(0.0) Roman numerals: A failed system
- Suppose that you want to invent language and mathematics from scratch. Where do you start?
(0.1) Arabic numerals: A successful system
- We would like to do better than the Romans. One way to think about this is: Why don't we systematize the way in which we compress quantities?
(0.1.0) Adding Arabic numerals
- The positional system offers another advantage: easy addition.
(0.1.1) Radix and Addition Exercises
- All exercises have solutions below. You are encouraged not to look at the solutions for as long as you can stand it. Struggling will make you a stronger mathematician.
Also, exercises are part of the text. Important discoveries, concepts, and definitions are introduced here, which we reference as established facts later. Because the exercises have solutions, this should not impede the reader in being able to progress through the text. But because of this, you should at least glance over the problems in order to see if they cover anything that you need for later.
Programming exercises
- Theoretical computer science is practically a subfield of mathematics, and the use of computers can aid education in many ways. On of the most important ways, is that when YOU the student start playing with programs and making things work, you learn a lot.
Solutions
(0.1.2) Constants and Variables
- In the previous exercises we saw the question: If 82 is one summand, what is the other, if the sum is 101? It would have been easier to state, if I could have asked: What is the solution to the equation $\square+82=101$? Here '$\Box$' is a symbol for an unknown quantity. Such symbol we call a constant symbol.
(0.1.3) Subtraction as inversion
- We can now define subtraction by the following equation: If x and y are any numbers where x>y, the expression $x-y$ is defined to be the number z which satisfies $y+z=x$. So for instance, $5-3=2$ because 2 is the number that satisfies $3+\Box=5$.
(0.1.4) Subtraction exercises
- Ok that’s cool. But how do you compute a difference?