A

1. Who didn’t consider 1 to be a number at all?

2. What number associates with negatives? Why?

3. What number is the dimension of the smallest magic square in which every row, column, and diagonal equals fifteen?

4. What numbers are considered to be perfect in Mathematics? Why?

5. What does the number 7 determine in China?

B

1. In what country is the number 4 considered to be unlucky?

2. Why was the number 5important to the Maya?

3. Why is a knot tied in the form of the pentagram called a lover’s knot in England?

4. What number leads to a few years of bad luck, if you break a mirror?

5.What did students pursue in medieval education?

DO YOU KNOW THAT…

· The number 8 is generally considered to be an auspicious number by numerologists. The square of any odd number, less one, is always a multiple of 8 (for example, 9 − 1 = 8, 25 − 1 = 8 × 3, 49 − 1 = 8 × 6), a fact that can be proved mathematically.

· The early inhabitants of Wales used ninesteps to measure distance in legal contexts; for example, a dog that has bitten someone can be killed if it is nine steps away from its owner’s house, and nine people assaulting one constituted a genuine attack.

· The number 20has little mystical significance, but it is historically interesting because the Mayan number system used base 20. When counting time the Maya replaced 20 × 20 = 400 by 20 × 18 = 360 to approximate the number of days in the year. Many old units of measurement involve 20 (a score), for example, 20 shillings to the pound in pre-decimal British money system.

V. Find information on the Internet and give a presentation of the number you are interested in (brings you good or bad luck).

Text 10

Reading and Speaking

NUMBER AND REALITY

Many aspects of the natural world display strong numerical patterns, and these may have been the source of some number mysticism. For example, crystals can have rotational symmetries that are twofold, threefold, fourfold, and sixfold but not fivefold − a curious exception that was recognized empirically by the ancient Greeks and proved mathematically in the 19 th century.

An especially significant number is the golden ratio, usually symbolized by the Greek letter ϕ. It goes back to early Greek mathematics under the name ‘extreme and mean ratio’ and refers to a division of a line segment in such a manner that the ratio of the whole to the larger part is the same as that of the larger part to the smaller. This ratio is precisely (1 + √5)/2, or approximately 1.618034. The popular name golden ratio, or golden number, appears to have been introduced by the German mathematician Martin Ohm in Die reine Elementarmathematik (1835; ‘Pure Elementary Mathematics’). If not, the term is not much older and certainly does not go back to ancient Greece as is often claimed.

In art and architecture the golden number is often said to be associated with elegance of proportion; some claim that it was used by the Greeks in the design of the Parthenon. There is little evidence for these claims. Any building has so many different lengths that some ratios are bound to be close to the golden number or for that matter to any other ratio that is not too large or small. The golden number is also often cited in connection with the shell of the nautilus, but this too is a misunderstanding. The nautilus shell has a beautiful mathematical form, a so-called logarithmic (or equiangular) spiral. In such a spiral each successive turn is magnified in size by a fixed amount. There is a logarithmic spiral associated with the golden number, and in this case the fixed amount is precisely ϕ. However, the spiral of the nautilus does not have the ratio ϕ. Logarithmic spirals exist with any given number as their ratio, and the nautilus ratio has no special significance in mathematics.

The golden number is, however, legitimately associated with plants. This connection involves the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…), in which each number, starting with 2, is the sum of the previous two numbers. These numbers were first discussed in 1202 by the Italian mathematician Leonardo Pisano, who seems to have been given the nickname Fibonacci (son of Bonaccio) in the 19th century. The ratio of successive Fibonacci numbers, such as 34/21 or 55/34, gets closer and closer to ϕ as the size of the numbers increases. As a result, Fibonacci numbers and ϕ enjoy an intimate mathematical connection.

Fibonacci numbers are very common in the plant kingdom. Many flowers have 3, 5, 8, 13, 21, or 34 petals. Other numbers occur less commonly; typically they are twice a Fibonacci number, or they belong to the ‘anomalous series’ 1, 3, 4, 7, 11, 18, 29… with the same rule of formation as the Fibonacci numbers but different initial values. Moreover, Fibonacci numbers occur in the seed heads of sunflowers and daisies. These are arranged as two families of interpenetrating spirals, and they typically contain, say, 55 clockwise spirals and 89 counterclockwise ones or some other pair of Fibonacci numbers.

This numerology is genuine, and it is related to the growth pattern of the plants. As the growing tip sprouts, new primordial − clumps of cells that will become special features such as seeds − arise along a generative spiral at successive multiples of a fixed angle. This angle is the one that produces the closest packing of primordial; and for sound mathematical reasons it is the golden angle: a fraction (1 − 1/ ϕ) of a full circle, or roughly 137.5 degrees.

I. Match the words (1–11) with the definitions/explanations (a– k):