Pi is the ratio of the circumference of a circle to its diameter.

It is a constant. The familiar value is 3.14159.

It is irrational – it cannot be written as a fraction using 2 integers, although, some approximations, like 22 / 7, are used.

It is transcendental – this means there is no algebraic equation for which it is the solution (algebraic equations look like this: x² – 2 = 0 or x² – x – 1 = 0 or 8x³ – 3 = 0), or, as Euler is claimed to have said, “They transcend algebra”.

The relationship between the circumference of a circle and its diameter has been known for thousands of years. The Babylonians knew it, as did the ancient Egyptians and the ancient Greeks – although, it wasn’t called Pi back then. The symbol π and the name Pi were introduced William Jones in 1706 – previously, it was always referred to as “the ratio between the circumference of a circle to its diameter”. The symbol π is taken from the first letter of the Greek word perimetros (περιμετρος) – for the perimeter (or circumference) of a circle (circumference comes from a Latin root, whereas perimeter comes from a Greek root).

Had Julius Caesar not managed to burn down the Library of Alexandria in 48BCE, our knowledge of ancient history, culture, mathematics and science would likely be more complete. As it stands, our knowledge is fragmentary, but it is pretty obvious that the ancients were curious about the world around them and sought to understand it (just as people do today).

One of the things they discovered was that the length of the edge of a circle (its circumference or perimeter) was always a little more than 3 times the length of the diameter of the circle. The ancient Egyptians estimated it at 256 / 81 (written in some texts as 4 x (8 / 9)²). Archimedes narrowed down the value to being somewhere between 3 + 10 / 71 and 3 + 1 / 7 (or 223 / 71 and 22 / 7).

It doesn’t matter how big or small the circle is, the length of the edge is always 3.14159… times longer than its diameter.

The ratio of the circumference to the diameter is not the only relationship we can find in a circle. There is also the ratio of the circumference to the radius (which leads to the concept of the radian) and the ratio of the diameter to the radius.

The ratio of the circumference to the radius is 6.28318… (or 2π). It doesn’t matter how big or small the circle is, the length of the edge (circumference or perimeter) of the circle will always be a little more than 6 times the length of the radius. (If mathematicians had been using the ratio of the circumference to the radius, then the value of π would have been 6.28318… instead of the familiar 3.14159…)

The ratio of the diameter to the radius is 2. Again, it doesn’t matter how big or small the circle is, the length of the diameter will always be twice as long as the length of the radius. (Let’s be thankful that we use the symbol 2 for this instead of some Greek letter, like ζ)

Empirically determining the value of Pi

You can do a simple experiment to determine the value of Pi:

1. Get a round cardboard tube (one that is reasonably round and not squished too much – a toilet paper roll is fine).
2. Measure the diameter of the tube using a ruler (it’s a good idea to take several different measurements across the tube and average them out).
3. Cut the tube lengthwise with a pair of scissors.
4. Lay the cut tube flat and measure the width.
5. Divide the width of the flat tube by the diameter of the tube.

Another experiment is:

1. Get something round (a tin can, a plastic tub, a CD, a roll of tape).
2. Measure the diameter of the object.
3. Make a small mark on the edge of the object.
4. Put the object, mark down, on a sheet of paper.
5. Mark the position on the paper.
6. Carefully roll the object along the paper until the mark once again touches the paper. (This “unrolls” the length of the edge on the paper)
7. Mark that position on the paper.
8. Measure the distance between the two marks.
9. Divide the distance between the two marks (which is equal to the length of the edge of the round object) by the diameter of the object.

Alternately, if you have a roll of masking tape:

1. Using a sharp knife, cut through several layers of the tape (at least two).
2. Measure the diameter of the roll.
3. Pull off a strip of tape (since the tape is cut, you will only be able to pull off a length equal to the circumference of the roll – well, the first strip might be shorter, so discard it).
4. Carefully stick the strip to a piece of paper.
5. Measure the length of the strip.
6. Divide the length of the strip by the diameter of the roll.

The diameter of my roll of masking tape was 127.7mm and the length of the strip of tape was 401.5mm:

401.5 / 127.8 = 3.144 – another very close empirical measurement.

The post Understanding Pi appeared first on Complete, Concrete, Concise.

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. – Wikipedia

A better (simpler) definition at Math Open Reference is:

One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

What they are saying is:

One radian is just shy of 57.3 degrees (57.295779513… degrees – the number is both irrational and transcendental).

Needless to say, radians seem a strange way to measure angles – especially since there aren’t even a whole number of radians in a circle (there are 2π radians in a circle). What’s wrong with using degrees?

How we divide a circle is arbitrary. We use 360 degrees for a circle because that’s how the ancient Egyptians divided their year (the remaining 5 days were set aside as a festival for the gods Osiris, Horus, Set, Isis and Nephthys – one day each). The 360 day cycle of the year became the 360 degrees of a circle.

A circle could just as well be divided into 4 or 8 or 16 parts. It would definitely be easier than dividing a circle into 360 parts. The French divided the circle into 400 parts (100 parts for each quarter circle), following the French Revolution, in their quest to make everything metric (this is the grad or gradian you see on most scientific calculators).

In some ways, radians is a more “natural” division of the circle. A radian is equal to rotating the edge of a circle through a distance equal to the radius, whereas one degree is equal to one-three-hundred-sixtieth of a rotation of the circle. The radian is defined in terms of the circle, whereas degrees are an arbitrary division of the circle.

Dividing a circle into radians has advantages:

If you have a wheel, converting from rotation (angular displacement) to distance (linear displacement) is very easy – multiply the number of radians rotated by the length of the radius to calculate the distance. Converting degrees rotated into distance is much less fun.

For example:

1 radian x 20 cm = 20 cm (radians, like degrees are dimensionless)

The same problem using degrees is not as easy:

1. the circumference of the circle (the length of the edge) is equal to 2π x length of the radius
2. the length of the circumference : 2π x 20 cm = 125.7 cm
3. multiply the length of the circumference by the fraction the wheel turned (57.3 / 360 = 0.1597)
4. 0.1597 x 125.7 cm = 20 cm (using values from the steps 1, 2, & 3)
5. or (57.3 / 360) x 2π x 20 cm = 0.1597 x 6.28318 x 20 cm = 20 cm (all the steps in one line)

Try another example:

The advantage of using radians is that it takes the term 2π out of the equation – thus simplifying calculations. There are 2π radians in a circle, consequently, one radian is equal to 1 / 2π of a circle.

We know that ~57.3° equals 1 radian which is equal to 1 / 2π of a circle. If, instead of using degrees, we use its equivalent in radians then we get: (some fraction of a radian) x (1 / 2π) x 2π x (length of radius). The terms 2π cancel out and we are left with (some fraction of a radian) x (length of radius).

In problems dealing with waves (electronics, audio, etc), working with radians instead of degrees makes the computations easier. The factor 2π appears frequently in these calculations and can be eliminated if we deal in radians. This (1) simplifies calculations and (2) reduces error in the calculations – both round off errors and the chance of making errors (consider constantly multiplying or dividing by 2π – or its approximation, 6.28318 – multiplication errors are possible, round off errors a definite possibility). If we need to convert back to degrees, then we can do so at the end.

The post Understanding Radians appeared first on Complete, Concrete, Concise.