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The Parabolic Free Fall of an Object

A "free falling" object is an object that is only submitted to the gravity force.

Strictly speaking, no object on the earth is "free falling", because of the air resistance force.

But it is said to be so if the resistance of the air is neglegtible compared to the gravity force.

Examples of "free falling" objects are a tennis ball, a socker ball and a gun bullet.

All these objects follow trajectories that are parabolic curves.

The proof of it relies on the solution of integral equations:...

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Find Prime Numbers with Scilab®: Programing of the Sieve of Erastothenes

The Sieve of Erasthotenes is a very old algorithm to find by elimination the prime numbers up to a given integer N.

The manual process is so:

  1. write down the numbers from 2 to N (say 100)
  2. stripe the multiples of 2, except 2
  3. stripe the multiples of 3 not yet striped, except 3
  4. iterate the process for the next unstriped number k: stripe the multiples of k not yet striped, except k.
  5. continue until you get no more unstriped multiples before N

The unstriped remaining numbers are the numbers...

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A (wrong) proof that the square root of 2 is equal to 2

That post is a wrong "proof". We know it because it "proves" that the square root of 2 is equal to 2, and thus, elevating both sides to the power 2, that 2=4!

We do it by the means of successive geometrical constructions, in order to obtain a broken line that "approximates" a straight line of length the square root of 2.

First, we obtain the square root of 2 as the diagonal of a rectangle triangle of equal sides 1.

Second, we split the sides 1 of the triangle by 2 each, and construct a...

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The square root of 2 is a non rational number

The rational numbers are the results of exact divisions of integers.

They are represented as fractions or, more exactly, by equivalence classes of fractions.

Namely, each rational number has an infinity of representations as fractions, deduced from a "canonical" one, with mutually prime numbers. The canonical representation of a rational number has a numerator and a denominator that do not share any common divisor. The other representations are so that their numerators and denominators are...

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Music and Math: how is the Major C scale built?

The Greeks discovered the link between the musical intervals and the length of a string. The simpler example is that half the string sounds at the octave of the whole string. We know now that this corresponds to twice the frequency of the vibration.

Thus, 2 sounds differing by a 2 ratio on their vibrations have an interval of an octave. This is why the octave sounds so well to our ears, and was chosen as the periodicity of the scale.

The next such "harmonic" is when we divide the string in...

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Sharing a Pizza in 6 with trigonometry!

You may share a pizza in 6 equal portions with the help of trigonometry laws!

The method is so:

  • share the pizza in 2
  • share the diameter in 2 radius
  • share one radius in 2
  • elevate the perpendicular to the radius up the the circle
  • cut from the center to the obtained point.

You obtain a portion that is exactly the 6th of the pizza!

Why is it so?

It is because of trigonometry!

Indeed, if you cut a circle into 6 equal portions from the center, the angles of the portions are 60° each. And...

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Infinite in Theory, Finite in Practice

When we start to count on our fingers and continue by heart, we only have a small part of all the possible "natural integers": these are all the possible numbers we may count with.

We know that we may count until very very big numbers, the numbers we may imagine, and even further… These very big numbers may be the size of a Galaxy, the size of the known universe, the number of cells in a human body, in all the human bodies, and so on.

The mathematicians construct the set of all "natural...

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Why real numbers are generally not real

The numbers defined in mathematics are in the increasing complexity order:

- The natural integers, from 0 to infinity, that we use to count

- The relative integers, that are the natural integers plus the negative integers, that we use for instance for accountancy "credit" or "debit" notions

- The rational numbers, that we use to share a quantity: they are the result of divisions

- The real numbers, that are all the possible finite or infinite combination of digits, with or without a...

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Zero Divide: why it is an error

If you try to divide any number by 0 on your calculator, it displays an ERROR. Try it with 1/0, 10,000/0 or even 0/0. Is that normal?

It will tell you the truth about it: it is completely normal. This is in the theory of division in maths and has something to do woth the multiplication by 0.

What is the problem? You should remember first that the division is the reverse operation of the multiplication: a/b is the number c so that b x c = a. For instance, 6/2=3, because 2 x 3 = 6.

Now,...

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Our Practical Math first course is on the track !

Hello!

We are very pleased to announce you our forthcoming first course of Practical Math.

It will be about the numbers and there operations. The numbers will be of progressively more and more advanced type.

These are the following, in that order:

  •     The positive integers to count
  •     The natural integers with 0, which we know on that blog that it is an important number
  •     The relative integers with the negative numbers, introduced by the subtraction
  •     The rational...
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