Whenever we do mathematics, we need to begin by identifying some foundational truths which we accept without proof—a starting point from which to prove any other statement. These foundational truths are usually referred to as axioms. The purpose of this page is to describe the starting point for MATH 2001, that is, the facts which we will take for granted throughout the semester. We are treating these facts as axioms not because they cannot be proven (they can!), but because building up these facts from even more basic axioms is an immense task; we would spend so much time thinking about why these facts are true that we wouldn’t get around to doing anything with them.

This is meant to be an exhaustive list of foundational mathematical facts that we will make use of in MATH 2001, but I will probably forget a few things. If you think of something that might belong on this list, let me know!

• There is a set of integers (denoted $\mathbb{Z}$). Addition, subtraction, and multiplication of integers are well-defined and satisfy all of the expected properties. For example:
  • The sum, difference, and product of two integers is an integer, i.e. if $a, b \in \mathbb{Z}$, then $a + b, a - b, ab \in \mathbb{Z}$.
  • Addition is associative, i.e. $(a + b) + c = a + (b + c)$ for all $a, b, c \in \mathbb{Z}$.
  • Addition is commutative, i.e. $a + b = b + a$ for all $a, b \in \mathbb{Z}$.
  • There is an integer called $0$ which is an additive identity element, i.e. $0 + a = a = a + 0$ for all $a \in \mathbb{Z}$.
  • Every integer has an additive inverse, i.e. for every $a \in \mathbb{Z}$ there is an element $-a \in \mathbb{Z}$ such that $a + (-a) = 0 = (-a) + a$.
  • Multiplication is associative, i.e. $(ab)c = a(bc)$ for all $a, b, c \in \mathbb{Z}$.
  • Multiplication is commutative, i.e. $ab = ba$ for all $a, b \in \mathbb{Z}$.
  • There is an integer called $1$ which is a multiplicative identity element, i.e. $1 \cdot a = a = a \cdot 1$ for all $a \in \mathbb{Z}$.
  • Multiplication distributes over addition, i.e. $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \in \mathbb{Z}$.
  • Subtraction is (by definition) the same as adding the inverse, i.e. $a - b = a + (-b)$ for all $a, b \in \mathbb{Z}$.
• The set of natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$ is a subset of $\mathbb{Z}$ and satisfies all of the same properties as integers (except the properties that refer to $0$ or negative integers).
• Similarly, the rational numbers $\mathbb{Q}$ and the real numbers $\mathbb{R}$ satisfy all of these properties, and also allow division:
  • If $a$ is any nonzero number, then $a$ has a multiplicative inverse, i.e. there is a number $a^{-1}$ such that $a \cdot a^{-1} = 1 = a^{-1} \cdot a$.
  • Division is defined as multiplication by the multiplicative inverse, i.e. $a/b = a \cdot b^{-1}$. (The quotient $a/b$ is also denoted $a \div b$ or $\frac{a}{b}$.)
• The sets $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ each have an ordering, i.e. a way to compare any two numbers. This ordering satisfies all of the properties that you’re familiar with. For example:
  • For any two numbers $x$ and $y$, exactly one of the following three statements is true: $x < y$, $x = y$, $x > y$.
  • If $x \leq y$ and $x \geq y$, then $x = y$. (This is a common way of showing that two numbers are equal.)
  • If $x < y$ and $y < z$, then $x < z$. Similarly, if $x \leq y$ and $y \leq z$, then $x \leq z$. (Similar statements apply for $>$ and $\geq$.)
  • If $x < y$, then $a + x < a + y$ for any $a$. (Similar statements apply for $\leq$, $>$, and $\geq$.)
  • A number $a$ is positive if $a > 0$, negative if $a < 0$, non-negative if $a \geq 0$, and non-positive if $a \leq 0$.
  • The product of two positive numbers is positive. The product of two negative numbers is positive. The product of a positive number and a negative number is negative.
  • If $x < y$, then $ax < ay$ if $a$ is positive, and $ax > ay$ if $a$ is negative. (A similar statement applies with all of the inequalities reversed.)
  • If $x \leq y$, then $ax \leq ay$ if $a$ is non-negative, and $ax \geq ay$ if $a$ is non-positive. (A similar statement applies with all of the inequalities reversed).
• There is a division algorithm for integers, meaning that integers can be divided with remainder. More precisely, if $a$ and $b$ are integers with $b > 0$, then there exist unique integers $q$ and $r$ such that $a = qb + r$ and $0 \leq r < b$. (The number $b$ is the divisor, $q$ is the quotient, and $r$ is the remainder.)
• Every natural number greater than 1 has a unique factorization into primes.