About Mathematics

Prime Generating Polynomials : The polynomial,  n 2 + n + 41 n 2 + n + 41  can be used to produce 40 primes for consecutive integer values 0≤n≤39.  This property was discovered by Euler . Similarly, the incredible formula,  n 2 − 79 n + 1601 n 2 − 79 n + 1601  was discovered, which produces 80 primes for the consecutive values 0≤n≤79! Kaprekar’s Constant : 6174 is known as Kaprekar’s Constant , after the Indian Mathematician D.R. Kaprekar. Take any four-digit number, using at least two different digits (Leading zeros are allowed.) Arrange the digits in descending and then in ascending order, to get two four-digit numbers, adding leading zeroes if necessary. Subtract the smaller number from the larger number and go back to step 2. The above process will always reach the fixed number, 6174, taking at most 7 iterations. Try it yourself! ( Note that after the first subtraction or the subsequent subtractions, the result obtained is always a multi...

How is tan (A+B) = [tan A + tan B]/ [1 - tan A tan B]? Method 1: Let A = 30 deg and B = 45 deg. LHS = tan (30+45) = tan 75 = 3.732050808 RHS = [tan A + tan B]/[1 - tan A tan B] = [tan 30 + tan 45]/[1 - tan 30 tan 45] = [0.577350269 + 1]/[1 - 0.577350269*1] = 1.577350269/0.42264973 = 3.732050808 = LHS Proved. Method 2: tan (A+B) = [tan A + tan B]/[1 - tan A tan B] RHS = [tan A + tan B]/[1 - tan A tan B] =[(sin A/cos A) + (sin B/cos B)]/[1-(sin A/cos A)(sin B/cos B) = [sin A cos B + cos A sin B]/[cos A cos B][1 - sin A sin B/(cos A cos B)] = sin (A+B)/{[cos A cos B][cos A cos B - sin A sin B]/(cos A cos B)} = sin (A+B)/[cos A cos B - sin A sin B = sin (A+B)/cos (A+B) = tan (A+B) = LHS. Proved. Thanks. source:Quara

Well ever wondered how the  Surface Area of a Sphere  was derived? Well here is a great visualisation to alter your perception. Step 1 : Cut the sphere in the following way. Step 2 : Spread the cut out part across the paper Step 3 : Collate the pieces together in the following way Step 4 : Spread the areas out separately to form a sine curve Step 5 : The area of the sine curve is the surface area of the sphere Here’s a  GIF file  for better understanding. There’s of course the generic method of calculation where the surface area is calculated by cutting the sphere into infinitesimally thin disks of varying radius stacked over one another and integrating them, but this above method is a different way to look at the same problem. Image Source: Google Images 220 Loves 284. Yes, this pair of numbers was considered as a symbol of love in medieval period, lovers used to send flowers, fruits to each other with these numbe...