What is unit , when we speak about ∏?

[pairofstrings]

What is "unit", when we speak about ∏?



The video says that we are taking a 1 unit circle and if we want to find the diameter of this unit circle then we just use the ruler as shown in the video. And now the length of the diameter of this circle is called as PI (∏), which is approximately 3.14 ! And it is constant value with no units!?
My question is:
1. What is this "unit" he is talking about?
2. How much is 1 unit? Can this unit be 1 cm in length?
3. How do we measure the length of this 1 unit?

 

[awkward]

Hi pairofstrings,

As used in the video, "unit" just means 1 unit in some system of measurement. If you are measuring things in centimeters, then it's one centimeter. If you are measuring things in light-years, then it's one light-year.

Yes, pi is a constant with no units, because it is a ratio-- the ratio of a circle's circumference to its diameter.

Note: The video refers to a circle whose diameter is one unit as a "unit circle". More often than not, mathematicians use "unit circle" to mean a circle whose _radius_ is one unit. So beware when you see this phrase used.

 

[pairofstrings]

But why are we dividing circumference by diameter. We can also divide circumference by radius if we want. And why only division, we can also add, subtract or multiply. Right?

 

[awkward]

Yes, we can divide the circumference by the radius if we want, and if we do the result will be about 6.28. But by tradition we call the circumference divided by the diameter "pi". If we divide by the radius then the result will be two times pi.

If we add the add the circumference to the diameter we don't get anything particularly useful. The same goes for subtracting. But if we multiply the circumference by the radius, this turns out (surprise!) to be twice the area of the circle.

 

[pairofstrings]

I have an impression that there is something happening when we think about this general equation.

The general equation is ∏ = Circumference/ Diameter.

That means, if we take any number of circles with different circumferences and divide it by Diameter we are getting the answer as ∏. Which is approximately(?) 3.14.

Now, I understood what your first paragraph meant.

But from what I understand, the second paragraph says this: " if we multiply the circumference by the radius, this turns out (surprise!) to be twice the area of the circle "

Now, from the above discussion, ∏ = C/ D.
Now, we know that circumference C of a circle is C = ∏ x D. But D = 2r.
So, now, C = ∏ x 2r, which is the circumference of a circle. If we multiply C with r,
C x r = (∏ x 2r) x r

We only get the same result, i.e., C = ∏ x 2r.
Okay, so, (∏ x 2r) x r = 2 ( ∏ x r² ). Right?
Okay, I think now I should look at why area of circle is ∏ x r².

Please tell me what's the surprise thing in Fibonacci numbers.
I know that in Fibonacci numbers the sum of last two numbers will determine the next number in the series. But why consider only the last two numbers?

Please tell me what's the surprise thing in PHI ∅.

I know that the ratio is 1: 1.6180... and it's the Golden ratio.
But what are they still calculating about 1.6180... ?
And also when we break open a unit circle on a ruler to measure it's circumference, shouldn't it give the accurate value of the circumference's length using the ruler? What are the scientists still calculating about 3.14... ?

 

[JHamm]

The area of a circle can be found using calculus, probably something not quite up your street yet.
With Fibonacci numbers you COULD add the last three, but that pattern isn't seen much (ever?) in any work so it wouldn't be very helpful or interesting to examine.
The golden ratio is irrational, that means that it has an infinite number of non repeating decimal numbers which cannot be written as a fraction of integers, it is aesthetically appealing in geometry.
You might find it difficult to break open a circle of known radius, but you are welcome to try.

 

[Mark44]

I have an impression that there is something happening when we think about this general equation.

The general equation is ∏ = Circumference/ Diameter.

That means, if we take any number of circles with different circumferences and divide it by Diameter we are getting the answer as ∏. Which is approximately(?) 3.14.

Now, I understood what your first paragraph meant.

When you use colored text, please put some thought into how it will display. The text in "plum" below is almost invisible against a light blue background.

But from what I understand, the second paragraph says this: " if we multiply the circumference by the radius, this turns out (surprise!) to be twice the area of the circle "

Now, from the above discussion, ∏ = C/ D.
Now, we know that circumference C of a circle is C = ∏ x D. But D = 2r.
So, now, C = ∏ x 2r, which is the circumference of a circle. If we multiply C with r,
C x r = (∏ x 2r) x r

We only get the same result, i.e., C = ∏ x 2r.
Okay, so, (∏ x 2r) x r = 2 ( ∏ x r² ). Right?
Okay, I think now I should look at why area of circle is ∏ x r².

Did you have a question about why C*r = 2$\pi$r²?

Suppose you had a square that was d units by d units. Its area would be given by the formula A = d².

The distance from the center of the square to any of the four sides would be d/2. Let's call this r. By simple geometry, we could write the area as A = 4r², which is the same as 4(d/2)².

Now draw a circle of radius r = d/2 inside the square. Since the circle is smaller than the square, we would expect the area of the circle to be less than the area of the square, say about 3/4 of the area of the square. It turns out that the area of the circle is slightly larger than this, namely A = $\pi$/4 (d/2)². The way you usually see this formula is A = $\pi$r², and the details are usually shown in a calculus class.

Please tell me what's the surprise thing in Fibonacci numbers.
I know that in Fibonacci numbers the sum of last two numbers will determine the next number in the series. But why consider only the last two numbers?

Because that's how the Fibonacci sequence is defined; namely, that the nth term in the sequence is obtained by adding the two previous terms together.

Please tell me what's the surprise thing in PHI ∅.

I know that the ratio is 1: 1.6180... and it's the Golden ratio.
But what are they still calculating about 1.6180... ?
And also when we break open a unit circle on a ruler to measure it's circumference, shouldn't it give the accurate value of the circumference's length using the ruler? What are the scientists still calculating about 3.14... ?

I don't understand what you are asking in the last two questions. For the first, what do you mean by "But what are they still calculating about 1.6180... ?"

And for the second, what does this mean? "What are the scientists still calculating about 3.14... ?"

 

[pairofstrings]

Sorry about the colors.

My understanding::

Suppose I have a thread(string). Let's say, it is 1 cm long. If I make a circle by attaching ends of this threads together then the circumference of this circle should be same as the length of this thread. Now I will take ratio of circumference to diameter and it will me a number. Let's call it number1.

Now I did another experiment, but this time I will vary the length of the thread, let's take 2 cms and again I will create a circle with this thread and take the ratio and again I will find that it's a number. Let's call it number 2.

But we found something. The two numbers, number1 and number2 have something to do with ∏. I mean they have something in common. Maybe, number2 is a multiple of number1. And number1 is ∏. Why is ∏ an irrational number of infinite number of digits after the decimal point? Is it because the ratio circumference/diameter is always going to give an irrational number? How about the case when I take the circumference of the circle such a way that when I divide the circumference by the diameter of a circle, it gives me a rational number. Possible? With this case in mind, why the circle only has to be a unit circle? ... Just to get ∏? I mean there must be some reason for considering using a unit circle, which is the question of this thread in some sense.

EDIT:
I found the answers to questions above.
My mistake: I thought, since ∏ = circumference/ diameter, I can take circumference of one circle and diameter of a different circle in the same ∏ equation, which is wrong.
The correct situation is that if you take circumference of a circle then diameter of the same circle should be taken, and if we do that, we always get ∏. The reason why we always get ∏ is, suppose if we increase the circumference of a circle the diameter of this circle also automatically increase. So, yes the observation that there is always a ∏ is correct.

Suppose you had a square that was d units by d units. Its area would be given by the formula A = d².

The distance from the center of the square to any of the four sides would be d/2. Let's call this r. By simple geometry, we could write the area as A = 4r², which is the same as 4(d/2)².

Now draw a circle of radius r = d/2 inside the square. Since the circle is smaller than the square, we would expect the area of the circle to be less than the area of the square, say about 3/4 of the area of the square. It turns out that the area of the circle is slightly larger than this, namely A = ∏/4 (d/2)². The way you usually see this formula is A = ∏r², and the details are usually shown in a calculus class.

A = ∏/4 (d/2)² ? How did you get 4 in the denominator?

My understanding about the Golden ratio::

We know that things are aesthetically appealing in geometry when some pattern is in the ratio 1:1.6180 ...
Please explain the following figure.

My question is: why the value 1.6180 ... is not accurately defined?
Same question could apply to the value of ∏. But I understand, that, if ∏ is a ratio of two other numbers, and their division is yielding an irrational number and hence we are in the process of calculating the exact number of ∏.
So, how about 1.6180 ... ? Why still inaccurate?

I am sorry but a teacher expect a lot from the students. They think students are suppose to figure out "few" things by themselves and I was not taught about these things. They write stuff on the board and give the "definition" and read it. And we are suppose to memorize it. Or do what I am doing now - that's because, I don't like using definitions and formulas without understanding the core concepts.

 

[JHamm]

The golden ratio is defined as $\frac{1+\sqrt{5}}{2}$, as has already been pointed out it is irrational which means you can never fully write it out without using root symbols.
With your questions on $\pi$, dividing the circumference of a circle by its diameter will always give you $\pi$, it doesn't matter how big you make your circle that ratio is always $\pi$.

 

[pairofstrings]

The golden ratio is defined as $\frac{1+\sqrt{5}}{2}$, as has already been pointed out it is irrational which means you can never fully write it out without using root symbols.
With your questions on ∏, dividing the circumference of a circle by its diameter will always give you ∏, it doesn't matter how big you make your circle that ratio is always ∏.

Please explain how did you get the following expression ?

$\frac{1+\sqrt{5}}{2}$ (if you are unable to read the expression, refresh the page)

Yes, I learned the reason why it's always ∏. I edited my original post.

Now, I need answer to above question, and
Post #8: figure explanation.
Post #8: A = ∏/4 (d/2)² ? How did you get 4 in the denominator?

 

[BloodyFrozen]

$\frac{1+\sqrt{5}}{2}$ comes from the definition of the golden ratio.

From Wikipedia:

Two quantities a and b are said to be in the golden ratio φ if:$$\frac{a+b}{a}=\frac{a}{b}=\phi$$

So, if we split apart the first fraction, we get:

$$\frac{a+b}{a}=1+\frac{b}{a}=\phi$$

Since $\phi$ was defined to be also $\frac{a}{b}$, then $\frac{b}{a}=\frac{1}{\phi}$
This leads to:

$$1+\frac{1}{\phi}=\phi$$

Now we multiply both sides by $\phi$.

$$\phi+1=\phi^{2}$$
$$\phi^{2}-\phi-1=0$$
$$\phi=\frac{1\pm\sqrt{5}}{2}$$

We are only considering the positive root, so
$$\phi=\frac{1+\sqrt{5}}{2}\approx 1.6180339887...$$

 

[awkward]

With Fibonacci numbers you COULD add the last three, but that pattern isn't seen much (ever?) in any work so it wouldn't be very helpful or interesting to examine.
[snip]

Actually, the series where you add the last three numbers is called the "tribonacci numbers", or more generally, if you add the last n numbers, it's the "n-step fibonacci numbers". These numbers have applications in probability theory and computer science.

http://mathworld.wolfram.com/Fibonaccin-StepNumber.html

 

[pairofstrings]

How did you get that fraction marked in blue? What that fraction means?

why a+b is not divided by b? I understood the later steps but I need to know how the first step came into existence.

 

[dalcde]

It's part of the definition (so the only explanation of why it is so is because we want it to be so).

 

[JHamm]

If you divide by b instead of a you end up with exactly the same number, this is a naming convention, call them "bert" and "ernie" if you please.

 

[pairofstrings]

You are saying that, if $\frac{a+b}{a}$ = $\frac{a}{b}$ = $\varphi$

Let's solve the above numbers from figure

$\frac{5+3}{5}$ = $\frac{8}{5}$ = 1.6

$\frac{a}{b}$ = $\frac{5}{3}$ $\approx$ 1.6

Which is again an observation made by the mathematicians just like the value of ∏.