If we divide 23 by 3 the answer will be 7 and the remainder 2. 3. Examples: Let’s say we have four natural numbers 2, 3, 4, 5 and the four prime numbers 2, 3, 5, 7 and we want to test Fermat’s little theorem: When a = 2 and p = 2 22 – 2 = 2 2 ÷ 2 = 1 (clearly 2 is divisible by 2 and has a remainder of 0) When a = 3 and p = 3 33 – 3 = 24 24 ÷ 3 = 8 (24 is divisible by 3 and has a remainder of 0) When a = 4 and p = 5 45 – 4 = 1020 1020 ÷ 5 =204 (1020 is divisible by 5 and has a remainder of 0) When a = 5 and p = 7 57 –... ... middle of paper ... ...is) as well as in number theory. The theorem is used in the encryption of data, which is the process of encoding information in such a way that only authorized parties can read it by unlocking the hidden information using a decryption key. 7.
Obviously Euclid’s The Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world’s oldest continuously used mathematical textbook. Very little information is known about the author, beyond knowing the fact he lived in Alexandria around 300 BCE. Subjects of works includes geometry, proportion and number theory. Euclid proved his concepts logically, using definitions, axioms, and postulates. Proclus Diadochus wrote a commentary on Euclid's Elements that kept Euclid's works in circulation.
Our second member would be Pythagoras, He was born in Ionian Greek in 570 to 495 B.C. He is also known to be the founder of the religious movement. There is not a lot of information about him, but most say he traveled to Egypt for Knowledge. What he was most famous for was proving his own theory called the Pythagoras theorem. Pythagoras theorem is used in geometry, it’s about a right angle triangle that is equal to the sum of an area.
Apollonius of Perga Apollonius was a great mathematician, known by his contempories as " The Great Geometer, " whose treatise Conics is one of the greatest scientific works from the ancient world. Most of his other treatise were lost, although their titles and a general indication of their contents were passed on by later writers, especially Pappus of Alexandria. As a youth Apollonius studied in Alexandria ( under the pupils of Euclid, according to Pappus ) and subsequently taught at the university there. He visited Pergamum, capital of a Hellenistic kingdom in western Anatolia, where a university and library similar to those in Alexandria had recently been built. While at Pergamum he met Eudemus and Attaluus, and he wrote the first edition of Conics.
Double Numbers Examples: 12 (5,7)= 12 à 5+7 à 5x7=35 (6,6)= 12 à 6+6 à 6x6=36 (4,8)= 12 à 4+8 à 4x8=32 I have found that 36 is the highest number so far that can be retrieved from 6 and 6 when the number is 12, in whole numbers. I will now try in decimal numbers if I can get a number higher than 36. (6.5,5.5)=12 à 6.5+5.5 à 6.5x5.5=35.75 (6.7,5.3)=12 à 6.7+5.3 à 6.7x5.3=35.51 (6.3,5.7)=12 à 6.3+5.7 à 6.3x5.7=35.91 (6.2,5.8)=12 à 6.2+5.8 à 6.
Around Two thousand five hundred years ago, a Greek mathematician, Pythagoras, invented the Pythagorean Theorem. The Theorem was related to the length of each side of a right-angled triangle. In a right-angled triangle, the square on the hypotenuse, the side opposite to the right angle, equals to the sum of the squares on the other two sides. (148, Poskitt) To know more about this famous theorem, we can look at the other forms of the Pythagorean Theorem, such as it can also be written as c^2-a^2=b^2 which is for reverse operations like finding side b with the data of a and c. Meanwhile, the proofs of the theorem can make us understand more about the invention of the theorem and how Pythagoras figured it out. And with the invention of this theorem, we shall look into where this theorem was used in these days and how important it is.
Pythagoras' Theorem I am going to study Pythagoras' theorem. Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle. For example, I will use 32 x 42 = 52 . This is because: 32 = 3 x 3 = 9 42 = 4 x 4 = 16 52 = 5 x 5 = 25 So.. 9 +16 = 25 For this table, I am using the term a, b, b + 1 Triangle Number (n) Length of shortest side Length of middle side Length of longest side Perimeter Area 1 3 4 5 12 6 2 5 12 13 30 30 3 7 24 25 56 84 4 9 40 41 90 180 5 11 60 61 132 330 6 13 84 85 183 546 7 15 112 113 240 840 8 17 144 145 296 1224 Formulas ======== Shortest side = 2n + 1, n being the triangle number Middle side = 2n2 + 2n.
In the 3:4:5 triangle, 5 is the hypotenuse. 3 and 4 are the legs. The angle opposite of the hypotenuse is a right angle. Since the hypotenuse squared equals the two legs squared, the equation should be: Example without the 3:4:5 triangle: Although the theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier he may have been the first to prove it (Maxwell, Seth) Of Pythagoras actual work nothing is known. It is hard to tell the difference between his work and the work of his students.
Thus, we can use the following algebraic proportion to represent definition 5.5: (m)a : (n)b :: (m)c : (n)d. However, it is necessary to be more specific because of the way in which the definition was worded with the phrase "the former equimultiples alike exceed, are alike equal to, or alike fall short of….". Thus, if we take any four magnitudes a, b, c, d, it is defined that if equimultiple m is taken of a and c, and equimultiple n is taken of c and d, then a and b are in same ratio with c and d, that is, a : b :: c : d, only if: (m)a > (n)b and (m)c > (n)d, or (m)a = (n)b and (m)c = (n)d, or (m)a < (n)b and (m)c < (n)d. Though, because magnitudes are continuous quantities, and an exact measurement of magnitudes is impossible, it is not possible to say by how much one exceeds the other, nor is it possible to determine if a > b by the same amount that c > d. Now, it is important to realize that taking equimultiples is not a test to see if magnitudes are in the same ratio, but rather it is a condition that defines it. And because of the phrase "any equimultiples whatever," it would be correct to say that if a and b are in same ratio with c and d, then any one of the three
e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa. You can see more about these numbers in the History topics article Perfect numbers. By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved.