User:Le coq d'or/Sandbox

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: $(ab)^{c}=a^{c}b^{c}.$
• 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 $ac<bc.$