Addition ($+$): Combining quantities. Sum.
- Commutative Law: $a+b=b+a$.
- Associative Law: $(a+b)+c=a+(b+c)$.
- Identity Element: $0$ ($a+0=a$).
Subtraction ($-$): Finding the difference between quantities.
- Not commutative or associative.
Multiplication ($\times$ or $\cdot$): Repeated addition. Product.
- Commutative Law: $a \cdot b = b \cdot a$.
- Associative Law: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
- Identity Element: $1$ ($a \cdot 1=a$).
- Zero Property: $a \cdot 0 = 0$.
Division ($\div$ or $/$): Splitting into equal parts. Quotient.
- Not commutative or associative.
- Division by zero is undefined.
Distributive Law: $a \cdot (b+c) = a \cdot b + a \cdot c$. Also $a \cdot (b-c) = a \cdot b - a \cdot c$.
Inverse Elements:
- Additive Inverse: For any number $a$, there is $-a$ such that $a + (-a) = 0$.
- Multiplicative Inverse (Reciprocal): For any non-zero number $a$, there is $1/a$ such that $a \cdot (1/a) = 1$.

Exponentiation: $a^n$ (read as "$a$ to the power of $n$") means $a$ multiplied by itself $n$ times.
- $a$ is the base, $n$ is the exponent or power.
- Rules: $a^m \cdot a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $(ab)^n = a^n b^n$, $a^0=1$ ($a \neq 0$), $a^{-n} = 1/a^n$.
Logarithms: The inverse operation to exponentiation. $\log_b x = y \iff b^y = x$.
- $b$ is the base, $x$ is the argument, $y$ is the logarithm.
- Common Logarithm: $\log_{10} x$.
- Natural Logarithm: $\ln x = \log_e x$.
- Rules: $\log_b (xy) = \log_b x + \log_b y$, $\log_b (x/y) = \log_b x - \log_b y$, $\log_b (x^k) = k \log_b x$, $\log_b b = 1$, $\log_b 1 = 0$.
- Change of Base: $\log_b x = \frac{\log_c x}{\log_c b}$.

Multiples: A multiple of an integer $n$ is any integer that can be divided by $n$ with no remainder. Formally, $kn$ for some integer $k$.
Factors (Divisors): An integer $d$ is a factor of $n$ if $n/d$ is an integer.
Prime Numbers: A natural number greater than 1 that has no positive divisors other than 1 and itself.
- Examples: $2, 3, 5, 7, 11, 13, \dots$.
- The number 2 is the only even prime number.
Composite Numbers: A natural number greater than 1 that is not prime (i.e., has more than two positive divisors).
Prime Factorization: The process of finding the prime numbers that multiply together to make a given number. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).
- Example: $12 = 2 \times 2 \times 3 = 2^2 \times 3$.
Number of Positive Divisors: If the prime factorization of $n$ is $p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}$, then the total number of positive divisors is $(e_1+1)(e_2+1)\dots(e_k+1)$.
Division Algorithm: For any integers $a$ (dividend) and $b$ (divisor) with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$, where $0 \le r < b$.