Arithmetic, the Science of Number

MAT-251 Lesson 001 – Arithmetic, the Science of Number

Welcome to Modern Arithmetic III, the third and final course in your study of Arithmetic. If you have completed the earlier courses, you already know how to count and write numbers of any size, how to add, subtract, multiply, and divide, how to work with fractions and decimals, with measures, percentage, and proportion. You have learned the whole of Arithmetic once, and then a second time more deeply. Why, then, should you study it a third time? The answer is the key to this entire course, and it is the subject of this first lesson.

In your earlier studies, you learned Arithmetic chiefly as an art. An art is the skill of doing some work well. A carpenter possesses the art of building; he knows how to measure, cut, and join wood so that the finished work is sound. In the same way, you possess the art of computing: give you two numbers, and you can find their sum, their difference, their product, or their quotient, quickly and correctly. That art is a real possession, and nothing in this course will take it away from you. But there is something higher than an art, and that is a science. A science is knowledge of things through their causes and reasons. The carpenter who has only the art can build what he has been shown to build; the man who also knows the science of building understands why a beam of a certain size will bear a certain weight, and he can therefore judge, design, and correct. The one knows how; the other knows why.

Arithmetic is the science of number. In this course, you will study number itself: what it is, what properties it has, and why the rules you have practiced for years must work as they do. Until now, when you learned a rule, you received it on trust. You were told that to divide by a fraction you invert the divisor and multiply, and you found that it always gave the true answer. But why does it? You were told that a number is divisible by 9 whenever the sum of its digits is divisible by 9. It is a wonderful fact, and you have used it—but why is it true? In this course, we will not receive such rules on trust. We will demonstrate them, which means we will prove them, step by step, from first principles, so that you see not only that they are true, but why they cannot be otherwise.

This is the method of every true science, and you must understand how it works. A science begins with definitions, which state exactly what its terms mean, and with axioms, which are truths so simple and evident that no one can doubt them—for example, that if equals are added to equals, the wholes are equal. From these definitions and axioms, the science reasons its way forward, proving one truth after another, each proof resting on what came before, until a great body of certain knowledge stands complete, like a building raised course by course upon a solid foundation. Nothing in it is guessed; nothing is taken merely on authority; everything is seen to be true by the light of reason.

There is also a practical purpose in this course, and you should keep it before your eyes. You are preparing for the regular high school course in Mathematics—for Algebra, Geometry, and the studies beyond them. Those subjects are demonstrative sciences through and through. Geometry, especially, is nothing but definitions, axioms, and proofs from beginning to end. The student who arrives knowing only the art of computing finds that ground strange and hard; the student who has already learned to demonstrate in Arithmetic finds it familiar, for he has been reasoning that way all along. This course is your bridge. In it you will master every computation Arithmetic contains, at full strength and full speed—and you will also learn to think as mathematicians think, proving what you assert and demanding the reason for every rule.

In our next lesson, we will lay the first stones of the foundation by restating, with new exactness, the three definitions on which all of Arithmetic rests: the unit, the number, and quantity.

Memory Work:

1. Arithmetic is the science of number.
2. An art is the skill of doing some work well.
3. A science is knowledge of things through their causes and reasons.
4. A demonstration is a proof which shows, from definitions and axioms, that a truth cannot be otherwise.

Return to MAT-251 Modern Arithmetic III

The lessons published on this site are freely available to read. If you wish to move from reading to formal study — with graded assessments, academic records, and the support of the Academy — you may enroll at any time. Enrollment is open year round, to students of all ages.

Enroll Now