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Exponentiation and the exponential increasing

The exponentiation is the action of multiplying a number by itself a given number of times.

For instance, 3 to the power 2 is 3x3=9, 10 to the power 3 is 10x10x10=1,000.

A sequence starting wit u(0)=C and continuing with the iterative formula u(n+1)=a x u(n) has for value u(n) equal to C multiplied by a to the power n.

Such a sequence is called a geometrical sequence of initial point C and "reason" a>0.

There are 3 cases for the behavior of the sequence:

  • if a>1, the sequence goes to...
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The Measurement of Time II: the time on the clock

The day is the smallest time unit that has an astronomic reference: the time by which the earth spins on itself by one turnaround.

But it is not the smallest duration we may evaluate: our clocks show us the hours and minutes, and even the seconds.

These are obtained from the day duration by successive divisions:

  • The hour is the 24th part of the day, that is that 1 day lasts 24 hours of same duration.
  • The minute is the 60th part of the hour: an hour lasts 60 minutes
  • The second is the...
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The Measurement of Time I: the dates in the calendar

As we saw in a previous post, the metric system for the measurement of lengths is a decimal system, and the time measurement system is not.

First, the basic time units are NOT the hour or the second, but the day, the month and the year:

  • The day is the time by which the earth spins on its axis completely.
    It used to be measured by the time between 2 instants at which the sun is at its highest position in the sky.
  • The year is the time by which the earth turns around the sun by a full...
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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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