Who invented even numbers

The second definition is still with us today: algebraically the formula for the odd numbers is 2n-1 where n is given the successive values 1, 2, 3…. In concrete terms, we have the sequence. The unit itself is something out on its own and was traditionally regarded as neither even nor odd. In practice it is often convenient to treat the unit as if it were odd , just as it is to consider it a square number, cube number and so forth, otherwise many theorems would have to be stated twice over.

Note that distinguishing between even and odd has nothing to do with counting or even with distinguishing between greater or less — knowing that a number is even tells you nothing about its size. It is significant that we do not have words for numbers which, for example, are multiples of four or which leave a remainder of one unit when divided into three.

If our species had three genders instead of two, as in the world described in The Player of Games , we would maybe tend to divide things into threes and classify all numbers according to whether they could be divided into three parts exactly, were a counter short or a counter over. This, however, would have made things so much more complicated that such a species would most likely have taken even longer to develop numbering and arithmetic than in our own case.

The integers can be separated out into so-called equivalence classes according to the remainder left when they are divided by a given number termed the modulus. All numbers are in the same class modulus 1 since when they are separated out into ones there is only one possible remainder : nothing at all. And the odd numbers are all 1 mod 2 i. What is striking is that although the distinction between even and odd, i. In concrete terms we can set up equivalence classes relative to a given modulus by arranging collections of counters in fact or in imagination between parallel lines of set width starting with unit width, then a width which allows two counters only, then three and so on.

This image enables us to see at once that the sum of any two or more even numbers is always even. We end up with the following two tables which may well have been the earliest ones ever to have been drawn up by mathematicians.

For any one of these assignments you could select the same person to study, Pythagoras. Usually, when we hear the name Pythagoras, or more formally Pythagoras of Samos, we think of right-angled triangles or the hypotenuse and maybe squares and things!

But there was a lot more to Pythagoras than his famous theorem. He was well known in his day, enough that statues of him were sculptured and drawings and paintings made. If you look for information about him at a maths history site, like the wonderfully informative one produced by St Andrew's University , you will discover that after Nash, Einstein and Newton, Pythagoras' is the most requested biography of a mathematician.

He lived from about BC to about BC in Greece, we can't be sure exactly, but it was a long time ago and we are as fascinated by him today as people in his time were. Little is known of Pythagoras' childhood. The only description of how he looked that is probably true is the description of a noticeable birthmark on his thigh! Information differs, some sources say that he had two brothers, although others state it was three.

What they agree on is that he was well educated; he was a fine musician, he played the lyre and used music to help people who were ill; he learned poetry and was able to recite famous and popular Greek writers like Homer. While he was a young student, three teachers who were philosophers greatly influenced Pythagoras. When he was betwee n 18 and 20 years old, Pythagoras left Greece and went to a town called Miletus, which is in the country we now call Turkey, and visited an old man named Thales.

Thales made a big impression on him and advised Pythagoras to travel to Egypt. While he was there he visited many temples and took part in discussions with the priests and learned from some of Thales' pupils about geometry and cosmology.

Because of the people he met and the experiences he had, Pythagoras became a philosopher like his teachers, but went on to make important discoveries in mathematics, astronomy, and the theory of music.

Pythagoras was taken prisoner and taken to Babylon where he continued his quest for learning new things. He was instructed in the sacred rites of the Bab ylonians and learnt about their mystical worship of the gods.

He perfected his skills in music and arithmetic as well as the other mathematical sciences taught by the Babylonians. Although he was a very important figure in the development of mathematical ideas, we don't know much about Pythagoras' actual mathematical achievements.

Hi, Raynel! How about Wonder Who Invented Numbers? Hi, Corvo! Hi, Bethany! That is a tricky question to answer, because the first ancient prehistoric people who likely developed simple methods of counting didn't leave any records behind to explain themselves.

If you're still interesting in this subject, we recommend that you take a Wonder JOurney to your local library to see what additional information you can discover! That's awesome, Reece! Have you seen our other History Wonders? Hi, tumba! Thank you for sharing your connection to this Wonder!

The Ishango Bone, an ancient artifact, was found in Africa in , and this bone had a series of lines on it that looked like tally marks! Thanks for asking, namebot! We ask that Wonderopolis be listed as the author. Since we do not list the publish date for our Wonders of the Day, you may put the date you accessed this page for information.

The following is how you would cite this page:. Accessed 30 Jan. We do!! We do have Wonder Who Invented the Toilet? Let us know what you think!! Of course we'll respond, darius!! We always love to hear from you, and we're glad that you liked this Wonder! We are undergoing some spring clearing site maintenance and need to temporarily disable the commenting feature. Thanks for your patience. Drag a word to its definition. You have answered 0 of 3 questions correctly and your score is:.

Want to add a little wonder to your website? Help spread the wonder of families learning together. We sent you SMS, for complete subscription please reply. Follow Twitter Instagram Facebook. Who invented numbers? Which famous mathematicians helped to develop numbers? What impact did numbers have on developing societies? Wonder What's Next? Try It Out Are you ready to count?

Don't forget to check out the following activities with a friend or family member: Can you believe that some of the oldest evidence of numbers was found on a bone?

Jump online to check out the Mathematical Treasure: Ishango Bone page to learn more about this interesting artifact. If you had found this bone, would you have had any idea what it showed and how important it was? Do you rely heavily on numbers on a daily basis? You might be surprised! Try going as long as you possibly can without using any numbers.

That means no television, since you need numbers to choose the proper channel. You also can't throw a snack in the microwave, since you'd need numbers to program it to cook for a certain amount of time. How else do numbers come up? Once you start thinking about it, you'll be amazed at how interwoven numbers are in your daily life! You're familiar with the normal base number system that uses , but did you realize there are a variety of other number systems out there, such as base-8, base-2, and even base?

Check out Number Systems online to learn how these other interesting number systems work! Did you get it? Test your knowledge. However, the Semicircle was more than just a school that studied intellectual disciplines, including in particular philosophy, mathematics and astronomy.

The latter is reflected in the Pythagorean motto: Number Rules the Universe. Pythagoreans consumed vegetarian dried and condensed food and unleavened bread as matzos, used by the Biblical Jewish priestly class the Kohanim , and used today during the Jewish holiday of Passover.

One reason for the rarity of Pythagoras original sources was that Pythagorean knowledge was passed on from one generation to the next by word of mouth, as writing material was scarce. Therefore, the true discovery of a particular Pythagorean result may never be known. Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths.

Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is 2 feet long will vibrate x times per second that is, hertz, a unit of frequency equal to one cycle per second , while a string that is 1 foot long will vibrate twice as fast: 2 x. Furthermore, those two frequencies create a perfect octave. The Pythagoreans were so troubled over the finding of irrational numbers that they swore each other to secrecy about its existence.

It is known that one Pythagorean did tell someone outside the school, and he was never to be found thereafter, that is, he was murdered, as Pythagoras himself was murdered by oppressors of the Semicircle of Pythagoras. Mesopotamia arrow 1 in Figure 2 was in the Near East in roughly the same geographical position as modern Iraq.

Mesopotamia was one of the great civilizations of antiquity, rising to prominence years ago. Thousands of clay tablets, found over the past two centuries, confirm a people who kept accurate records of astronomical events, and who excelled in the arts and literature.

Only a small fraction of this vast archeological treasure trove has been studied by scholars. The great majority of tablets lie in the basements of museums around the world, awaiting their turn to be deciphered and to provide a glimpse into the daily life of ancient Babylon.

The marks are in wedge-shaped characters, carved with a stylus into a piece of soft clay that was then dried in the sun or baked in an oven. They turn out to be numbers, written in the Babylonian numeration system that used the base In this sexagestimal system, numbers up to 59 were written in essentially the modern base numeration system, but without a zero.

Units were written as vertical Y-shaped notches, while tens were marked with similar notches written horizontally. What is the breadth? Its size is not known. And 5 times 5 is You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? The number along the upper left side is easily recognized as The conclusion is inescapable. This was probably the first number known to be irrational.

Two factors with regard to this tablet are particularly significant. First, it proves that the Babylonians knew how to compute the square root of a number with remarkable accuracy. The unknown scribe who carved these numbers into a clay tablet nearly years ago showed a simple method of computing: multiply the side of the square by the square root of 2. But there remains one unanswered question: Why did the scribe choose a side of 30 for his example?

Probably, 30 was used for convenience, as it was part of the Babylonian system of sexagesimal, a base numeral system. To Pythagoras it was a geometric statement about areas. It was with the rise of modern algebra, circa CE , that the theorem assumed its familiar algebraic form. In any right triangle, the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares whose sides are the two legs the two sides that meet at a right angle.

An area interpretation of this statement is shown in Figure 5. The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. Ancient Egyptians arrow 4, in Figure 2 , concentrated along the middle to lower reaches of the Nile River arrow 5, in Figure 2 , were a people in Northeastern Africa. The ancient civilization of the Egyptians thrived miles to the southwest of Mesopotamia.

The two nations coexisted in relative peace for over years, from circa BCE to the time of the Greeks. As to the claim that the Egyptians knew and used the Pythagorean Theorem in building the great pyramids, there is no evidence to support this claim. Egypt has over pyramids, most built as tombs for their country's Pharaohs. Egypt arrow 4, in Figure 2 and its pyramids are as immortally linked to King Tut as are Pythagoras and his famous theorem. King Tut ruled from the age of 8 for 9 years, — BC.

He was born in BC and died some believe he was murdered in BC at the age of Elisha Scott Loomis — Figure 7 , an eccentric mathematics teacher from Ohio, spent a lifetime collecting all known proofs of the Pythagorean Theorem and writing them up in The Pythagorean Proposition , a compendium of proofs.

The manuscript was prepared in and published in Loomis received literally hundreds of new proofs from after his book was released up until his death, but he could not keep up with his compendium.

As for the exact number of proofs, no one is sure how many there are. Surprisingly, geometricians often find it quite difficult to determine whether certain proofs are in fact distinct proofs. He died on 11 December , and the obituary was published as he had written it, except for the date of his death and the addresses of some of his survivors.

According to his autobiography, a preteen Albert Einstein Figure 8. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity and the sense that it reveals the connection between length and area that is at the heart of the theorem.