Math ᶠᵃᶜᵗˢ ᵃⁿᵈ ᵠᵘᵉˢᵗᶦᵒⁿˢ

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Facts

Did you know 

1. Rhombicosidodecahedron  is an Archimedean solid, which has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, and 120 edges.
2. The equation (x - h)^2 + (y - k)^2 = r^2, in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (0, 0), h and k are both 0. (To understand in detail view my blog Graph of a circle )
3. A hypersphere is the four-dimensional analog of a sphere. Although a sphere exists in 3-space, its surface is two-dimensional. Similarly, a hypersphere has a three-dimensional surface which curves into 4-space.
4. A dekeract, in geometry, is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.
5. The Mandelbrot set, named after mathematician Benoit Mandelbrot, is a complex mathematical object that is formed by iterating a simple equation over and over again. The set is known for its intricate and never-ending patterns and is a popular subject of study in the field of fractal geometry.
6. In number theory, Fermat's Last Theorem states that there are no non-zero integers a, b, and c that satisfy the equation a^n + b^n = c^n for any value of n greater than 2. The theorem was famously unsolved for over 350 years until it was finally proven by Andrew Wiles in 1994.
7. The concept of imaginary numbers, which are numbers that cannot be expressed as real numbers, was first introduced by mathematician Rafael Bombelli in the 16th century. Imaginary numbers are represented by the letter "i" and are the square root of -1. They are used in many areas of mathematics, including complex numbers, quantum mechanics, and electrical engineering.
8. The concept of non-Euclidean geometry, which is a branch of mathematics that explores the properties of geometries that do not follow the traditional postulates of Euclidean geometry, was first introduced by mathematicians János Bolyai and Nikolai Lobachevsky in the 19th century. Non-Euclidean geometry is used in many areas of mathematics, such as general relativity and cosmology.
9. The concept of infinity, which is the idea of an unbounded quantity, was first introduced by mathematician Georg Cantor in the late 19th century. Cantor's work laid the foundation for set theory and the study of infinite sets, which is an important branch of mathematics with applications in many areas such as computer science, physics, and engineering.
10. The Riemann Hypothesis is a unsolved problem in number theory, proposed by Bernhard Riemann in 1859, which states that all non-trivial zeroes of the Riemann zeta function, which encodes the distribution of prime numbers, lie on the critical line of 1/2. The solution of this problem would have important implications in number theory, cryptography, and physics.
11. The Euler's formula, named after mathematician Leonhard Euler, states that e^(i*pi) + 1 = 0, where e is the base of the natural logarithm, i is the imaginary unit and pi is the ratio of a circle's circumference to its diameter. It's widely used in physics and engineering, and connects five of the most important numbers in mathematics: 0, 1, i, pi, and e.
12. The Pythagorean Theorem, states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is not only found in Euclidean geometry but also in non-Euclidean geometries, like the hyperbolic geometry.
13. The Fourier Transform, named after Joseph Fourier, is a mathematical tool that breaks down a function into its individual frequencies. It is used in many areas of science and engineering, such as signal processing, image compression, and quantum mechanics.
14. The Game theory, which is a mathematical framework for modeling decision-making in strategic situations, was invented by John von Neumann and Oskar Morgenstern in 1944. It's widely used in many fields such as economics, political science, biology, and computer science.
15. The Lucas numbers are a sequence of integers that are defined recursively, with the initial terms 2 and 1, and the recurrence relation L_n = L_{n-1} + L_{n-2}. The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, ...                                                                                                      

Questions

1. Which is the coolest number? And why?

Ans:- one reason why the best number is 73 is that it's the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3. In binary 73 is a palindrome, 1001001, which backwards is 1001001. 

Another reason why 73 may be considered "cool" is its prevalence in mathematics. For example, 73 is a prime number, and it is the 21st Mersenne prime (a prime number that is one less than a power of 2). In addition, the number 73 is the 21st Lucas number, and it appears in many other mathematical sequences and patterns.

73 also appears in various other fields. For example, in computer science, the ASCII code for the letter 'I' is 73. In astronomy, the Saros cycle of lunar eclipses is about 73 years. In literature, J.R.R. Tolkien's The Lord of the Rings has 73 chapters.

One reason why some people might consider 73 to be a "cool" number is that it has some interesting properties. For example, 73 is the 21st Lucas number, which is a sequence of integers that appears in many different areas of mathematics, including number theory, combinatorics, and the Fibonacci sequence. Additionally, 73 is a prime number, which means it can only be divided by 1 and itself.

2. Prove that there are infinitely many prime numbers.

Answer: 

Euclid's proof, also known as the Euclid's theorem, is a mathematical proof that states that there are infinitely many prime numbers. The proof is based on the assumption that there are only finitely many prime numbers, and uses this assumption to prove the existence of at least one more prime number.

The proof begins by assuming that there are only finitely many prime numbers, and listing them as p1, p2, ..., pn. Next, the proof considers the number N, which is defined as the product of all the prime numbers listed plus one. This number, N, is clearly not divisible by any of the prime numbers listed, p1, p2, ..., pn. This is because if N were divisible by any of the prime numbers, it would mean that N would be divisible by that prime number and therefore would not be prime.

However, since N is greater than 1, it must be divisible by some prime number. Let's call this prime number q. This contradicts the assumption that the prime numbers p1, p2, ..., pn were all the prime numbers, since q is a prime not in the list. Therefore, there must be infinitely many primes.

In simpler terms, Euclid's proof uses the fact that if there are only a finite number of prime numbers, we can multiply them all together and add 1, resulting in a number that is not divisible by any of the primes we started with. But since it is greater than 1, it must be divisible by some prime, which contradicts the assumption that the primes we started with were all the primes there are. Therefore, there must be infinitely many primes.

Euclid's proof is a very important mathematical concept that has been used for centuries to demonstrate the existence of infinitely many prime numbers. It is also a fundamental concept in number theory, and is used in many other mathematical proofs and theories.

3. Prove that the square root of 2 is irrational.

Answer: Suppose the square root of 2 is rational, say a/b where a and b are integers and b is not zero. Then we can square both sides to get 2 = (a/b)^2 = a^2/b^2. This means that a^2 = 2b^2, so a^2 is even. This means that a is even, say a = 2c. Substituting, we get 2b^2 = (2c)^2 = 4c^2, so b^2 = 2c^2. This means that b is even, say b = 2d. But this means that a and b have a common factor of 2, which contradicts the assumption that they are relatively prime. Therefore, the square root of 2 is irrational.

4. Prove that the sum of the angle in a triangle is 180 degrees

Answer: Draw a triangle with vertices A, B, and C. Now, draw an altitude from vertex A to the line BC. Call the point where the altitude intersects BC D. Now, we have a right triangle ADC with right angle at D. The angle ADC and angle BAC are complementary since they share an angle at D. Similarly, angle ADC and angle ACB are also complementary. Therefore, angle BAC and angle ACB add up to 90 degrees. By symmetry, angle BAC and angle CBA add up to 90 degrees as well. Therefore, angle BAC + angle ACB + angle CBA = 90+90 = 180 degrees.

5. A farmer has a rectangular field that measures 8 acres. He wants to divide the field into two smaller fields with the same area. What dimensions should the two smaller fields have?

Answer: Let the length of the smaller fields be x and the width be y. We know that the area of the field is 8 acres, so we can set up the equation:

xy = 8

Since the two smaller fields have the same area, we can divide the 8 acres equally between them, which means that each field will have an area of 8/2 = 4 acres. So, we can set up the equation:

(x/2)(y/2) = 4

To find the dimensions, we can cross-multiply and combine like terms to get:

xy = 16

To find the dimensions, we can either set x=4,y=2 or x=8,y=1 because both are satisfy the equation xy =16

6. A company wants to build a fence around a rectangular storage area. The storage area has a length of 60 feet and a width of 40 feet. The company wants to use the least amount of fencing material possible. What is the length of the fence? 

Answer: The perimeter of the rectangular storage area is the total length of the fencing material that is needed. To find the perimeter, we add up the lengths of all four sides of the rectangle. The perimeter is given by the formula:

P = 2L + 2W

where P is the perimeter, L is the length, and W is the width. Substituting in the given values, we get:

P = 2(60) + 2(40) = 120 + 80 = 200 feet

So, the company needs 200 feet of fencing material to build the fence around the storage area.

7. If a contestant is presented with three doors, behind one of which is a valuable prize and behind the other two are goats, and the contestant chooses one door but before the door is opened, the host Monty Hall, who knows what is behind each door, opens one of the other two doors to reveal a goat. The host then gives the contestant the option to stick with their original choice or to switch to the remaining door. What is the probability of winning the prize if the contestant sticks with their original choice and what is the probability of winning the prize if the contestant switches to the remaining door?

Answer: The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was first posed (and solved) by Steve Selvin in 1975.

The Monty Hall problem is a probability puzzle that goes as follows:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Let's call the door you picked door A, the door with the car door C, and the remaining door door B.

When you pick a door, there is a 1/3 chance that you have picked door C, and a 2/3 chance that you have picked a goat.

When Monty shows you a goat, he eliminates one of the goats, and the probability that the car is behind door A reduces to 1/2 and the probability that the car is behind door B becomes 1/2.

So, the probability of winning if you switch is 1/2, and the probability of winning if you stick with your original choice is 1/3. Therefore, it is more likely that you will win the car if you switch.

This is a famous problem and is known as Monty Hall problem which is a counter-intuitive probability problem, where people get confused with the winning probability based on the choice of switching or staying.