The value of
is a little larger than 3.14. There have been many estimates of its value through history. The ancient Egyptians used a value of 
, but Archimedes, a Greek mathematician who lived
from 287?BC to 212BC, calculated two much better estimates of . He showed that is bigger than 
and smaller than 
.
His method involved calculating the areas/perimeters of regular 96-sided figures, one just big enough to contain the circle and one just small enough to fit inside the circle. His technique was
an extension of a
method discovered by Antiphon, who began by sandwiching a circle of diameter 2
between two hexagons. The circle has radius 1 and circumference 2 . The inner hexagon has a smaller circumference than the circle and the outer one has a larger circumference. Both can be worked out by using Pythagoras' Theorem. In the diagram (left), the sides of the inner (green) hexagon are each equal to the radius of the circle. Pythagoras' Theorem enables us to calculate the length of the sides of the outer hexagon as 
. The inner hexagon has circumference 6 and the outer one
. Halving these numbers gives the estimate of as between 3 and 
. But Archimedes didn't stop there. Using Pythagoras'
Theorem again enabled him to calculate the side lengths of enclosing 12-sided,
24-sided, 48-sided and 96-sided shapes. Each gives a closer estimate, enabling
Archimedes to prove that is between 
and 
. The last number was used in schools for many years
until calculators became widely available. The value of was calculated to 6,442,450,938 decimal places by Takahashi and Kanada in October 1995.
The number 
or e
The number denoted by (or more often by e) is close to 2.71828. It is important in calculus, the main topic in A-level mathematics. The two basic problems dealt with by calculus are to determine the gradient (or steepness) of a curve at each point and to determine the area enclosed by a curve and three straight lines, as illustrated left. The diagram shows the area between the x-axis, the vertical lines 
and 
, and the curve with equation 
. The special property of the number e is that it indicates where to place the right hand boundary to ensure that the area enclosed is exactly 1. The other important property of e is related to the gradient of the exponential curve 
. This curve has the property that the gradient at each point is also equal to 
. The exponential curve is useful for understanding physical processes like radioactive decay and cooling and statistical processes like the lifetime of electrical components. The number e is also transcendental. Expressed as a decimal, the digits go on forever without recurring. The best known formula for e is the infinite series
.
The classical problems and transcendental numbers
The classical Greek problem of ‘squaring the circle'
calls for a method of constructing a square with the same area as a given
circle, using only a compass and straight-edge. It amounts to calculating the
value of . The problem is now known to be impossible, though it continues to attract the interest of mathematical cranks. Compass and straight-edge constructions are the geometrical equivalent to solving a system of linear or quadratic equations with whole-numbered coefficients. For example, 
, is a linear equation whose solution 
is an example of a rational number. In 1761 it was proved by Johann Heinrich Lambert that is
irrational (ie not rational), meaning that it is not the solution of a linear
equation with whole-numbered coefficients. This tells us that the decimal digits
in do not recur, since any recurring decimal can easily be shown to be rational. In 1767, Lambert proved that e is also irrational. However, Lambert's proofs left
the possibility that (or e) might be the solution of a quadratic equation.
Quadratic equations, like 
, have two solutions (in this case 
), but they needn't be rational. For example, the equation 
does not have any rational solutions. The two solutions are the irrational numbers 
, which we will return to later. Equations can also involve higher powers of x. Those involving powers up to 
are respectively called cubic, quartic and quintic equations. The collective term for such equations is polynomial equations. Any number that is the solution of a polynomial equation with whole-number coefficients is called an algebraic number (it can be rational or irrational). Those that are not the solution of any polynomial equation are called transcendental numbers. In 1873, Charles Hermite proved that e is
transcendental and, nine years later, Ferdinand Lindemann presented a paper,
Uber die Zahl , in which
he proved that is transcendental: it is
not the solution of any polynomial equation with whole-numbered coefficients. In
particular, this tells us that is not the solution of a quadratic equation, which means that it is impossible to ‘square the circle' by means of a straightedge and compass construction. The two other great problems of antiquity, to find a straightedge and compass construction to ‘trisect the angle' or ‘duplicate the cube' are also impossible, since they involve solving cubic equations rather than quadratics.
The cylinder and sphere The hand is holding a cylinder that encloses a sphere. Archimedes (again) discovered that the volume of a sphere is two thirds of the smallest cylinder that will enclose it. He imagined the sphere and the cylinder being sliced into thin discs. The disc sliced from the cylinder (shown in turquoise) has cross-sectional area 
. At a distance a above the centre, the radius of the disc sliced from the sphere (shown in orange) is 
(by Pythagoras' Theorem) so it has cross-sectional area 
. The difference between these areas is 
, the cross-sectional area of a disc of radius a. But a stack of discs where the disc at height a has radius a forms a cone, which Archimedes knew to have volume one third of the enclosing cylinder. In other words, the volume of the ‘spare' space inside the cylinder but outside the sphere is equal to the volume of a cone with the same height and radius as the cylinder. This means that the volume of the sphere is two thirds that of the cylinder.
The
pentagram and the golden ratio, 
The pentagram design shows a pentagon with straight line segments connecting its five vertices, forming a smaller pentagon in the middle. Another famous number is involved in this diagram. The ratio of the length of a diagonal (yellow) to the length of one of the edges (red) is the golden ratio, 
.
Occurrences of the golden ratio are endemic in the diagram. The ratio of the length of the edge of the outer pentagon (red) to the length of one arm of the star (blue) is 
, and the ratio of the length of one arm of the star (blue) to the length of the edge of the inner pentagon (green) is also 
. This is because the three isosceles triangles indicated in the diagram by the two adjacent sides of the same colour are all similar. If the red segments have length 1 and we denote the length of the blue segments by x, we see that the green segments have length 
. Because they are similar, the ratio of the long side of the isosceles triangles to the short side is the same in each case. This means 
and therefore 
, all of which reduce to the same quadratic equation 
. This equation has two solutions, only one of which is positive, i.e. 
. Since 
, this means that 
. It also follows that the ratio of the sides of the outer and inner pentagons is 
and the ratio of their areas is 
.
Further reading
There are many books that explain more about the mathematics referred to in the crest.
1. Beckman P, A history of pi, The Golem Press (1982).
2. Berggren L, Borwein J, Borwein P, Pi: a source book, Springer-Verlag (2000).
3. Blatner D, The joy of pi, Allen Lane The Penguin Press (1997).
4. Boyer C B, A history of mathematics, , John Wiley & Sons (1978).
5. Klein F, prep TÄgert O, trans Beman W W, Smith D E, revised Archibald R C, Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle, Dover Publications (1956).
6. DÖrrie H, trans Antin D, 100 great problems of elementary mathematics: their history and solution, Dover Publications (1965).
7. Fauvel J, Gray J, The history of mathematics: a reader, 1996, Macmillan and Company.
8. Huntley H E, The divine proportion: a study in mathematical beauty, Dover Publications (1970).
9. Maor E, e: the story of a number, , Princeton University Press (1998).
STEVE ABBOTT
The Latin inscription on The Mathematical Association's coatof arms is Tibi creditum debes; taken from an Ode to Virgil, by Horace. In translation this becomes: What has been entrusted to you, you owe.
The coat of arms features an open book, signifying learning. The symbols 