Why does pi start with 3 and not any other number?

Question

Thank you.

Answer 1

Pi is defined as the ratio between the circumference of a circle, and the distance across a circle (the diameter).

Let's start with a hexagon. Notice how you can fit a regular hexagon inside a circle. Look at the "radius" of a hexagon (distance to a vertex). Let's call that r. Double that is 2r. And the distance around the hexagon is 6r. So the ratio of perimeter to "diameter" is exactly 3.

Now if you try polygons with larger numbers of sides (better approximating a circle), the number gets bigger. Octagons have a ratio of 3.0616. A decagon will have a ratio of a little more than that and a dodecagon more than that. But it starts approaching a limit. Certainly never more than 4... actually never more than pi, until you get all the way to an infinite sided polygon which would be a circle.

The ratio will approach pi (3.141592653589...) as a limit. A hexagon starts at 3.0, other shapes add more to the decimal portion, but never make it exceed pi. So it will never jump to 4 or go below 3.

So that is why pi starts with 3.

Answer 2

Consider the following:

π is the ratio of a circle's circumference to its diameter. Now, imagine a square circumscribed about a circle - the length of a side of the square will be the diameter of the circle, and the perimeter of the square will be four times the length of the side - i.e. four times the diameter of the circle. Now clearly the circumference of the circle is less than the perimeter of the circumscribed square, so π must be less than 4.

Now instead of a circumscribed square, consider an inscribed regular hexagon. Connecting each vertex to the center divides the hexagon into six equilateral triangles, so each side of the hexagon will be equal to the distance from the center to a vertex of the hexagon, which is the radius of the circle. Thus, the perimeter of the hexagon is 6 times the radius, and therefore three times the diameter of the circle. Since the circumference of the circle is greater than the circumference of the circle, π must exceed 3.

Thus we have the inequality 3 < π < 4, and every number in this range starts with 3. And that's why π starts with 3 and not some other number.

Answer 3

PI is the ration of a circle circumference to is diameter. That is, if the diameter is 1 (inch, cm, meter, or whatever), the circumference will be 3.1415...

It is not a manufactured number, is it just the way the world works.

Answer 4

The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = 25⁄8, which is within 0.5% of the true value.

The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.

It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular.

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 223⁄71 and 22⁄7. The average of these two values is roughly 3.1419.

The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.

The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 62832⁄20000 = 3.1416, correct when rounded off to four decimal places. He also acknowledged the fact that this was an approximation, which is quite advanced for the time period.

The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355⁄113 and 22⁄7, in the 5th century.

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of π⁄4 into the form

\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π⁄4, he was able to compute π to an accuracy of 13 decimal places.

The Persian astronomer Ghyath ad-din Jamshid Kashani (1350-1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2π = 6.2831853071795865

By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.

In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).

Answer 5

Pi is a number referred to in figuring the area of a circle. It is approximately equal to 22/7 or about 3.142857.........

Answer 6

when pi is written as 22/7 it does not start with 3

Answer 7

If you write PI in another base (binary or hexadecimal) it doesn't start with 3.

Answer 8

Because that is the value of pi. That's like asking 'why did this sentence start with a "T"?'. Because that is how you spell it.

Answer 9

24/7 is less than 4 but a little more than 3 that is why it is what it is.

Answer 10

yes its because it is the only way you could use it to find the circumference of a circle, the area of a circle, or for radians. you pretty much wasted 5 points for that question