them (and other prerequisites for other proofs). Mathematical proofs are also often very dense, economical (they use as little space as possible) and therefore hard to read anyone who isn't a mathematician.
> What are integers?
Numbers like 0, 1, 2, -4, -7, 2000, -39875243, 19, 55, -3 are integers.
To review: Natural numbers are all counting numbers, the numbers we all first learn as a child: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... all the way to infinity (although infinity is a concept, not a number)
Integers include all natural numbers. But they also include the number 0. And the negative versions of all natural numbers. So they also include -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, ... all the way to negative infinity (which also is a concept, not an actual number).
Does that answer your question? :-)
> Does ∑ basically mean the same as x+y=z?
A bit more than just that. It's a notation for summation: en.wikipedia.org/wiki/Summation#Capital-sigma_notation
Example: ∑_(k=0)^10 k = 0 + 1 + 2 + 3 + 4 + 5 +6 + 7 + 8 + 9 + 10 = 55
Some people also write sum_(k=0)^10 k = 55 and that's the same.
k is the "Index of summation". It increases by one with each step, until it reaches the upper bound. The bounds are written below and above the ∑ symbol. The (k=0) means that you start your index of summation at 0. The 10 above the ∑ symbol is the upper bound. That means you stop at index 10. The term behind the ∑ symbol is what you're actually summing up. In this case it's very easy, you just sum up the indices of summation k themselves. So you go from k=0 to 1 in the next step, 2 in the next and so on until you arrive at the upper bound 10.
That's how this weird notation works.
Another example:
∑_(k=0)^6 k^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 0 + 1 + 4 + 9 + 16 + 25 + 36 = 91
Here you sum up the square numbers k^2: en.wikipedia.org/wiki/Square_number
Another example:
∑_(k=0)^4 (1/2)^k = (1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 + (1/2)^4 = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 31/16
Here you sum up the powers of one half.
> I did understand what the three cases are, tho
>
> Integers are closed under addition and multiplication, therefore q*v + p*u is an integer and p*v is also an integer. Therefore z is a quotient of two integers, i.e. a rational number.
>
> I could understand the rule with rational+rational = rational, but in what way does this prove it?...
All it shows is that finite decimal expansions (like 1/8 = 0.125) can only come from rational numbers. And that infinitely long rational expansions with finitely many digits that don't infinitely recur, followed by a sequence of periodically recurring digits that extend out to infinity (like 1/12 = 0.08333333333333333333333333333333333333333 ... and 3s all the way to infinity) can also only come from a rational number.
The finite decimal expansion (case 1) can be understood as a special case of the infinitely repeating decimal expansion in case 2). It's just that the sequence of periodically recurring digits is all zeros. 1/8 = 0.125 = 0.1250000000000000000000000000000000000 ...
Makes sense?
There's an "If and only if" (en.wikipedia.org/wiki/If_and_only_if) relation between a number's rationality and the case 2) nature of its decimal expansion. If it's rational, it has to have a case 2) decimal expansion. And vice versa, if it has a case 2) decimal expansion, it HAS TO be rational.
Therefore irrational numbers (numbers that are not rational) cannot have periodic decimal expansions (case 2)) and must have case 3). Irrational numbers must have decimal expansions that extend out to infinity and never repeat periodically.
> My thought was that if you just randomly throw numbers , you'd somewhen get a repeating pattern. And because we use the decimal system, there are only 10 letters the number could be made of...
Good thinking! Of course you get finite sequences of numbers in the decimal expansion of Pi that come up more than once (in fact, they come up an infinite amount of times, if I recall correctly).
So you might have something like π = 3.1415.....123456789........123456789....123456789......
The finite sequence "123456789" turns up several times (in fact infinitely many times, if I recall correctly) in the decimal expansion of π. But the spacing between these occurrences is not regular. Maybe the first occurrence is from the 100th decimal place to the 109th decimal place. But then the second occurrence is between the 510th decimal place and the 519th decimal place. But the third occurrence happens much faster, from the 531th to the 540th decimal place. Then you have to wait more than 10000 decimal places until you see that particular sequence of numbers again, let's say from the 10875th decimal place to the 10884th decimal place.
So you are right, you can find finite sequences of numbers that show up over and over again in the decimal expansion of π. But they don't show up periodically. They show up randomly. There are arbitrarily long sequences of one number repeated over and over again in the decimal expansion of π. You might find π = 3.1415................82437500000000000000000000000000000.....
And get real exited because you thing π might be rational after all.
But at some point (after finitely many zeros) the string of zeros always ends, crushing your dreams and aspirations and you're back to seemingly random numbers. It might continue:
π = 3.1415................82437500000000000000000000000000000.....000000013489327598.....
There's no regular periodic pattern in there. If there were, pi would be a rational number. But it's been proved to be irrational.
> God you wrote so much and I understood so little
Don't beat yourself up, my explanation might not have been that great. An actual mathematician would probably be able to give you a better explanation. Also keep in mind that understanding the irrationality of pi is not particularly easy. Maths is hard.
Sorry that this post wasn't actually any shorter than my first post!
What Pi value is used in scientific calculator? How many decimal places?
This topic has been archived and can no longer be replied to.