"Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e^iπ = -1, — a formula which fuses together four of the most important concepts of the science — the logarithmic base, the transcendental ratio π, and the imaginary and negative units." (David E Smith, "The Poetry of Mathematics", The Mathematics Teacher, 1926)
"Other questions must be answered as well. Why should e^πi equal, of all things, -1? e^πi has an imaginary number in it; wouldn't you therefore expect the result to be imaginary, not real? e is about differentiation, about change, and π is about circles. What do the ideas involved in change and in circles have to do with the answer? e and n are both transcendental numbers - numbers that are not roots of any algebraic equation. If you operate on one transcendental number with another and then operate on the result with an imaginary number, why should you get a simple integer like -1?" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"What a wealth of insight Euler’s formula reveals and what delicacy and precision of reasoning it exhibits. It provides a definition of complex exponentiation: It is a definition of complex exponentiation, but the definition proceeds in the most natural way, like a trained singer’s breath. It closes the complex circle once again by guaranteeing that in taking complex numbers to complex powers the mathematician always returns with complex numbers. It justifies the method of infinite series and sums. And it exposes that profound and unsuspected connection between exponential and trigonometric functions; with Euler’s formula the very distinction between trigonometric and exponential functions acquires the shimmer of a desert illusion."
"I think e^iπ+1=0 is beautiful because it is true even in the face of enormous potential constraint. The equality is precise; the left-hand side is not 'almost' or 'pretty near' or 'just about' zero, but exactly zero. That five numbers, each with vastly different origins, and each with roles in mathematics that cannot be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler's formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan)
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." (Keith Devlin)
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