A number born from circles
Pi, written π, is the ratio of a circle’s circumference to its diameter. The circumference is the distance around the circle; the diameter is a line segment through its centre joining two opposite points on the circle.
Whether the circle is a small coin or an enormous wheel, this ratio stays the same in ordinary plane geometry. Dividing the circumference by the diameter always gives π, approximately 3.14159.
π = circumference ÷ diameter
A circle with diameter 1 has circumference π.
Measuring a circle
Knowing π connects a circle’s dimensions. If its radius is r, its diameter is 2r. Its perimeter, or circumference, is therefore 2πr. The area of the disc enclosed by the circle is πr².
For a circle with radius 1 metre, the circumference is about 6.28318 metres and the area about 3.14159 square metres. The units differ: length is measured in metres, area in square metres.
P = 2πr · A = πr²
r is the radius, P the circumference and A the area of the disc.
How much precision?
The symbol π is a letter of the Greek alphabet. It denotes a specific number, not a unit of length: the ratio does not change when you switch from centimetres to metres.
The value 3.14 is enough for some everyday estimates, but it is not the exact value of π. How many decimal places to keep depends on the precision you need; rounding too early can magnify a calculation error.
The fractions 22/7 and 355/113 are useful approximations, but neither equals π.
Digits without end
π is irrational: it cannot be written exactly as the quotient of two integers. Its decimal expansion therefore never ends or becomes periodic. No finite block of digits repeats forever from some point onwards.
Local repetitions are still possible. The same sequence can appear several times, two matches can overlap, and identical digits can occur in a row. These repetitions do not make the number rational.
π = 3.141592653589793238462643…
The ellipsis shows that the decimal expansion continues forever.
Proofs and conjectures
π is also transcendental: it is not a root of any non-zero polynomial with integer coefficients. Like its irrationality, this is a proven property.
π is conjectured to be normal in base ten: each block of k digits would then have a limiting frequency of 1/10ᵏ. This property has not been proved.
The irrationality of π alone does not guarantee that every finite digit sequence appears in its decimal expansion.
Explore without jumping to conclusions
This book explores a finite dataset of decimal digits of π. The decimal explorer shows the size of the installed dataset. Search for a sequence of 1 to 30 digits, keep leading zeros and browse the matches. Position 1 means the first digit after the decimal point: the initial 3 is not counted.
If a sequence does not appear in this excerpt, that does not mean it is absent from the infinite decimal expansion of π. Likewise, an apparently regular distribution in a large excerpt does not prove a property of all its digits.
Search covers the finite contents of this book, not all the decimal digits of π.