simplified and solved
A cyclic number is a number of “n” digits that when multiplied by 1, 2, 3, …, n results in the same digits but in a different order.
Let’s find out whether 142857 is a cycling number or not
Here the number of digits is equal to 6, Therefore we check whether the given number is cyclic or not by multiplying it by 1, 2, 3, 4, 5, 6.
Yes it is.
Hippasus, a Greek philosopher who was an early disciple of Pythagoras, is credited with being the first person to discover the existence of irrational numbers.
However, many of Pythagoras’s disciples couldn’t wrap their heads around the concept of irrational numbers and also couldn’t contradict Hippasus with reasoning.
He was drowned to death upon Pythagoras’ command.
The proof is by contradiction.
First let us assume that √2 is rational.
This means that, x/y = √2, where x and y have no common factors.
Squaring on both sides, we get:
x2 /y2 = 2
x2 = 2y2 (The number is even on the right-hand side. Because any number multiplied by 2 is always even.)
The only way the equation to be true is that x itself should be even.
Thus x2 is even and it is divisible by 4.
Therefore, x and y are even numbers with common factors. This contradicts our assumption that x and y have no common factors.
Therefore, √2 cannot be rational.
The square root of 2 is written as √2. The square root of 2 is often known as root 2, radical 2, or the Pythagoras constant.
Additionally, it is the first irrational number to ever be identified.
Geometrically,
Consider a square with side length 1, and the need to determine the diagonal length.
The formula for the diagonal of a square is derived using the Pythagoras theorem.
The length of a diagonal across a square with sides of one unit of length is equal to the √2.
The value of √2 up to 15 decimal places is 1.414213562373095…The value of √2 is currently known to 1 trillion decimal places.
An intriguing fact about paper sizes is that they are dependent on √2.
The international paper size standard ISO 216 (International Organisation for Standardisation) is the standard that paper sizes are all based on.
In the ISO paper size system, the height-to-width ratio of all pages is the square root of two (1.4142 : 1). In other words, the width and the height of a page relate to each other like the side and the diagonal of a square.
When bees construct honeycombs, they do it in a hexagonal layout.
The honeycomb’s hexagonal structure, which can be used to store honey, pollen, or eggs, is created out of wax by the bees.
Hexagons are a common shape used by bees. So why not go with a more simple shape like a circle, triangle, or square?
A tiling is any pattern that repeats but doesn’t overlap on a flat surface. A tessellation is another word for tiling.
The three geometrical figures with equal sides can fit together on a flat surface without leaving gaps: equilateral triangles, squares, and hexagons.
Of these three Hexagon has the least perimeter. That’s why nature has chosen the hexagon shape.
The number 6174 is called the Kaprekar constant. It was discovered by the Indian Mathematician Kaprekar.
We get this constant if we perform certain routine calculations to any four-digit number.
The rule is
1. Take any 4 digit number.
2. Arrange the digits of the 4 digit number in descending and ascending order.
3. Subtract the smaller number from the bigger number
4. Repeat the steps from 2 for the answer obtained also.
Repeat the steps till you reach 6174.
We always end up with 6174. Once you reach 6174, the process will continue generating the same number.
Let us solve an example to understand this..
A Kaprekar number is
A number’s square divided into two parts such that the sum of its parts is equal to the original number.
An example of this Kaprekar Number is explained in the following video.
D.R.Kaprekar was a Nashik School Math teacher, He discovered several classes of Numbers and constants in the field of Number theory, which are named after him as Kaprekar Number and Kaprekar Constant.
Other discoveries include Self Numbers, Harshad Number and Demlo Number.
He became popular when Martin Gartner wrote about him on his article for ‘Mathematical Games for Scientific Americans’.
Nicolaus Copernicus was an astronomer who first proposed that the Earth and the other planets orbit around the Sun and also rotate on their own axis.
During Copernicus’ times, it was believed the Earth was at the center of the solar system and the Sun, the Moon and the planets revolve around the Earth.
His discovery of heliocentric theory with Sun at the center of the solar system was not accepted by the scholars of Philosophy, Astronomy and Religious fields during his time.
Galileo and Kepler developed and spread the Copernican theory.
By the late 18th century, the Copernican view of the solar system was almost universally accepted.
umasundresh