What are some interesting math numbers

Lessons (<45 min) school year 8-10

Anne Hilgers

Irrational numbers

The most important facts about the number pi

Every year ... Pi day is celebrated on March 14th. In honor of the circle number there are traditionally round cakes, which are called "pie" in English. We've put together some interesting and fun facts about this number. By the way, Pi Approximation Day is celebrated on July 22nd.

The circle number Pi is the ratio of the circumference and diameter of a circle. This geometric relationship is easy to understand. Whereby one can be amazed that this ratio is really always the same, for all conceivable circles. If you don't believe it, you can measure and calculate with a wide variety of round objects with a thread and ruler. Measurements are of course not exact, but that's another topic. So what do we know about Pi?

Pi is an irrational number

This means that π cannot be represented as a fraction and has neither a finite nor a periodic decimal fraction expansion. This was proven in 1761 by Johann Heinrich Lambert using continued fractions, but his proof was not complete. Adrien-Marie Legendre proved the necessary proposition about continued fractions in his Éléments de géométrie.

Pi is a transcendent number

This means that there is no polynomial with integer coefficients that has π as a zero. (Not every irrational number is also a transcendent number, so about √2 is the root of the polynomial x2 - 2). It is a popular game to find your date of birth in the infinite number of decimal places of π. However, it is unclear whether every conceivable sequence of digits actually appears there in the same way. If the digits of π followed one another like random numbers, then this should be the case. In particular, all digits from 0 to 9 should occur statistically with the same frequency. If you want to check it: There are many decimal places of π to be found here.

Pi and the squaring of the circle

The (unsolvable) problem is to construct a square with the same area for a circle, using a compass and a ruler. To do this, one would have to be able to construct a segment of length π · d from a segment d. In 1882 Ferdinand von Lindemann proved that π is a transcendent number and therefore especially not constructible with a compass and ruler.

Pi is a Youtube star

The song from π has over 8 million views, in which each digit in the decimal fraction is assigned a note (there are several videos on this). The result (with appropriate accompaniment by chords) can be heard. The well-known math rapper DorFuchs has also dedicated some interesting videos to the number. So he proves Pi is irrational and explains what the sum of the reciprocal values ​​of the square numbers has to do with π (it is π² / 6).

Pi makes you creative

Artists have always been inspired by the number pi. François Morellet, for example, created numerous images from the series of digits, including Pi rococo no.4. And the Canadian Martin Krzywinski publishes posters on the occasion of the Pi-Tag that visualize the decimal places in different ways, as Alexander Salle shows in his blog piistgenaudrei.

Pi is popular

Pi-Day was introduced in 1988 at the Exploratorium, a science museum in San Francisco. On March 14, 2015, the number pi was particularly close, because this date is noted in the Anglo-Saxon region as 3/14/15. The Pi-Tag is becoming more and more popular with us as well. On March 11, 2018, the taz printed almost 8,000 decimal places of pi and wrote: “There are mathematicians who spend years of their lives looking for the last place after the decimal point in pi.” Is that really true? Claus Michael Ringel also discovered a reference to Pi Day in the DBmobil magazine of Deutsche Bahn. There they asked: “How many decimal places does the number of numbers have? So far, supercomputers have determined 22 trillion. ”We know that the supercomputers can continue to calculate for as long as they want - they will never find a last digit.

For further reading

Pi is a number, and numbers can be calculated. But: what actually is √3 · π? Can you imagine this calculation? If yes how? You can find out more about pi and other irrational numbers in the Irrational Numbers booklet.

Lessons (<45 min) school year 8-10