by Chris Waring
The formula for these numbers is n × (n+1) × (n+2) ÷ 6. To find the total number of gifts your generous true love has given to you, set the following:
n = 12
12 × 13 × 14 ÷ 6 = 364
That’s a pretty decent haul!
Return to the party
Back to the Birthday problem: in order to find out the number of possible combinations of birthdays shared between two people in a group of 23 people you need to know the 22nd triangular number:
½ × 22 × 23 = 253 combinations
With this many pairings of 23 people, it now seems more reasonable that there is a 50% chance of two of them sharing a birthday.
To show the exact probability here, it is actually easier to find the opposite – the chance that no two people share a birthday – and exploit the fact that in probability the chance of something happening and the chance of something not happening have a sum of 100%.
Alan’s birthday can fall on any of the 365 days of the year, leaving 364 alternative days on which Blaise’s birthday could fall. In turn, there are 363 days on which Carl’s birthday could fall in order for it not to be shared with either Alan or Blaise. By the time the 23rd guest, Walter, steps into the room there are 343 possible days on which his birthday could fall without it being shared by anyone else in the room. If you write each one of these numbers as a probability out of 365, and then multiply them together, the total probability generated is:
(365 × 364 × 363 × 362 × ... × 345 × 344 × 343) ÷ 36523 = 49.3%
which means that the probability of two people sharing a birthday when there are 23 people in the room is 50.7%.
THE ALIENS HAVE LANDED
In 1960 American astronomer Frank Drake (1928–) was the first person to use radio telescopes to search for signals, messages or other evidence of intelligent life in the universe. This spawned what is now known as SETI– the Search for Extra-Terrestrial Intelligence.
Drake developed an equation to calculate the number of civilizations in the Milky Way that we should be able to communicate with:
number of civilizations = R* × fp × ne × fl × fi × fc × L
R* is the number of new stars made in the galaxy each year; fp is the fraction of new stars that will have planets; ne is the number of potentially life-supporting planets; fl is the fraction of the life-supporting planets that are known to have life on them; fi is the fraction of planets that have intelligent life; and fc is the fraction of planets that emit some kind of evidence of their civilization, such as radio waves. L represents how long such evidence emits for.
At the time of writing, many of these factors are pure conjecture, and scientists have come up with widely varying answers. Why not try your own!
A Foreign Language
It has been suggested that, if we do make contact with an alien civilization, numbers may be one of the first ways in which we communicate. In 1974 the Arecibo radio telescope in Puerto Rico beamed a radio message in the direction of a galaxy 25,000 light years away. Much of the information contained was numerical: the numbers from 1 to 10, the atomic numbers of the elements that make up DNA, the height of a man and the population of the earth.
The Future of Mathematics
We’re not done yet. Of course, the use of new computer methods to solve numerical problems that were previously deemed impossible to solve have boosted enormously developments in technology, science, medicine and engineering. However, there still remain thousands of unsolved problems in mathematics and science that will keep the experts busy for some time to come...
THE MARCH FORWARD
In 1900 at the International Congress of Mathematicians German mathematician David Hilbert (1862–1943) posed twenty-three mathematical problems that he felt were key to the development of the subject. Since the congress ten of Hilbert’s problems have been solved, seven have been solved to some extent or have been shown not to have a solution, three were too vague to be solved and three remain unsolved.
Posing these problems had exactly the effect that Hilbert wanted – the competition spurred mathematicians to strive to tackle them and in the process forge into new areas of research. In 2000, in much the same vein as Hilbert, the Clay Mathematics Institute issued another seven problems, now known as the Millennium problems.
So far, only one of the problems has been solved – the Poincaré conjecture, which relates to the topology of spheres. It was solved by an extraordinary Russian mathematician called Grigori Perelmann, who has declined not only a Fields Medal (the highest accolade in mathematics) but also the $1 million dollar prize from the Clay Institute.
One of the problems posed in both Hilbert’s problems and the Millennium problems is the Riemann hypothesis, a problem that many mathematicians feel is the most important in mathematics. It concerns the distribution of prime numbers. The Goldbach conjecture (see here) tells us approximately where the prime numbers should be; the Riemann hypothesis would help us to know how far away from the expected place the prime should actually be.
WHAT NEXT?
The future of mathematics depends very much on mathematicians who are, as I write, children, or as yet unborn. In order to cultivate the best possible mathematicians and scientists to help solve the world’s problems we need people with excellent mathematical training, which is quite an educational investment. In our current educational system, every schoolchild is taught numbers and arithmetic through to algebra and geometry so that by the onset of adulthood they have the tools necessary to enter a technical career path, should they so choose.
The majority of people, however, do not enter a technical career and therefore do not necessarily need mathematics taught beyond primary school. Most people use calculators or, more frequently, mobile telephones with built-in calculators, to do the mundane arithmetic that is all the maths needed in everyday life.
So, should we continue to make mathematics a compulsory subject until the age of sixteen? There are clearly those who enjoy maths and those who do not. Perhaps we could just teach basic arithmetic and everyday maths to younger children and save the harder, more interesting stuff as an optional course for older children who show a particular inclination and aptitude towards the subject? Well, if it worked for the Ancient Greeks...
The fundamental theories of how the universe works – as discovered by, among others, Newton, Einstein, Feynman and Hawking – have been made through the creation of a mathematical model and the pursuit of the mathematical conclusions that ensue. These models are then tested by experiments in the real world to check the accuracy of the model.
As time goes by, it seems that in order to generate the best possible rate of advancement in science, we need mathematicians who understand the latest developments in that field, and scientists who understand the latest developments in mathematics too.
As Galileo said:
Mathematics is the language with which God has written the universe.
Bibliography
50 Mathematical Ideas You Really Need to Know
by Tony Crilly (Quercus)
A History of Mathematics
by Carl Boyer and Uta Merzbach (John Wiley & Sons)
Descartes: Key Philosophical Writings
by Rene Descartes trans. Elizabeth Haldane and G. Ross (Wordsworth Classics)
Elements of Geometry by Euclid trans. Richard Fitzpatrick
Fermat’s Last Theorem by Simon Singh (Fourth Estate)
How Mathematics Happened: The First 50,000 Years
by Peter Rudman (Prometheus)
Mathematics: From the Birth of Numbers
by Jan Gullberg (WW Norton & Co.)
Number: From Ancient Civilisations to the Computer
by John McLeish (Flamingo)
The Code Book by Simon Singh (Fourth Estate)
The History of Mathematics: A Very Short Introduction
by Jacqueline Stedall (OUP)
The Psychology of Learning Mathematics
by Richard Skemp (Pelican)
&
nbsp; Index
abacus ref 1, ref 2
abundant numbers ref 1
accountants, dawn of ref 1
Al-Khwarizmi ref 1
Al-Kindi ref 1
Alcuin of York ref 1
Alexandria ref 1
algebra ref 1, ref 2
Boolean ref 1, ref 2
Boolean, as Venn diagram ref 1
and geometry ref 1
algorithms ref 1
alien civilizations ref 1
Analytical Engine ref 1
Ancient Egypt ref 1, ref 2, ref 3, ref 4
fractions used by ref 1
Ancient Greeks ref 1
alphabet of ref 1
lasting mathematical legacy of ref 1
philosophers in ref 1; see also individual names
Appel, Kenneth ref 1
Archimedes ref 1
Aristotle ref 1
arithmetic:
early ref 1, ref 2
fundamental theorem of ref 1
modular ref 1
symbols and operators for ref 1
see also mathematics; number systems
Arithmetica (Diophantus) ref 1, ref 2
Artificial Intelligence ref 1
astronomy and astrology ref 1, ref 2, ref 3
Australian Aboriginals ref 1
Avogadro’s constant ref 1
Babbage, Charles ref 1
Bede, Venerable ref 1
Berlin Papyrus ref 1
Bernoulli, Daniel ref 1
Bernoulli, Jacob ref 1, ref 2
Bernoulli, Johann ref 1
Bertrand, Joseph ref 1
binary numbers ref 1, ref 2
binomial coefficients ref 1
binomial theorem ref 1
Birthday problem ref 1, ref 2
Blanusa, Danilo ref 1
Bletchley Park ref 1
Bombe ref 1
Book of Addition and Subtraction According to the Hindu Calculation (Al-Khwarizmi) ref 1
Boole, George ref 1
Brahmagupta ref 1
calculator ref 1, ref 2, ref 3, ref 4, ref 5, ref 6
Leibniz’s ref 1
Pascal’s ref 1
calculus ref 1
Fundamental Theorem of ref 1
calendars:
in Dark Ages ref 1
Gregorian ref 1
Julian ref 1, ref 2
Mayans ref 1
Cantor, Georg ref 1
Cardano, Gerolamo ref 1
Cartesian graphs ref 1
catenary curve ref 1
Chaos Game ref 1
chaos theory ref 1
and weather ref 1
Chaucer, Geoffrey ref 1
China ref 1, ref 2
Compendious Book on Calculation . . ., The (Al-Khwarizmi) ref 1
complex numbers ref 1
compound interest ref 1
computers:
and Babbage ref 1
and chaos ref 1
and four-colour theorem ref 1
and fractals ref 1
and Turing ref 1
conic sections ref 1
continuum hypothesis ref 1
coordinates ref 1
Copernicus, Nicolaus ref 1
cosine ref 1
counting:
bone markings for ref 1
on fingers ref 1, ref 2
sheep ref 1
see also mathematics
cryptography ref 1
and frequency analysis ref 1
cubic equations ref 1
cyclic quadrilateral ref 1
De sectionibus conicis (Wallis) ref 1
decimal point ref 1
decimal system ref 1, ref 2, ref 3
deficient numbers ref 1
Descartes, René ref 1, ref 2
Description of the Wonderful Rule of Logarithms (Napier) ref 1
Difference Engine ref 1
difference of two squares ref 1, ref 2
differential equations ref 1, ref 2
differentiation ref 1
Diophantus ref 1, ref 2
‘Epitaph’ of ref 1
distributed computing ref 1
Drake, Frank ref 1
Dresden Codex ref 1, ref 2
economy, beginnings of ref 1
Einstein, Albert ref 1
Elements (Euclid) ref 1, ref 2
Enigma ref 1
see also cryptography
equations ref 1, ref 2, ref 3, ref 4, ref 5, ref 6
and Cartesian graphs ref 1
cubic ref 1
differential ref 1, ref 2
Euler’s identity ref 1
linear ref 1
quadratic ref 1
Eratosthenes ref 1
sieve of ref 1, ref 2
Euclid ref 1, ref 2, ref 3
fifth postulate of ref 1
Euler, Leonhard ref 1, ref 2
and Königsberg bridges ref 1, ref 2
Europe, Middle Ages ref 1
exhaustion, method of ref 1
Exiguus, Dionysius ref 1
extraterrestrial intelligence ref 1
Fermat, Pierre de ref 1, ref 2
Feynman, Richard ref 1
Fibonacci ref 1
fluid mechanics ref 1
four-colour theorem ref 1
fractals ref 1
ultimate ref 1
fractions ref 1, ref 2, ref 3, ref 4
frequency analysis ref 1
Galileo ref 1
Gauss, Carl ref 1
geometry ref 1, ref 2, ref 3
and algebra ref 1
and Euclid ref 1
and pyramids ref 1, ref 2
Goldbach, Christian ref 1
Goldbach conjecture ref 1, ref 2
golden mean ref 1, ref 2
Google/Googol ref 1
gradients ref 1
graph theory ref 1
Guthrie, Francis ref 1
Haken, Wolfgang ref 1
Hall, Monty ref 1
Hardy, Godfrey ref 1
Hawking, Stephen ref 1
Hero ref 1
hexagrams ref 1
hieroglyphics and hieratic writing ref 1
Hilbert, David ref 1, ref 2
Hippasus ref 1
hunter-gatherers ref 1
Huygens, Christiaan ref 1, ref 2
Hypatia ref 1
I Ching ref 1
imaginary numbers ref 1, ref 2
India ref 1
infinity ref 1, ref 2, ref 3
integration ref 1
irrational numbers ref 1
Ishango bone ref 1
Islam ref 1
iteration ref 1, ref 2
Jevons, William ref 1
Jia Xian ref 1
Kanada, Yasumasa ref 1
Kasner, Edward ref 1
Kerckhoff, Auguste ref 1
Key to Mathematics, The (Oughtred) ref 1
Khayyám, Omar ref 1
Koch, Helge von ref 1
Königsberg bridges ref 1, ref 2
Lambert, Johann ref 1
Large Hadron Collider ref 1
large numbers ref 1
Leibniz, Gottfried ref 1, ref 2, ref 3, ref 4, ref 5, ref 6
Leonardo da Vinci ref 1
Leonardo of Pisa (Fibonacci) ref 1
Let’s Make a Deal ref 1
logarithms ref 1, ref 2
natural ref 1, ref 2
and slide rule ref 1
Lorenz, Edward ref 1
Lovelace, Ada ref 1
magic squares ref 1
Mandelbrot, Benoit ref 1
Mandelbrot set ref 1
mathematics:
Ancient Egyptian ref 1, ref 2, ref 3
Ancient Greek ref 1
beginnings of ref 1
in Bronze Age, see Ancient Egypt; Ancient Greeks; Mayans; Mesopotamia
Chinese ref 1
in digital age ref 1
in early civilized era ref 1
future of ref 1
Indian ref 1
> Islamic ref 1
language of the universe ref 1
Mayan ref 1
in Mesopotamia ref 1, ref 2, ref 3
in Middle Ages Europe ref 1
modern ref 1
to prove theories ref 1
in Renaissance and after ref 1
Roman ref 1
in Stone Age ref 1
of turbulence ref 1
see also counting
Mayans ref 1, ref 2
calendar of ref 1
and placing of digits ref 1
and Spanish invaders ref 1
Meditations on First Philosophy (Descartes) ref 1
Mesopotamia ref 1, ref 2, ref 3, ref 4
number system of ref 1
Metrodorus ref 1
Millennium problems ref 1
Mises, Richard von ref 1
modular arithmetic ref 1
see also arithmetic
Mohammed ref 1
morphogenesis ref 1
Moscow Mathematical Papyrus ref 1
multiplication ref 1
musical notes and mathematics ref 1
Napier, John ref 1, ref 2
Napier’s bones ref 1
Newton, Isaac ref 1, ref 2, ref 3, ref 4, ref 5, ref 6, ref 7
Nine Chapters on the Mathematical Art ref 1
normal distribution ref 1
number systems:
base-60 (sexagesimal) ref 1
Chinese ref 1
Greek ref 1
