Mathematical Properties

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This small lesson will introduce you to the math properties. To see what you can expect to read, check "Contents", which shows the list of sections that are present on this page. No extra materials are required for this lesson, but you may do so out of your own will.

The targeted audience for this lesson is young learners (8-16).

Properties are the laws of math that state that a mathematician must follow these rules [the properties] to solve a math problem. Every math subject, such as Geometry and Algebra, follow these properties. A math that doesn't follow, for example, with the commutative property in: ${\displaystyle a}$ • ${\displaystyle b}$ = ${\displaystyle b}$ • ${\displaystyle a}$ is not, in simple terms, math. However, it is a rule, a math rule but it's called a property because math follows it. Thus, it is essential for every mathematician to, not only memorize, but apply these properties as well.

• Commutative Property of Addition
• Commutative Property of Multiplication
• Associative Property of Addition
• Associative Property of Multiplication
• Additive Identity Property
• Multiplicative Identity Property
• Additive Inverse Property
• Multiplicative Inverse Property
• Multiplicative Property of Zero
• Additive Property of Zero
• Substitution Property
• Distributive Property
• Division Property
• Inverse Property of Inequality/Equality
• addition property of equality

1. Commutative Property of Addition/Multiplication - The word, commutative, from the French word, commuter (to switch), means to "move around". Such as in problem, ${\displaystyle a}$ • ${\displaystyle b}$ = ${\displaystyle b}$ • ${\displaystyle a}$, the ${\displaystyle a}$ and the ${\displaystyle b}$ move around about the equal sign, but nonetheless, still equal to the same sum (in this situation, we can say ${\displaystyle c}$). Now, let's implement this on real numbers, such as:

• ${\displaystyle 6}$ + ${\displaystyle 4}$ = ${\displaystyle 4}$ + ${\displaystyle 6}$

No matter the order, 6 + 4 and 4 + 6 will ALWAYS equal to 10. When these two numbers are added, regardless of the order, the sum is the same. The order does not make any difference in the result. So:

Order does not affect results

This is the same case with multiplication.

The commutative property does not have a "commutative property of subtraction", because ${\displaystyle 6}$ - ${\displaystyle 4}$ does not equal ${\displaystyle 4}$ - ${\displaystyle 6}$.

2. Associative Property of Multiplication/Addition - Etymology here: The word, associative, is derived from the word associate, which comes from the Latin word, associo, which means to unite together, associate. This property basically reflects the same rules as the Commutative property: No matter the order of the parentheses, you will get the same result. The parenthesis can go wherever you like it to be! So:

order (does not affect) result

Example:

• ${\displaystyle 4}$ • (${\displaystyle 6}$ • ${\displaystyle 5}$) = (${\displaystyle 4}$ • ${\displaystyle 6}$) • ${\displaystyle 5}$

It does not matter where the paranthesis are at, the same result you will get on both sides (following the Order of Operations) is a ${\displaystyle 120}$.

3. Identity Property - Specifically the Additive Identity Property and the Multiplicative Identity Property, is when the total (number) does not change throughout the equation. The word, identity, comes from the Latin word for idem, which means "the same". So, you can think of it as the number is still "the same" after the math equation. In ${\displaystyle 5}$ + ${\displaystyle 0}$ = ${\displaystyle 5}$: The total (number) stayed the same throughout the math problem--It is still 5! For the Additive Identity Property, it is always zero (this works for subtraction as well)--For the Multiplicative Identity Property, it is always one (this works for division as well).

4. Inverse Property, also the Additive Inverse Property and the Multiplicative Inverse Property, state that any number to added to its opposite counterpart (${\displaystyle 5}$ and ${\displaystyle -5}$) will equal either zero or one. This "zero or one" depends on which Inverse Property you are using. If you are using the Additive Inverse Property, you should be getting zero. If you are using the Multiplicative Inverse Property, you should be getting one. The Additive Inverse Property works with subtraction as well, and the Multiplicative Inverse Property works with division as well. In the Additive Inverse Property, the number must to be added to its negative counterpart, such as ${\displaystyle 29}$ and ${\displaystyle -29}$, ${\displaystyle 6}$ and ${\displaystyle -6}$, ${\displaystyle 91}$ and ${\displaystyle -91}$, ${\displaystyle 47}$ and ${\displaystyle -47}$, etc.. In the Multiplicative Inverse Property, the number must be multiplied to its reciprocal, which is the number found by flipping the numerator and the denomerator of a fraction (${\displaystyle 2}$ = ${\displaystyle {\tfrac {2}{1}}}$), such as ${\displaystyle {\tfrac {3}{4}}}$ and ${\displaystyle {\tfrac {4}{3}}}$, ${\displaystyle {\tfrac {9}{2}}}$ and ${\displaystyle {\tfrac {2}{9}}}$, ${\displaystyle {\tfrac {82}{93}}}$ and ${\displaystyle {\tfrac {93}{82}}}$, ${\displaystyle 2}$ and ${\displaystyle {\tfrac {1}{2}}}$, etc..

• ${\displaystyle 4}$ + ${\displaystyle -4}$ = ${\displaystyle 0}$ (Additive Inverse)
• ${\displaystyle 5}$ • ${\displaystyle {\tfrac {1}{5}}}$ = ${\displaystyle 1}$ (Multiplicative Inverse)

5. Property of Zero - Specifically the Multiplicative Property of Zero and the Additive Property of Zero, is when zero plays a role in a math equation. Specifically in the Multiplicative Property of Zero, anything multiplying zero is zero. This rule is true no matter what number it is. Big or small, this rule works in every and all cases of problems that follow the Multiplicative Property of Zero. For example:

• ${\displaystyle 6}$ • ${\displaystyle 0}$ = ${\displaystyle 0}$

The "starting number", ${\displaystyle 6}$, multiplies the number ${\displaystyle 0}$. As per the rules of the property of zero (multiplication), the answer is and will always be ${\displaystyle 0}$.

The Additive Property of Zero is part of the "property of zero" group and the "identity" group, which will be explained later in this section. The Additive Property of Zero states that any number adding zero will remain the same (thus, "identity"): ${\displaystyle r}$ + ${\displaystyle 0}$ = ${\displaystyle r}$. Here are a few problems to help explain:

• ${\displaystyle 4}$ + ${\displaystyle 0}$ = ${\displaystyle 4}$
• ${\displaystyle 9}$ + ${\displaystyle 0}$ = ${\displaystyle 9}$
• ${\displaystyle 9827}$ + ${\displaystyle 0}$ = ${\displaystyle 9827}$

As per the Additive Property of Zero rule, the "starting number" is the sum. So, ${\displaystyle 4}$ being added to ${\displaystyle 0}$, is ${\displaystyle 4}$--and likewise for the other problems.

6. Substitution property is the act of substituting factors with the answer. An example is ${\displaystyle -5x+8+x=-4x+8}$. The substitution property simply takes ${\displaystyle -5x}$, and adds it to ${\displaystyle x}$, which equals ${\displaystyle -4x}$ and the constant: ${\displaystyle 8}$ (since there are no like terms).

7. Distributive Property - A separate page for the Distributive Property is in this link with its own quiz (2 quizzes).

Search for Mathematical Properties on Wikipedia.

• Khan Academy - Associative property of multiplication review
• Mathwarehouse - Associative Property
• Identity Property of Addition
• Mulitiplicative Identity
• Identity Property as a whole - Study.com
• Property of Zero
• Substitution Property of Equality - YT