Light Weight (*)

Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a radial displacement of the star images is found. But the amount predicted by Newton’s laws is only half the observed value.

Solar-Eclipse Continue reading ‘Light Weight (*)’

Waring’s Problem & Lagrange’s Four-Square Theorem

\displaystyle \mathrm{num}\ = \square+\square+\square+\square

Introduction

We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.

More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.

Continue reading ‘Waring’s Problem & Lagrange’s Four-Square Theorem’

Old Octonions may rule the World

This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.

Multiplication table for octonions, of the formz=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

Continue reading ‘Old Octonions may rule the World’

Triangular Numbers: EYPHKA

The maths teacher was at his wits’ end. To get some respite, he set the class a task:

Add up the first one hundred numbers.

That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians. Continue reading ‘Triangular Numbers: EYPHKA’

Algebra in the Golden Age

This week’s That’s Maths column in The Irish Times (TM054, or search for “thatsmaths” at irishtimes.com) is about the emergence of algebra in the Golden Age of Islam. The Chester Beatty Library in Dublin has several thousand Arabic manuscripts, many on mathematics and science.

Left: Societ stamp commemorating al-Khwārizmī's 1200th birthday. RIght: A page from al-Khwārizmī's Al-Jebr.

Left: Soviet Union postage stamp (1983) commemorating al-Khwārizmī’s 1200th birthday. RIght: A page from al-Khwārizmī’s Al-Jebr.

Continue reading ‘Algebra in the Golden Age’

Curves with Singularities

Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it does not have a unique value there, the point is singular.

Slinky traces a smooth helical curve in three dimensions.

Slinky traces a smooth helical curve in three dimensions.

Generally, if we zoom in close to a point on a curve, the curve looks increasingly like a straight line. However, at a singularity, it may look like two lines crossing or like two lines whose slopes converge as the resolution increases. Continue reading ‘Curves with Singularities’

How Big was the Bomb?

By a brilliant application of dimensional analysis, G.I.Taylor estimated the explosive energy of the first atomic blast, the Trinity Test (see this week’s That’s Maths column in The Irish Times, TM053, or search for “thatsmaths” at irishtimes.com).

US army soldiers watching the first test of an atomic weapon, the Trinity Test.

US army soldiers watching the first test of an atomic weapon, the Trinity Test.

Continue reading ‘How Big was the Bomb?’


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