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Laws of elementary algebra

 * Addition
 * Associativity: When performing multiple additions, it does not matter which one is done first:
 * $$ (a + b) + c = a + (b + c) \ $$


 * Commutativity: When adding two numbers, it does not matter which number is given first. $$ a + b = b + a \ $$
 * Identity element: Adding a number to zero returns that number. $$ a + 0 = a \ $$
 * Inverse element: Adding a number to its negative returns zero. $$ a + (-a) = 0 \ $$
 * Inverse function: Subtraction is the inverse of addition. $$ a + b = c \equiv c - b = a $$
 * Multiplication
 * Associativity: When performing multiple multiplications, it does not matter which one is done first. $$ (a \times b) \times c = a \times (b \times c) \ $$
 * Commutativity: When multiplying two numbers, it does not matter which number is given first. $$ a \times b = b \times a \ $$
 * Distributivity
 * Identity element: Multiplying a number by one returns that number. $$ a \times 1 = a \ $$
 * Inverse element: Multiplying a number by its reciprocal returns one. $$ a \times \frac{1}{a} = 1 \ $$
 * Inverse function: Division is the inverse of multiplication. $$ a \times b = c \equiv c \div b = a $$

Laws of elementary algebra

 * Addition is a commutative operation (two numbers add to the same thing whichever order you add them in).
 * Subtraction is the reverse of addition.
 * To subtract is the same as to add a negative number:
 * $$ a - b = a + (-b). \ $$
 * Example: if $$5 + x = 3$$ then $$x = -2.$$


 * Multiplication is a commutative operation.
 * Division is the reverse of multiplication.
 * To divide is the same as to multiply by a reciprocal:
 * $$ {a \over b} = a \left( {1 \over b} \right). $$


 * Exponentiation is not a commutative operation.
 * Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
 * Examples: if $$3^x = 10$$ then $$x = \log_3 10 .$$ If $$x^{2} = 10$$ then $$x = 10^{1 / 2}.$$
 * The square roots of negative numbers do not exist in the real number system. (See: complex number system)
 * Associative property of addition: $$(a + b) + c = a + (b + c).$$
 * Associative property of multiplication: $$(ab)c = a(bc).$$
 * Distributive property of multiplication with respect to addition: $$c(a + b) = ca + cb.$$
 * Distributive property of exponentiation with respect to multiplication: $$(a b)^c = a^c b^c .$$
 * How to combine exponents: $$ a^b a^c = a^{b+c} .$$
 * Power to a power property of exponents: $$ (a^b)^c = a^{bc} .$$

Laws of equality

 * If $$a = b$$ and $$b = c$$, then $$a = c$$ (transitivity of equality).
 * $$a = a$$ (reflexivity of equality).
 * If $$a = b$$ then $$b = a$$ (symmetry of equality).

Other laws

 * If $$a = b$$ and $$c = d$$ then $$a + c = b + d.$$
 * If $$a = b$$ then $$a + c = b + c$$ for any c (addition property of equality).
 * If $$a = b$$ and $$c = d$$ then $$ac$$ = $$bd.$$
 * If $$a = b$$ then $$ac = bc$$ for any c (multiplication property of equality).
 * If two symbols are equal, then one can be substituted for the other at will (substitution principle).
 * If $$a > b$$ and $$b > c$$ then $$a > c$$ (transitivity of inequality).
 * If $$a > b$$ then $$a + c > b + c$$ for any c.
 * If $$a > b$$ and $$c > 0$$ then $$ac > bc.$$
 * If $$a > b$$ and $$c < 0$$ then $$ac < bc.$$