Most people learn "$(-1)(-1) = 1$" as a rule to memorize. But it can be proven using only the distributive property and the fact that anything times zero is zero. No circular reasoning needed.
We'll prove the core case $(-1)(-1) = 1$ first. The general case follows immediately since $(-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)(ab)$.
We know that any number plus its additive inverse equals zero:
Multiplying both sides of an equation by the same number preserves equality:
Anything times zero is zero:
Apply the distributive property $a(b+c) = ab + ac$:
We know $(-1)(1) = -1$. Substituting:
The only number that makes this true is $(-1)(-1) = 1$.
For any $a, b > 0$: $(-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1) \cdot ab = 1 \cdot ab = ab$. The proof rests on just three axioms: the distributive property, the existence of additive inverses, and $a \cdot 0 = 0$.
These two factoring formulas seem to appear from nowhere. But they can be verified by simply multiplying the right side back out and watching all the middle terms cancel.
Proof of the difference of cubes (the sum follows the same pattern):
Distribute $a$ across the trinomial, then distribute $(-b)$:
Watch the magic: $+a^2b$ cancels with $-a^2b$, and $+ab^2$ cancels with $-ab^2$:
All that survives is:
The trinomial $a^2 + ab + b^2$ is engineered so that when you multiply by $(a - b)$, the cross terms appear in equal-and-opposite pairs. The same trick works for higher powers: $a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})$ for any positive integer $n$.
The distance formula isn't a new idea — it's the Pythagorean Theorem in disguise. And the equation of a circle follows immediately from it.
Given two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$, draw a horizontal line from $P_1$ and a vertical line from $P_2$. They meet at the point $(x_2, y_1)$, forming a right triangle. The horizontal leg has length $|x_2 - x_1|$ and the vertical leg has length $|y_2 - y_1|$.
The distance $d$ between $P_1$ and $P_2$ is the hypotenuse:
Taking the positive square root:
A circle is the set of all points $(x, y)$ that are exactly distance $r$ from a fixed center $(h, k)$. In other words, for every point on the circle:
Replace "distance" with the formula from Part A:
The equation of a circle isn't a separate formula to memorize — it literally is the distance formula with $d$ set equal to $r$. If you know the Pythagorean Theorem, you know the equation of a circle.
This is one of the oldest and most famous proofs in mathematics, dating back to ancient Greece. It uses proof by contradiction: assume the opposite of what you want to prove, then show that assumption leads to a logical impossibility.
Suppose, for the sake of contradiction, that $\sqrt{2}$ is rational. Then we can write:
where $p$ and $q$ are integers with no common factors (the fraction is fully reduced).
Since $p^2 = 2q^2$, we see that $p^2$ is even (it's $2$ times something). But if $p^2$ is even, then $p$ itself must be even. (If $p$ were odd, $p^2$ would also be odd.)
Since $p$ is even, we can write $p = 2m$ for some integer $m$.
Replace $p$ with $2m$ in the equation $p^2 = 2q^2$:
By the same reasoning as Step 3, since $q^2 = 2m^2$, we know $q^2$ is even, so $q$ must be even too.
We've shown both $p$ and $q$ are even. But in Step 1, we said $p/q$ was fully reduced — meaning $p$ and $q$ share no common factors. If both are even, they share the factor $2$. Contradiction.
Our original assumption must be false. Therefore, $\sqrt{2}$ is irrational.
This proof shook ancient Greek mathematics. The Pythagoreans believed all quantities could be expressed as ratios of whole numbers. The discovery that the diagonal of a unit square ($\sqrt{2}$) cannot be so expressed was a genuine philosophical crisis. The same technique proves $\sqrt{3}$, $\sqrt{5}$, and $\sqrt{p}$ for any prime $p$ are all irrational.
The three log laws — product, quotient, and power — aren't separate rules. They are the exponent rules, viewed through the lens of the logarithm. Each proof is just a few lines.
Let $\log_b M = m$ and $\log_b N = n$. This means $b^m = M$ and $b^n = N$.
Multiply $M$ and $N$ in exponential form:
Now take $\log_b$ of both sides:
Divide $M$ by $N$ in exponential form:
Take $\log_b$:
Raise $M$ to the power $k$ in exponential form:
Take $\log_b$:
Every log law comes from the same three-step template: (1) rewrite using $b^m = M$, (2) apply the corresponding exponent rule, (3) convert back to logarithmic form. Multiplication becomes addition, division becomes subtraction, and powers become multiplication — logarithms are a bridge between the world of multiplication and the world of addition.