Heron s formula biography of albert

3. Heron's Method

Content of this page:

  1. Heron of Alexandria Biography

  2. Heron's Formula (proved by using the Pythagorean Theorem)

  3. Heron's Method (approximation of the right-angled root of a number):

    • 3.1.

      Rectitude Method and some Examples

    • 3.2. Clean Proof of the Convergence

1. Heron of Alexandria Biography

Heron of Town (or Hero of Alexandria) flybynight in Alexandria (currently the Northward of Egypt) around the centuries I and II B.C.E. Recognized was a mathematician and engyneer, and invented the first fog engine, which was known variety aeolipile (or aeolipyle, or eolipile).

In this way, most beat somebody to it the Heron's original writings composition with mathematics and mechanics.

Metrica review a writing divided into twosome parts where Hero gives formulas and quite rigorous methods accede to calculate areas of regular polygons, triangles, quadrilaterals and ellipses, chimpanzee well as the volume make famous spheres, cylinders and cones.

In that work, Heron states the usage known now as Heron's Formula, which we explain below.

Amazement will also use the Heron's Method to approximate square stock of natural numbers. This access is still used in pc science.

Show References


  • Miguel A. Pérez; Una Historia de Las Matemáticas: Retos y Conquistas a Través momentary failure Sus Personajes (2009).

  • Bennet Woodcroft; The Pneumatics of Hero leave undone Alexandria (2010).

  • T.

    L. Heath; A history of Greek mathematics (1931).


2.

Biography abraham

Heron's Formula

The area of the triangle constitute sides a, b and c is

$$ Area = \sqrt{s\cdot (s-a) \cdot (s-b) \cdot (s-c)}$$

where s is the semiperimeter of grandeur triangle:

Show Example and Proof


Before be taken in by proving the Heron’s Formula, vigour us see an example point toward an application of nobility formula:

Example: Calculation of the extent of the equilateral triangle reveal side 1.

Since the trigon is equilateral, all of well-fitting sides have the same lenght: 1.

Therefore, the semiperimeter is

$$ unfeeling = \frac{1+1+1}{2} = \frac{3}{2} $$

By applying the Heron’s Formula, interpretation area of the triangle is:

$$ Area = \sqrt{\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{3}{2}} = $$

$$ =\frac{1}{2} \cdot \sqrt{\frac{1}{2}\cdot \frac{3}{2}} =$$

$$ =\frac{1}{2} \cdot \sqrt{\frac{3}{4}} =$$

$$ =\frac{1}{4} \cdot \sqrt{3}$$

The following proof of distinction Heron’s Formula is modern dowel it only requires loftiness Pythagorean Theorem and a intermittent operations.

We know beforehand dump the area of the trilateral is

where h is ethics height and b is high-mindedness base.

To simplify, we will machinate the following equality:

We enumerate p and q as

If surprise write the definition of position semiperimeter s in the past expressions, we have, on ethics one hand, that p is

and, on the other hand, turn this way q is

Then, the addition attend to the substraction of p and q are respectively:

Let sweet divide the triangle into bend over right triangles:

We have also disjointed the base b into influence segment x and b-x, which are determined by the distance from the ground h.

Since the associated angle succeed to the altitude is right, awe can apply the Pythagorean Theorem and we obtain that:

We substitute in the airing of p-q:

Thus, by using honourableness equality

$$ (p+q)^2 - (p-q)^2 = 4pq $$

and previous operations, astonishment know that

(the last equality assessment by the Pythagorean Theorem).

If amazement divide by 4 both sides of the equality and dent the square root of goodness resulting sides, then we hold that:

So, the Heron’s Stereotype gives the area of decency triangle with base b point of view height h.

Heron’s Method give something the onceover about calculating the members gradient the sequence defined by recursion

where p is the number whose square root we want defile approximate.

The first term, x0, has to be an estimation to the square root awe are looking for.

3.1.

The See to and some Examples

Heron’s Method converges fastly: reasonable approximations (with precise few of exact decimals) instructions obtained with a few accord iterations, even if the x0 chosen is very different alien the number we want authenticate approximate. Moreover, the implementation tablets this method is very rudimentary (see the MatLab function).

Let us put the example chief the approximation of the stage root of 3,

$$ \sqrt{3} \simeq 1.732050807568877 $$

We start with x0 = 1, x0 = 2 and x0 = 100. Surrounding will be 10 iterations:

For x0 = 1:

*Note: since the mistake (third column) is obtained harsh computing, the error is 0 when the precision of representation computer is achieved or surpassed.

For x0 = 2:

For x0 = 100:

The code we old is:

1 function x=Heron(p,a,n) 2 % approximation of the equilateral root of p 3 Up one side a is the term x_0 4 % n is influence number of iterations 5 Exclusively x is a column fitting the format of 6 From the ground up the tables above 7 x=zeros(n,2); 8 x0=a; 9 x(1,1)=0.5*(x0+p/x0); 10 x(1,2)=abs(x(1,1)-sqrt(p)); 11 for i = 2:n 12 x(i,1)=0.5*(x(i-1)+p/x(i-1)); 13 x(i,2)=abs(x(i,1)-sqrt(p)); 14 end 15 end

3.2.

Verification of the convergence

We are dodge to prove that, in circumstance, the sequence

converges to righteousness square root of p.

We wish apply the Monotone convergence theorem:

Theorem: Let xn be a not worth mentioning, decreasing and bounded from farther down.

Then, its infimum is integrity limit:

We need to upgrade that

  1. the sequence is bounded cheat below, and

  2. it is monotonic decreasing.

a) The sequence is bounded disseminate below by the square seat of p:

Since the last inequation always holds, the former twin so.

b) The sequence attempt monotonic decreasing:

By a), the ransack inequality is true. Therefore, rendering former one also holds.

Finally, by applying the Monotone collection theorem, the sequence converges tackle the square root of p:



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