I believe that Einstein once said: “intuition is more important than knowledge.” If I were to define “intuition,” I would define it as follows:

Intuition: is the ability to translate between two worlds (often times one world being abstract and the other concrete) and to infer new things about one world from observations in the second world.

Intuitive math and science then takes mathematical formulas and concepts, and maps them to a different world. Being in this new world should help with making new observations and inferring new scientific or mathematical facts.

What is an example of such a translation then?

The PLUS and MINUS Operators.

A simple example is the intuitive meaning of plus (+) and minus (-) for real numbers.

There are many ways of translating these abstract operators into concrete facts. For example, plus could represent money in the pocket and minus could represent money owed. Or, it could be translated as weight gained and weight lost. (Or could it? Is it clear what having a negative weight means?) My favorite intuitive translation, for plus and minus, is obtained from using the line of real numbers. Plus translates to moving to the right and minus translates to moving the the left. Thus

+5-7

translates to: From zero (always the starting point), go to the right five steps, followed by going to the left 7 steps. The result is 2 steps to the left of zero, or -2, as shown below.

Explain this plus and minus concept to a 6 year old, and they will be able to do some pretty fancy math. The intuitive math took an abstract concept of plus and minus and translated it to a concrete sequence of movements. This type of “intuitive translation” is usually not taught the first time plus and minus is introduced in schools. The first time I learned about plus and minus, I can still recall how, my first grade teacher taught me that solving:

2-3

was impossible! The concept and intuitive understanding of negative and positive numbers was left out. The biggest problem with leaving out the “intuitive explanation” is that it gives the wrong impression: that math has to be memorized! This, often times, turns people away from math. Instead, I believe that math needs to be lived.

The idea behind these blog posts is to discuss topics that present intuitive interpretations of simple and complex mathematical and scientific concepts.

– D. Darian Muresan