- What are transcendent numbers?
- The number π
- The number e
- Formulas where the transcendent number π appears
- The perimeter of the circumference
- Area of a circle
- Surface of a sphere
- Volume of the sphere
- Exercises
- - Exercise 1
- Solution
- - Exercise 2
- Solution
- References
The transcendental numbers are those that can not be obtained as a result of a polynomial equation. The opposite of a transcendent number is an algebraic number, which are solutions of a polynomial equation of the type:
a n x n + a n-1 x n-1 + …… + a 2 x 2 + a 1 x + a 0 = 0
Where the coefficients a n, a n-1,….. a 2, a 1, a 0 are rational numbers, called the coefficients of the polynomial. If a number x is a solution to the previous equation, then that number is not transcendent.
Figure 1. Two numbers of great importance in science are transcendent numbers. Source: publicdomainpictures.net.
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 2 - 2x + 1 = 0 is √2 -1, so that number by definition is not transcendent.
e) Neither is √2 because it is the result of the equation x 2 - 2 = 0. By squaring √2 it results in 2, which subtracted from 2 equals zero. So √2 is an irrational number but it is not transcendent.
What are transcendent numbers?
The problem is that there is no general rule to obtain them (we will say a way later), but some of the most famous are the number pi and the Neper number, denoted respectively by: π and e.
The number π
The number π 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 measurement, for all of them, large or small, the perimeter will always be P = 3.14… = π, as can be seen in the animation in figure 2.
Figure 2. The length of the perimeter of a circle is pi times the length of the diameter, with pi being approximately 3.1416.
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 conclusion is that the decimals of the quotient have no end and never repeat themselves, so the number π in addition to being transcendent 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.
Figure 3. The transcendent numbers are irrational, but the converse is not true.
The number e
The transcendent number e is the base of natural logarithms and its decimal approximation is:
and ≈ 2.718281828459045235360….
If you wanted to write the number e exactly, it would be necessary to write infinite decimals, because every transcendent number is irrational, as stated before.
The first ten digits of e 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 e is as follows:
This means that the exact value of e is obtained 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 e, since no matter how large the number n is placed, a greater n can always be found.
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 coincides with 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 that match 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 will be to the true value. But that true value will only have when n is infinite.
Figure 4. It is shown graphically how the higher the value of n, the closer to e, but to arrive at the exact value n must be infinite.
Other important numbers
Apart from these famous numbers there are other transcendent numbers, for example:
- 2 √2
-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)
For when n is very very big. To have the exact value of the Gamma number, it would be necessary to do the calculation with n infinity. Something similar to what we did above.
And there are many more transcendent numbers. The great mathematician Georg Cantor, born in Russia and living 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 2 = ¼ π D 2
Surface of a sphere
S = 4 π R 2.
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 R 3
Exercises
- Exercise 1
The “EXÓTICA” pizzeria sells pizzas of three diameters: small 30 cm, medium 37 cm and large 45 cm. A boy is very hungry and he realized that two small pizzas cost the same as one large one. What will be better for him, to buy two small pizzas or one large one?
Figure 5.- The area of a pizza is proportional to the square of the radius, pi being the constant of proportionality. Source: Pixabay.
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:
Area of the large pizza = ¼ π D 2 = ¼ ⋅3.1416⋅45 2 = 1590.44 cm 2
Area of the small pizza = ¼ π d 2 = ¼ ⋅3.1416⋅30 2 = 706.86 cm 2
Therefore two small pizzas will have an area of
2 x 706.86 = 1413.72 cm 2.
It is clear: you will have a greater amount of 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 one would you choose?
Figure 6.- The surface of a hemisphere is twice the circular surface of the base. Source: F. Zapata.
Solution
As mentioned in the previous section, the surface of a sphere is four times that of a circle of the same diameter, so a hemisphere 30 cm in diameter will have:
30 cm hemispherical pizza: 1413.72 cm 2 (twice a circular of the same diameter)
Rectangular pizza: (30 cm) x (40 cm) = 1200 cm 2.
The hemispherical pizza has a larger area.
References
- Fernández J. The number e. Origin and curiosities. Recovered from: soymatematicas.com
- Enjoy math. Euler's number. Recovered from: enjoylasmatematicas.com.
- Figuera, J. 2000. Mathematics 1st. Diversified. CO-BO editions.
- García, M. The number e in elementary calculus. Recovered from: matematica.ciens.ucv.ve.
- Wikipedia. PI number. Recovered from: wikipedia.com
- Wikipedia. Transcendent numbers. Recovered from: wikipedia.com