Why is pi not a definite number?

The original question was: If the diameter of a circle is a fixed number, say 10cm, why does Pi go on for billions of numbers even though it extrapolates from the diameter’s value? Why is it not a definite number?

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Mathematician: Basically, a diameter of 10cm leads to a circumference that is an irrational number. There is a very simple relationship between diameter and circumference, given by circumference = diameter. It just so happens that the proportionality constant is an irrational number, which means that it has no [finite length] patterns in its digits that repeat forever. This occurs because, well, that is a property that circles have. is however a “definite” number, in the sense that it is a single fixed number that is well defined. It’s just that since its digits don’t have a single pattern, there is no way to write them out nicely. Note that when you write 10 this is the same thing as 10.00000000000000…. with an infinite number of zeros. We can write this just as 10 because it’s understood that this implies that all other digits after the decimal point are 0. So, it’s not that 10 doesn’t have an infinite number of digits (like ), it’s just that those digits are zero (unlike ) which makes it easy to come up with a notation for ten (namely, 10).

The definition of Pi. This image is from http://www.mathsisfun.com/geometry/circle.html

Circle

A circle is easy to make: Draw a curve that is "radius" away from a central point. And so: All points are the same distance from the center.

You Can Draw It Yourself Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!

Radius, Diameter and Circumference

The Radius is the distance from the center to the edge.
The Diameter starts at one side of the circle, goes through the center and ends on the other side.
The Circumference is the distance around the edge of the circle.
And here is the really cool thing:

When you divide the circumference by the diameter you get 3.141592654...
which is the number π (Pi)

So when the diameter is 1, the circumference is 3.141592654...

We can say:

Circumference = π × Diameter

Example: You walk around a circle which has a diameter of 100m, how far have you walked?

Distance walked = Circumference = π × 100m = 314m (to the nearest m)

Also note that the Diameter is twice the Radius:

Diameter = 2 × Radius

And so this is also true:

Circumference = 2 × π × Radius

Remembering

The length of the words may help you remember:

• Radius is the shortest word
• Diameter is longer (and is 2 × Radius)
• Circumference is the longest (and is π × Diameter)

Definition

The circle is a plane shape (two dimensional): And the definition of a circle is:

The set of all points on a plane that are a fixed distance from a center.

Area

The area of a circle is π times the radius squared, which is written:

A = π × r²

Or, in terms of the Diameter:

A = (π/4) × D²

Example: What is the area of a circle with radius of 1.2 m ?

A = π × r²
A = π × 1.2²
A = π × (1.2 × 1.2)
A = 3.14159... × 1.44 = 4.52 (to 2 decimals)

Area Compared to a Square

A circle has about 80% of the area of a similar-width square.
The actual value is (π/4) = 0.785398... = 78.5398...%

Names

Because people have studied circles for thousands of years special names have come about.

Nobody wants to say "that line that starts at one side of the circle, goes through the center and ends on the other side" when a word like "Diameter" would do.

So here are the most common special names:

A line that "just touches" the circle as it passes by is called a Tangent.

And a part of the circumference is called an Arc.

Slices There are two main "slices" of a circle The "pizza" slice is called a Sector. And the slice made by a chord is called a Segment.

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

	Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle.

Inside and Outside

A circle has an inside and an outside (of course!). But it also has an "on", because you could be right on the circle. Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle.

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10

Activity: Approximate Value For Pi

Lines A line that goes from one point to another on the circle's circumference is called a Chord. If that line passes through the center it is called a Diameter.