Imagine drawing a pencil line across a circle so that it cuts the circle in half. Now imagine cutting a string the length of that line across the circle, or its **diameter**. If you took that string and tried to wrap it around the outer edge of the circle, or its **circumference**, you would find the string went a little less than a third of the way around.

No matter how big or little your circle is, the **ratio**, or fraction, of the circle's circumference to its diameter is a little more than three. The exact ratio, however, is **irrational**. It can't be written exactly as a fraction or decimal number, and we call this ratio **pi** (????) (????).

## Approximations For Pi

You may have heard that pi is about 3.14, 22/7, or 333/106. While those numbers are close to the true value of pi, they are not exact. The true value of pi is an unending decimal number. This picture shows you the first 1000 digits of pi.

Even the early Greeks were fascinated with trying to get correct digits for pi. Archimedes came up with a clever approach.

He showed that if you draw two **polygons**, or figures with at least three straight sides, one inside the circle and one outside it, then pi would be somewhere in between.

He imagined creating polygons with more and more sides until you got an approximation for pi that was precise enough.

Since then, scientists and mathematicians have come up with a variety of ways that make it possible to calculate pi to trillions of decimal places. One approach can be visualized by a **continued fraction** – a fraction that has nested layers that go on forever.

In modern days, computers are used to find more digits of pi. Practically speaking, there is no need for us to know pi to trillions of digits. But, mathematicians find it fun to break records by finding more digits.

### Uses of Pi

Pi is not just theoretically interesting, but also useful to mathematicians. The simplest place pi is used is in finding the circumference or area of a circle. These formulas are probably familiar from math class:

The first formula tells you that the circumference, or the distance around, a circle can be found by multiplying the circle's diameter, which is 2 times the radius, by pi. The second formula shows you how to find the area of a circle by multiplying pi by the radius squared.

## Why is Pi (actually) Important

Other than being able to find your birthday in Pi,

why is Pi important enough to deserve a day of its own — and a search engine to draw attention to it? Well, let's start with the basics…

**Pi (π) is the ratio of the circumference of a circle to its diameter.**

It doesn't matter how big or small the circle is – the ratio stays the same.

Properties like this that stay the same when you change other attributes are called *constants*. Because it's so easily observed (you can measure it with a piece of string!), Pi has been popular for centuries.

### Pi is easy to observe, but hard to compute accurately by hand

- You can measure Pi by constructing a physical wheel and rolling it out – but you won't get more than a digit or two of accuracy.
- You can measure Pi by bracketing a circle with polygons. This is easy with polygons with small numbers of sides, but it gets harder as you add sides. Archimedes used this trick with 96 sided polygons to correctly estimate Pi to about two digits (3.14), proving 3.1408 < Pi < 3.1428. You can try this trick with squares:

The outer square has a side length of 1 (the same as the circle's diameter). The square we stuffed*inside*the circle has a*diagonal*length of 1. Importantly, it's a right triangle (the corner angle is 90 degrees), so we can usee the Pythagorean theorem a2 + b2 = c2 to determine the length of the sides: c2 = 1, and a and b have the same length, so 2 * side2 = 1. The length of the side, then, is

√1/2. From the side lengths, we can compute the circumference of the two squares. The outer square obviously has a circumference of 4, and the inner square has a circumference of 4 * √1/2, which is roughly 2.828. So we've proved some very loose bounds on Pi – and now you can see why Archimedes had to go up to a 96-sided polygon before he started getting good results!

Take a circle whose diameter (all the way across) is 1. If you roll it until you get back to the start, it will measure out Pi units. (Image from Wikipedia) |

### Methods to estimate Pi demonstrate several significant advances in mathematics

These days, we have better tricks for estimating Pi than the circumscribed-circle method used by Archimedes, but progress was mind-numbingly slow – and came due to advances in calculus and computing infinite series. See the Wikipedia Chronology of Computation of Pi for more cool detail.

YearDigitsValue26th century BC – Ancient Egypt | 3 | 22/7 = 3.14… |

250 BC – Archimedes | 3 | 3.1408 < Pi < 3.1428 |

150 AD – Ptolemy | 4 | 3.141666… |

480 AD – Zu Chongzhi | 6 | 3.1415926 < Pi < 3.1415927 |

1400 – Madhava of Sangamagrama | 11 | |

1424 Jamshid al-Kashi | 16 | |

1621 Ludolph van Ceulen + student | 35 | |

1699 | 71 | |

1706 | 100 | |

1794 | 137 | |

1841 | 152 | |

1844 | 200 | |

1853 | 440 | |

1874 – last before calculators | 527 |

### And computing Pi demonstrated advances in computing..

After the invention of the calculator and then computer, the records jumped drastically. In 1949, Ferguson and Wrench computed 1,120 digits using a desk calculator.

The first computer attempt, in 1949 on the ENIAC (the first general-purpose electronic computer), took 70 hours and computed 2037 decimal places. By 1967, the record stood at half a million digits, and in 2009, Takahashi et al.

used a supercomputer to compute 2.5 *trillion* digits of Pi. But it didn't stop there…

The first computational results used massive computers. But on the last day of 2009, Fabrice Bellard used a home computer – running an Intel Core i7 CPU similar to what you might be using today in 2011 to read this website – to compute 2.

7 trillion places. And the most recent record of 10 trillion digits of Pi was computed by Alexander J. Yee and Shigeru Kondo in 2011 using a fast, but not crazy, dual processor Intel Xeon-bsed machine with a huge amount of hard disk space.

Today, you can even compute thounsands of digits of Pi on your iPhone, something that might just have blown the mind of mathematicians 2000 years ago.

Click here to see our set of some programs to play with Pi on your own

### And Pi shows up everwhere that circles do..

Back to the Pi Search Page or…

Learn more about how the Pi Searcher works

## What is Pi (π) and what is it good for?

If you have a straight line but want a circle, you’re going to need some Pi.

Image via Max Pixel.

I’m talking about the number, not the delicious baked good. It’s usually represented using the lowercase Greek letter for ‘p’, ‘π’, and probably is the best known mathematical constant today. Here’s why:

### The root of the circle

Pi is the ratio of a circle’s diameter to its circumference. No matter the size of a circle, its diameter will always be roughly 3.14 times shorter than its circumference — without fail. This ratio, π, is one of the cornerstones upon which modern geometry was built.

**Bear in mind that (uppercase) ∏ is not the same as (lowercase) π in mathematics.**

For simplicity’s sake, it’s often boiled down to just two digits, 3.14, or the ratio 22/7. In all its glory, however, pi is impossible to wrap your head around.

It’s is an irrational number, meaning a fraction simply can’t convey its exact value. Irrational numbers include a value or a component that cannot be measured against ‘normal’ numbers.

For context, there’s an infinite number of irrational numbers between 1.1 and 1.100(…)001. They’re the numbers between the numbers.

## What is Pi?

*“Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi”*

**~William L. Schaaf, Nature and History of Pi**

Pi (often represented by the lower-case Greek letter π), one of the most well-known mathematical constants, is the ratio of a circle’s circumference to its diameter. For any circle, the distance around the edge is a little more than three times the distance across.

Typing π into a calculator and pressing ENTER will yield the result 3.141592654, not because this value is exact, but because a calculator’s display is often limited to 10 digits. Pi is actually an irrational number (a decimal with no end and no repeating pattern) that is most often approximated with the decimal 3.14 or the fraction (frac{22}{7}).

This brings up a rather interesting question: *If pi is the number of diameter lengths that fit around a circle, how can it have no end*?

**Pi: A Perennial Puzzle**

Pi has interested people around the world for over 4,000 years.

Many mathematicians – from famous ones such as Fibonacci, Newton, Leibniz, and Gauss, to lesser well-known mathematical minds – have toiled over pi, calculated its digits, and applied it in numerous areas of mathematics. Some spent the better parts of their lives calculating just a few digits. Here is a sampling of the many milestones in the life of pi.

## How Pi Works

Advertisement

Pi has mesmerized mathematicians for 4,000 years. It's the rarest of mathematical constants, an unfailingly accurate ratio that's also neverending. The digits of Pi have been calculated out to more than 22 trillion decimal places without ever repeating (that's called an “irrational number”).

The definition of pi is simple: It's the ratio of a circle's circumference divided by its diameter. But what's remarkable is that no matter the size of the circle you are measuring, that ratio of circumference to diameter will always equal 3.1415926535897, usually shortened to 3.14.

Divide the circumference of a tennis ball by its diameter and you get 3.14. Divide the circumference of the planet Mars by its diameter and you get 3.14. Divide the circumference of the known universe by its diameter — you get the point. As one mathematician put it, “Pi is part of the nature of the circle.

If the ratio was different, it wouldn't be a circle.”

The following figure shows how the circumference of a circle with a diameter of 1.27 inches (32.35 millimeters) is equal to a linear distance of 4 inches (10.16 centimeters):

As you might imagine, 4.0 (the circumference) / 1.27 (the diameter) = 3.14.

Pi is critical to several basic calculations in geometry, physics and engineering, including the area of a circle (πr2) and the volume of a cylinder (πr2)h. When the ancient Babylonians attempted to measure the precise areas of circles back in 1900 B.C.E., they assigned a value to pi of 3.125.

The ancient Egyptians came up with 3.1605. The Greek mathematician Archimedes (287-212 B.C.E.) and the Chinese mathematician Zu Chongzhi (429-501 C.E.

) are co-credited with calculating the most accurate approximations of pi before calculus and supercomputers gave us the definitive answer [source: Exploratorium].

In 1706, the self-taught Welsh mathematician William Jones assigned the Greek letter π to this magical number without end, possibly because π is the first letter of the Greek words for periphery and perimeter. The symbol's use was later popularized by 18th-century Swiss mathematician Leonhard Euler but wasn't adopted worldwide until 1934.

The fact that pi can be found everywhere — not only in circles, but in arcs, pendulums and interplanetary navigation — and that it's infinitely long has inspired a cult following that includes plenty of geeky tattoos and even its own national holiday. Keep reading to learn how you, too, can celebrate National Pi Day.

## What Is Pi, and How Did It Originate?

Succinctly, pi—which is written as the Greek letter for p, or π—is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle's size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3.14. But pi is an irrational number, meaning that its decimal form neither ends (like 1/4 = 0.

25) nor becomes repetitive (like 1/6 = 0.166666…). (To only 18 decimal places, pi is 3.141592653589793238.) Hence, it is useful to have shorthand for this ratio of circumference to diameter.

According to Petr Beckmann's A History of Pi, the Greek letter π was first used for this purpose by William Jones in 1706, probably as an abbreviation of periphery, and became standard mathematical notation roughly 30 years later.

Try a brief experiment: Using a compass, draw a circle. Take one piece of string and place it on top of the circle, exactly once around. Now straighten out the string; its length is called the circumference of the circle. Measure the circumference with a ruler.

Next, measure the diameter of the circle, which is the length from any point on the circle straight through its center to another point on the opposite side. (The diameter is twice the radius, the length from any point on the circle to its center.

) If you divide the circumference of the circle by the diameter, you will get approximately 3.14—no matter what size circle you drew! A larger circle will have a larger circumference and a larger radius, but the ratio will always be the same.

If you could measure and divide perfectly, you would get 3.141592653589793238…, or pi.

Otherwise said, if you cut several pieces of string equal in length to the diameter, you will need a little more than three of them to cover the circumference of the circle.

Pi is most commonly used in certain computations regarding circles. Pi not only relates circumference and diameter.

Amazingly, it also connects the diameter or radius of a circle with the area of that circle by the formula: the area is equal to pi times the radius squared.

Additionally, pi shows up often unexpectedly in many mathematical situations. For example, the sum of the infinite series

1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n2 + … is π2/6

The importance of pi has been recognized for at least 4,000 years. A History of Pi notes that by 2000 B.C., “the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π,” recognizing that every circle has the same ratio of circumference to diameter.

Both the Babylonians and Egyptians had rough numerical approximations to the value of pi, and later mathematicians in ancient Greece, particularly Archimedes, improved on those approximations. By the start of the 20th century, about 500 digits of pi were known.

With computation advances, thanks to computers, we now know more than the first six billion digits of pi.

## Pi (π)

Definition: Pi is a number – approximately 3.142

It is the circumference of any circle divided by its diameter.

The number Pi, denoted by the Greek letter π – pronounced 'pie', is one of the most common constants in all of mathematics. It is the circumference of any circle, divided by its diameter. Nobody knows its exact value, because no matter how many digits you calculate it to, the number never ends. For most practical uses, you can assume it is 3.142.

Some people have written computer programs and calculated it to an astonishing accuracy. For example some have calculated pi to 200 million digits.

### Pi to a few dozen digits

3.

1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273 724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609…

Most calculators have a button to enter the value of Pi directly, so you don't usually have to remember all the digits.

But it's handy to remember pi = 3.142 anyway.

** Using the calculator Pi button is better**, because it inputs Pi to the most number of decimal places that the calculator is capable of. This will result in your calculations having the best possible accuracy.

### Approximating Pi

If you just need a rough value for Pi you can assume it is 22/7, but this is only accurate to two decimal places. However it may be accurate enough for some uses when you don't have a calculator handy.

An even more accurate approximation is 355/113. This is corect to 6 decimal places.

Pi | 3.1415927 |

22/7 | 3.1428571 |

355/113 | 3.1415929 |

- Introduction to numbers
- The number line

(C) 2011 Copyright Math Open Reference. All rights reserved

## What Is Pi?

Before we can learn more about pi, it will help if we review a bit of geometry. In particular, we need to brush up on circles. Why? Well, we'll get around (pun intended!) to that in a second…

The circumference of a circle is its perimeter or the length around it. The distance from the center of a circle to its edge is the radius. The distance from one side of a circle to the opposite side (twice the radius) is the diameter. The area of a circle is the number of square units inside the circle.

Since circles can vary in size, yet they all retain the same shape, ancient mathematicians knew there had to be a special relationship amongst the elements of a circle. That special relationship turns out to be the mathematical constant known as pi.

Pi is the ratio of a circle's circumference to its diameter. Regardless of the size of the circle, pi is always the same number. So, for any circle, dividing its circumference by its diameter will give you the exact same number: 3.14159…or pi.

Pi is also an irrational number, which means that its value cannot be expressed exactly as a simple fraction. As a result, pi is an infinite decimal. Although 22/7 gives a result that is close to pi, it is not the same number.

Since mathematicians can't work with infinite decimals easily, they often need to approximate pi. For most purposes, pi can be approximated as 3.14159. Some people even shorten it to 3.14, which is why Pi Day is celebrated on March 14 (3/14).

Interestingly, there can be no “final” digit of pi, because it's an irrational number that never ends. Mathematicians have also proved that there are no repeating patterns in the digits of pi.

Computers have calculated pi to over three trillion digits. Here are a few representations of pi to different numbers of digits (past the decimal):

- Pi to 10 digits: 3.1415926535
- Pi to 100 digits: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
- Pi to 1000 digits: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989

Pi is an important part of many mathematical formulas. Most geometry students first encounter pi when they study circles and learn that the area of a circle is equal to pi times the square of the length of the radius. This formula — A=πr2 — is sometimes described as “area equals pi r squared,” which is the basis of the old joke about pies being round, not square.

You may have noticed in the equation above and in many other places, pi is represented by (and takes its name from) the Greek letter pi (π). The Greek letter π was first used to represent pi by William Jones in 1706, because π was an abbreviation of the Greek word for perimeter: “περίμετρος.”

## What Is Pi?

Understanding pi is as easy as counting to one, two, 3.1415926535…

OK, we'll be here for a while if we keep that up. Here's what's important: Pi (π) is the 16th letter of the Greek alphabet, and is used to represent the most widely known mathematical constant.

By definition, pi is the ratio of the circumference of a circle to its diameter. In other words, pi equals the circumference divided by the diameter (π = c/d).

Conversely, the circumference of a circle is equal to pi times the diameter (c = πd). No matter how large or small a circle is, pi will always work out to be the same number. That number equals approximately 3.

14, but it's a little more complicated than that. [10 Surprising Facts About Pi]

### Value of pi

Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because pi is what mathematicians call an “infinite decimal” — after the decimal point, the digits go on forever and ever.

## Where Did The Word"Pi" Come From?

Pi is the Latin name of the sixteenth Greek letter, **π**. (Mathematic notation borrows from a multitude of alphabets and typefaces.

) The first recorded use of **π** as a mathematical symbol comes from the Welsh mathematician William Jones in a 1706 work called Synopsis Palmariorum Matheseos, in which he abbreviated the Greek περιϕέρεια, (meaning “circumference,” or “periphery”) to its first letter: **π**.

### What does pi mean in mathematics?

The mathematical pi is defined as “the ratio of the circumference of a circle to its diameter.” It’s also known as Archimedes’ Constant, after the ancient Greek mathematician of the same name, who, in addition to coming up with an algorithm for calculating pi, also invented an early type of irrigation pump called the Archimedian screw. Very medieval-sounding, but we digress.

What makes pi so magical is that it doesn’t matter how big or small the circle may be: the pi ratio remains the same.

Pi is what’s known as an irrational number, which means, in part, that “it can never terminate or repeat when written out in decimal form.” As far as we can tell, it goes on forever, which is a bit mind-boggling. Computers have calculated pi to decimal places in the trillions.

It is also a transcendental number, a concept that exceeds the scope of this post but believe us it’s very cool.

### What is Pi Day?

Pi Day is the March 14th holiday commemorating the mathematical constant **π** (pi), written numerically as 3.141592+, and pronounced “pie.” And yes: lots of people mark the occasion by eating pie because why would you pass up an occasion to eat pie?

### World record pi?

Memorizing as many digits of pi as possible has become an obsession for many.

The Guinness World Record for memorizing digits of **π** is held by a man named Lu Chao, who set the record in November 2005 in the Shaanxi province of China.

It took him 24 hours and 4 minutes to recite the 67,890th decimal place of π without a mistake. Kind of makes us feel inadequate for all the times we’ve forgotten our 4-digit PIN.

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