I imagine that many of you are familiar with this remarkable mathematical equation that incorporates what are arguably the five most important mathematical constants into one equation. Yes, phi is missing from this equation. I've been reading this fascinating book, Where Mathematics Comes From. Wikipedia has an article about the book. The book seeks to found a cognitive science of mathematics. While the book is very philosophical and abstract in many places, what fascinated me were the very rich metaphors discussed for a number of common mathematical operations and concepts. These metaphors really helped me to see the conceptual basis for some mathematical processes and operations I took for granted and really helped to ground and deepen my understanding.
Some examples of metaphors:
1. Addition and subtraction. Moving to the right or left along the number line
2. Multiplication of a positive number by a positive number. Moving to the right along the number line by a factor.
3. Multiplication by -1. Rotation to the symmetry point of n.
4. A simple fraction (1/n). "Starting at 1, find a distance d such that by moving distance d toward the origin repeatedly n times, you will reach the origin. 1/n is the point-location at distance d from the origin."
5. Exponential function. The mapping of sums onto products. This explains what it means to raise numbers to non-integer and to negative powers.
The book spends several chapters building up the conceptual framework for the famous formula that includes the five constants. What really impressed me was the idea that a number of branches of mathematics are intimately connected to this formula. You'll have to read the book to see the connections. I can't do justice to the incredible treatment in the book so I'll just touch on some of its ideas. Consider these ties:
1. What does e have to do with this formula? What is special about the constant e? e has to do with rates of change and e is the basis for natural logarithms. So, the study of logarithms is involved here.
2. Also, e^x is a function that is its own derivative; its rate of change is identical to itself. So, the calculus of differentiation is involved in this formula.
2. What about pi? Pi has to do with circles, and angles, and with rotations. So, now we're including geometry in this discussion.
3. The imaginary number, i, is part of the formula. i involves complex numbers, which are represented on the Cartesian plane. Multiplication by i results in a 90 degree rotation of a number. Multiplication of complex numbers involves parallelograms and Euclidean geometry.
4. The Taylor series (more calculus) and trigonometry are involved in showing that e^(y*i) = cos(y)+i*sin(i) which is key to showing that e^(pi*i) = -1.
I hope you get that there's so much richness in how multiple branches of mathematics contribute and build upon one another to create this formula that a college freshman or a high school student with a little bit of knowledge of calculus can understand and enjoy.