Draft:Math Properties

Math Properties

Distributive Property visualized as the area of rectangles

Math Properties are formulas that describe something true. It contains a math symbol (operation) and type of property. For example, the commutative property of multiplication is $1=a\times b=b\times a$. They are very useful for understanding how operations work. They are used in problems so that way you don't have to solve the problem, but you just have to think of math properties. For example, you want to find if $(a+b)+c=a+(b+c)$. You don't need to plug in many values to find if they are equal, but you just need to remember these properties. Some operations' properties are equal and some are not.

All Properties

Properties	Addition	Subtraction	Multiplication	Division	Exponentiation
Associative	$(a+b)+c=a+(b+c)$	$(a-b)-c\neq a-(b-c)$	$(a\times b)c=a(b\times c)$	$(a/b)\div c\neq a\div (b/c)$	$(a^{b})^{c}\neq a$^$(b^{c})$
Commutative	$a+b=b+a$	$a-b\neq b-a$ but $a-b=-(b-a)$	$ab=ba$	$a\div b\neq b\div a$ but $a/b=1\div (b/a)$	$a^{b}\neq b^{a}$
Distributive	$x(a+b)=xa+xb$	$x(a-b)=xa-xb$	$x(ab)=(xa)(xb)$	$x(a/b)\neq (xa)/(xb)$	$x(a^{b})\neq (xa)$^$(xb)$
Identity	$a+0=a$	$a-0=a$	$a\times 1=a$	$a\div 1=a$	$a^{1}=a$
Zero	$a+0=a$	$a-0=a$	$a\times 0=0$	$a\div 0=undefined$	$a^{0}=1$

Here are some extra properties that we discovered.

1. $a$^$(b+c)=a^{b}\times a^{c}$
2. $a^{2}=(a-1\times a+1)+1$

And some extra properties that you learn in elementary school.

1. $a/a=1$ if $a\neq 0$
2. $a/0=undefined$
3. $a/(1/b)=a\times b$

Examples

1. Solve for $5\times (32+35)=5\times 32+5\times 35$.
   1. Solve for the left side.
      1. $32+35=67$ so we can turn the equation into $5\times 67=5\times 32+5\times 35$.
      2. $5\times 67=335$ so we can turn the equation into $335=5\times 32+5\times 35$.
   2. Solve for the right side.
      1. $5\times 32=160$ so we can turn the equation into $335=160+5\times 35$.
      2. $5\times 35=175$ so we can turn the equation into $335=160+175$.
      3. $5\times 35=175$ so we can turn the equation into $335=160+175$.
      4. $160+175=335$ so we can turn the equation into $335=335$.
   3. The equation is true since it ends up with 335=335.
2. Solve for $37^{2}=(37-1\times 37+1)+1$
   1. Solve for the left side.
      1. $37^{2}=1369$ so we can turn the equation into $1369=(37-1\times 37+1)+1$.
   2. Solve for the right side.
      1. $37-1=36$ so we can turn the equation into $1369=(36\times 37+1)+1$.
      2. $37+1=38$ so we can turn the equation into $1369=(36\times 38)+1$.
      3. We no longer need the parentheses so we can turn the equation into $1369=36\times 38+1$.
      4. $36\times 38=1368$ so we can turn the equation into $1369=1368+1$.
      5. $1368+1=1369$ so we can turn the equation into $1369=1369$.
   3. The equation is true since it ends up with 1369=1369.

--GelatinPlayz (talk) 04:52, 7 September 2024 (UTC)

By Zhen Bui