April 2024 - Analytic Geometry, Logical-Thinking and Problem-Solving!

Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space, is one of the oldest branches of mathematics; it has evolved from the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) to analysing the most abstract thoughts and images which might be represented and developed in geometric terms. The main branches of geometry are Euclidean Geometry, Analytic Geometry, Projective Geometry, Differential Geometry, Non-Euclidean Geometries, and Topology.

What can entrepreneurs learn from each of these six branches of geometry?

Analytical Geometry is a combination of algebra and geometry. In analytical geometry, we aim at presenting the geometric figures using algebraic equations in a two-dimensional coordinate system or in a three-dimensional space.

This field is often deemed as the next level of mathematics, and is often claimed to help with logical thinking and analytical thinking, both of which improve problem-solving skills – which is a fundamental requirement for successful entrepreneurs!

Let’s explore a little further.

Logical thinking focuses on drawing conclusions based on a set of rules or pre-confirmed evidence, whereas analytical thinking involves breaking down complex information and synthesizing it to arrive at conclusions. Both types of thinking are valuable and can be used in combination to enhance problem-solving skills.

More generally in analytic geometry, formulas (which are nothing more than statements), involving some variables are used. Using the convention of capital letters to designate formulas, if the truth of a formula depends on the values of, say, x, y and z, we will use notation like P(x,y,z) to denote the formula. Such formulas are either true or false when we assign particular values to each of the variables.

Example - If P(x,y) is "x²+y=12'', then P(2,8) and P(3,3) are true, while P(1,4) and P(0,6) are false. If Q(x,y,z) is "x+y<z'', then Q(1,2,4) is true and Q(2,3,4) is false.

Complicated formulas are put together from simpler ones, using a small number of logical operations. Just a handful of these operations are used.

If P is a formula, then "not P '' (written symbolically as ¬P) is another formula. Of course, ¬P is false if P is true and vice versa. Also, many derivative instances such as conjunction of formulas P and Q denoted as P∧Q, disjunction of formulas denoted by as P∨Q and conditional formulas written as P⇒Q (implying "if P then Q'' or "P implies Q'') are all almost logically derived.

We can easily see the parallels in the world of business. Three core areas of success in entrepreneurship, we all know, involve Effective Marketing, Operational Excellence and Leadership Competence. In fact, each of these three can often broken down into formula-type of statements involving variables that impact their outcomes. The following are illustrative.

Effective Marketing –

 

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The well-known 4Ps can be represented by statements that correlate the dynamic linkages of variables. Product is a function of “who needs” and “why it is needed”; Price is a function of “real and perceived values”, “costs”, “special discounts”, ”margins”, “competition”; Place is a function of “location characteristics”; Promotion is a function of “visibility”, “aspirational demand”, “comfort demand” and more.

Operational Excellence –

 

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This is a function of “Quality” and “Acceptability”. Hiqh quality projects and processes that are low on acceptance by stakeholders will end up low on final impact excellence.

Leadership Competence –

                                        

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The competence will be an outcome of the dynamics of functional convergence of the “potential”, “motivation” and “development” of individuals in the organisation.

The successful enterprise is a result of an efficient problem-solving capability. This is often  be facilitated by the logical and analytical thinking across multiple paradigms that govern the truth (or otherwise) of the markets, as well as the possibilities from conjunction and disjunction of the functional elements of the enterprise ecosystems.

Yes, analytical geometry and the world of business are indeed linked by the skillsets of logical and analytical thinking!