Transcendent numbers what are they, formulas, examples, exercises

The transcendent numbers are those that cannot be obtained as a result of a polynomial equation. The opposite of a transcendent number is a algebraic number, which are solutions of a polynomial equation of the type:

to_n xⁿ + to_n-1 x^n-1 +… + A_two x^two + to₁ x + a₀ = 0

Where the coefficients a_n, to_n-1,… to_two, to₁, to₀ are rational numbers, called the coefficients of the polynomial. If a number x is a solution of the previous equation, then that number is not transcendent.

We will analyze a few numbers and see if they are transcendent or not:

a) 3 is not transcendent because it is a solution of x - 3 = 0.

b) -2 cannot be transcendent because it is a solution of x + 2 = 0.

c) ⅓ is a solution of 3x - 1 = 0

d) A solution of the equation x^two - 2x + 1 = 0 is √2 -1, so this number by definition is not transcendent.

e) Neither is √2 because it is the result of the equation x^two - 2 = 0. Squaring √2 yields 2, which subtracted from 2 equals zero. So √2 is an irrational number but it is not transcendent.

Article index

• 1 What are transcendent numbers?
  • 1.1 The number π
  • 1.2 The number e
• 2 Formulas where the transcendent number π appears
  • 2.1 The perimeter of the circumference
  • 2.2 Area of ​​a circle
  • 2.3 Surface of a sphere
  • 2.4 Volume of the sphere
• 3 Exercises
  • 3.1 - Exercise 1
  • 3.2 - Exercise 2
• 4 References

What are transcendent numbers?

The problem is that there is no general rule to obtain them (later we will say a way), but some of the most famous are the number pi and the Neper number, denoted respectively by: π Y and.

The number π

The number π It appears naturally by observing that the mathematical quotient between the perimeter P of a circle and its diameter D, regardless of whether it is a small or large circle, always gives the same number, called pi:

π = P / D ≈ 3.14159…

This means that if the diameter of the circumference is taken as the unit of measure, for all of them, large or small, the perimeter will always be P = 3.14… = π, as can be seen in the animation of figure 2.

In order to determine more decimals, it is necessary to measure P and D with greater precision and then calculate the quotient, which has been done mathematically. The bottom line is that the decimals of the quotient have no end and never repeat themselves, so the number π besides being transcendent it is also irrational.

An irrational number is a number that cannot be expressed as the division of two whole numbers. 

It is known that every transcendent number is irrational, but it is not true that all irrational numbers are transcendent. For example √2 is irrational, but it is not transcendent.

The number e

The transcendent number e is the base of natural logarithms and its decimal approximation is:

e ≈ 2.718281828459045235360… .

If you wanted to write the number and exactly, it would be necessary to write infinite decimals, because every transcendent number is irrational, as said before.

The first ten digits of and are easy to remember:

2,7 1828 1828 and although it seems to follow a repetitive pattern, this is not achieved in decimals of order greater than nine.

A more formal definition of and is the next:

Which means that the exact value of and is achieved by performing the operation indicated in this formula, when the natural number n tends to infinity.

This explains why we can only obtain approximations of and, since no matter how large the number n is placed, you can always find a n higher.

Let's look for some approximations on our own:

-When n = 100 then (1 + 1/100)¹⁰⁰ = 2.70481 which hardly coincides in the first decimal with the “true” value of e.

-If you choose n = 10,000 you have (1 + 1 / 10,000)^10,000 = 2.71815 which matches the “exact” value of e in the first three decimal places.

This process would have to be followed infinitely in order to obtain the "true" value of e. I don't think we have time to do it, but let's try one more:

Let's use n = 100,000:

(1 + 1 / 100,000)^100,000 = 2.7182682372

That only has four decimal places matching the value considered exact.

The important thing is to understand that the higher the value of n chosen to calculate e_n, the closer it is to the true value. But that true value will only be had when n is infinite.

Other important numbers

Apart from these famous numbers there are other transcendent numbers, for example:

- two^√2

Every algebraic number, other than 0 or 1, raised to an irrational exponent will be a transcendent number.

-The Champernowne number in base 10: 

C_10 = 0.123456789101112131415161718192021… .

-The Champernowne number in base 2:

C_2 = 0.1101110010110111… .

-The Gamma number γ or Euler-Mascheroni constant:

γ ≈ 0.577 215 664 901 532 860 606

Which is obtained by doing the following calculation:

γ ≈ 1 + ½ + ⅓ + ¼ +… + 1 / n - ln (n)

When n be very very big. To have the exact value of the Gamma number you would need to do the calculation with n infinite. Something similar to what we did above.

And there are many more transcendent numbers. The great mathematician Georg Cantor, born in Russia and who lived between 1845 and 1918, showed that the set of transcendent numbers is much greater than the set of algebraic numbers.

Formulas where the transcendent number π appears

The perimeter of the circumference

P = π D = 2 π R, where P is the perimeter, D the diameter, and R the radius of the circumference. It should be remembered that:

-The diameter of the circumference is the longest segment that joins two points of the same and that always passes through its center,

-The radius is half the diameter and is the segment that goes from the center to the edge.

Area of ​​a circle

A = π R^two = ¼ π D^two

Surface of a sphere

S = 4 π R^two.

Yes. Although it may not seem like it, the surface of a sphere is the same as that of four circles of the same radius as the sphere..

Volume of the sphere

V = 4/3 π R³

Training

- Exercise 1

The “EXÓTICA” pizzeria sells pizzas of three diameters: small 30 cm, medium 37 cm and large 45 cm. A child is very hungry and he realized that two small pizzas cost the same as one large one. What will be better for him, buy two small pizzas or one large?

Solution

The larger the area, the greater the amount of pizza, for this reason the area of ​​a large pizza will be calculated and compared with that of two small pizzas:

Large pizza area = ¼ π D^two = ¼ ⋅3.1416⋅45^two = 1590.44 cm^two

Small pizza area = ¼ π d^two = ¼ ⋅3.1416⋅30^two = 706.86 cm^two

Therefore two small pizzas will have an area of 

2 x 706.86 = 1413.72 cm^two .

It is clear: you will have more pizza buying a single large one than two small ones.

- Exercise 2

The “EXÓTICA” pizzeria also sells a hemispherical pizza with a radius of 30 cm for the same price as a rectangular one measuring 30 x 40 cm on each side. Which would you choose?

Solution

As mentioned in the previous section, the surface area of ​​a sphere is four times that of a circle of the same diameter, so a hemisphere 30 cm in diameter will have:

12 '' hemispherical pizza: 1413.72 cm^two (twice a circular of the same diameter)

Rectangular pizza: (30 cm) x (40 cm) = 1200 cm^two .

The hemispherical pizza has a larger area.

References

1. Fernández J. The number e. Origin and curiosities. Recovered from: soymatematicas.com
2. Enjoy math. Euler's number. Recovered from: gustolasmatematicas.com.
3. Figuera, J. 2000. Mathematics 1st. Diversified. CO-BO editions.
4. García, M. The number e in elementary calculus. Recovered from: matematica.ciens.ucv.ve.
5. Wikipedia. PI number. Recovered from: wikipedia.com
6. Wikipedia. Transcendent numbers. Recovered from: wikipedia.com