Math titbits
umasundresh
For example:
If we consider the three consecutive numbers 7,8 and 9
82 = 64
7 X 9 = 63.
umasundresh
Five Postulates of Euclidean Geometry
The elements of geometry, an exquisite and influential geometry treatise written around 300 B.C by the Greek mathematician Euclid, explains Euclidean geometry.
The five postulates form the foundation of euclidean geometry.
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Are there only 64 squares present on the chess board?
This is a trick question because it is possible for someone to get confused and immediately calculate the total number of squares in a chessboard with eight rows and eight columns by using the formula (number of rows X number of columns)
= 8 X 8 = 64.
Let’s start by reducing the complexity of the situation.
The total number of 1X1 squares is presented in eight rows by eight columns on the chess board.
= 8 X 8 =64.
However, if you start thinking about 2X2 squares, 3X3 squares, and so on up to 8X8 squares, you can figure out how to answer the question.
The total number of 2X2 squares is presented in seven rows by seven columns on the chess board.
= 7 X 7 =49.
The total number of 3X3 squares is presented in six rows by six columns on the chess board.
= 6 X 6 =36.
The total number of 4X4 squares is presented in five rows by five columns on the chess board.
= 5 X 5 =25.
The total number of 5X5 squares is presented in four rows by four columns on the chess board.
= 4 X 4 = 16.
The total number of 6X6 squares is presented in three rows by three columns on the chess board.
= 3 X 3 = 9.
The total number of 7X7 squares is presented in two rows by two columns on the chess board.
= 2 X 2 = 4.
The total number of 8X8 squares is presented in one row by one column on the chess board.
= 1 X 1 = 1.
Therefore, the total number of squares presented on the chess board is found by summing up all the values obtained by 1X1, 2X2, 3X3, 4X4, 5X5,6X6,7X7 and 8X8 squares.
i.e., 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204.
Therefore, the total number of squares on the chess board is 204.
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Suitcase lock problem
You have recently purchased a suitcase that features a combination number lock. Each slider provides a selection of ten numbers, ranging from 0 to 9. In order to prevent anyone from opening your suitcase, you will need to come up with a secret number combination. Do you know how many different possible choices there are for secret codes?
Solution:
You have an option of ten different digits, ranging from zero to nine, for each of the digits in the slider.
Therefore, the number of different combinations that can be made are
10 X 10 X 10 = 1000 options.
If there is a condition that the digit in the slider should not be repeated, then the following applies:
You will have a choice of 10 numbers for the first slider, and 9 for the second slider and 8 for the third slider.
Which means, we have
10 X 9 X 8 = 720 options.
The following video will explain the given problem:
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Attoseconds
Atto means eighteenth in Danish, Attosecond means 1/1,000,000,000,000,000,000th of a second.
In one attosecond, light can travel the length of 3 hydrogen atoms.
Generally, light can travel 7.5 times around the earth in 1 second. However, in an attosecond it can barely move from one end of the molecule to the other end.
Another way to think about this is:
one attosecond : one second = one second : 32 billion years.
The Flash, a superhero from a comic book who has the ability to run faster than the speed of light, has the ability to see events that occur for less than one attosecond, which, for humans, is less time than it takes to blink an eye.
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Fictional Mathematicians
Get to know some mathematicians whose whole existence was conjured up by the authors’ vivid imaginations.
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Platonic Solids
Platonic solids are three-dimensional geometrical objects that have been studied for millennia due to their symmetry and beauty. Euclid, a Greek Mathematician proved that there are exactly five such solids.
They are a tetrahedron, a cube, an octahedron, a dodecahedron, and an icosahedron.
Regular polygons are those with all of their sides equal. Since a Regular Polygon has equal sides, it also has equal angles.
| Number of sides ‘n’ | Regular Polygon |
| 3 | Equilateral Triangle |
| 4 | Square |
| 5 | Pentagon |
| 6 | Hexagon |
and so on…
A Platonic solid is a three-dimensional shape, each face is a regular polygon, and the same number of polygons intersect at each vertex.
There are only five Platonic solids exist.
1. Cube
- The cube consists of 6 squares
- Three squares meet at each vertex
- 8 vertices
- 12 edges
2. Tetrahedron
- The Tetrahedron consists of 4 Equilateral triangles.
- Three triangles meet at each vertex.
- 4 Vertices
- 6 Edges
3. Octahedron
- The Octahedron consists of 8 triangles
- 4 triangles meet at each vertex
- 8 Faces
- 6 Vertices
- 12 Edges
4. Dodecahedron
- The dodecahedron consists of 12 Pentagons.
- 3 pentagons meet at each vertex
- 20 Vertices
- 30 Edges
5. Icosahedron
- The Icosahedron consists of 20 Equilateral triangles.
- 5 triangles meet at each vertex
- 12 Vertices
- 30 Edges
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Seconds in 6 weeks=10!
We know that, for any positive integer n, n! is defined as the product of all positive integers less than or equal to n.
i.e., n! = 1 x 2 x 3 x … x (n-1) x n
so, 10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800
we see that, how seconds in 6 weeks make 10!
6 weeks = 6 x 7 days = 6 x 7 x 24 hours = 6 x 7 x 24 x 60 minutes = 6 x 7 x 24 x 60 x 60 seconds = 3,628,800 seconds
This video verifies that seconds in 6 weeks = 10!
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Cut a cake into eight equal pieces by using only three cuts.
1.Two cuts should make a cross on the surface of the cake, therefore splitting it into four equal halves
2. Third cut as a horizontal slit through the cake’s center.
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Applications of Logarithm in real life
In science, logarithm is used for many different things.
Some of the most common uses are to measure loudness (in decibels), earthquake intensity (on the Richter scale), radioactive decay, and acidity (pH = -log10[H+]).
Logarithms are also used to solve problems that involve exponential growth.
The following video summarizes the applications of logarithm in real life.
