Inventing Mathematical Formulas: A Step-by-Step Guide

Mathematics is a field where innovation and imagination can lead to new discoveries. Often, the process of inventing mathematical formulas can seem daunting, but it can be approached methodically. This article will guide you through the process of inventing a formula, as demonstrated with the area of a triangle in terms of its altitudes, and the Pythagorean theorem for right triangles. Let's dive into the process and explore the logic behind these inventions.

1. The Process of Inventing a Formula

The process of inventing a formula is more than just solving a problem; it's about coming up with a problem that is both original and solvable. Here’s how it works:

Identify a Problem: The first step is to identify a problem that hasn't been fully addressed yet. This could be a twist on an existing formula or a new perspective on a well-known concept. For example, finding the area of a triangle in terms of its altitudes. Setting Up the Problem: Once you have the problem, set it up mathematically. This involves defining the variables and relationships involved. For instance, let's define the sides of the triangle as a, b, and c, the area as Delta;, and the altitudes as d, e, and f. Apply Known Theorems: Use well-known theorems as a starting point. Start with a less well-known theorem to increase the chances of coming up with a novel formula. The Archimedes' Theorem is an example here. Manipulate and Derive the Formula: From the theorem, manipulate the equation to derive the desired formula. Follow the steps to solve the problem.

2. Deriving the Formula for the Area of a Triangle

Let's walk through the process used to derive the area of a triangle in terms of its altitudes. We start with Archimedes' Theorem and Archimedes' formula regarding the area of a triangle.

According to the theorem and the formula:

16Delta;2 4a2b2 - a2b c2a2b2c2 - 2a4b4c4 Using the given side lengths in terms of the altitudes: a 2Delta;/d, b 2Delta;/e, c 2Delta;/f.

Substituting these into the formula and simplifying:

16Delta;2 4(2Delta;/d)2(2Delta;/e)2 - (2Delta;/d)2(2Delta;/e)2 - (2Delta;/f)2 Simplifying further: frac16Delta;2 frac4d2e2 - left(frac1d2frac1e2 frac1f2right)

Thus, we came up with a novel formula for the area of a triangle in terms of its altitudes:

Theorem. A triangle with altitude lengths d, e, f and area Delta; satisfies:

frac1Delta;2 frac4d2e2 - left(frac1d2 frac1e2 - frac1f2right)

3. Deriving a Pythagorean Theorem for Right Triangles

Another classic problem we can tackle is the Pythagorean theorem for right triangles. We can use the same method to derive a formula that confirms a triangle is a right triangle, independent of the order of the sides.

Start from the obvious form: a2b2c2a2b2c2a2b2c2a2b2c2 0. Expand and solve in a symmetric manner, similar to Heron's formula.

Following this, we can derive:

Theorem: A triangle with sides a, b, and c is right-angled precisely when:

4a4b4 a4b4 - c4

Or, for a more symmetric look:

a4b4c4 2a8b8c8

These novel forms of the Pythagorean theorem demonstrate that innovation in mathematics is within reach when we approach problems methodically and use existing theorems as a base.