# Properties

## Terms

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- additive inverse
- Adding the opposite the equal zero, and used to praove an opposite. (a+-a=0, a-a=0, and used to prove that if a is real, negative a is real)
- multiplicative property of zero
- To multiply by zero. [a(0)=0]
- commutative of addition
- To change the order of an equation that uses addition or subtraction. (3+4=4+3)
- transitive
- Usually used at the end of a proof. To sum up. (if a=b, and b=c, then a=c)
- definition of division
- Rules for dividing fractions. [3/4=(3/4)/1=3(1/4)=3/4]
- definition of subtraction
- Just like switch-switch! [a+(-b)=a-b] Just Asking; doesn't switch switch go along with yellow-yellow?
- distributive
- To multiply a value through parenthesis. Only works if only additon and subtraction are in the parenthesis. [a(b+c)=ab+ac
- multiplicative property of negative one
- To multiply by negative one. [-1(a)=-a]
- substitution
- To replace a value or actually complete the given operations. ( if a=3, and a=b, then 3=b)
- opposites in products
- The rules of multiplying by integers. [-a(b)=-ab, a(b)=ab, and -a(-b)=ab]
- hypothesis
- To prove that any variable is real. (a, b, and c are real numbers.)
- axiom of closure
- If (ab) is real, then a+b is real. ( if 3+4=7 s real, 3(4)=12 is real)
- identity of addition
- To add zero. (a+0=a)
- associative of addition
- in an equation using additon and subtraction, the order will remain the same but parenthesis groupings will change. [(1+2)+3=1=(2+3)]
- multiplication property of equality
- Multiplying the same quantity to both sides of an equation. (If a=b, then ac=bc)
- symmetric
- You can flip both sides of an equal side to still have a true eqation. (3=a, then a=3)
- multiplicative inverse
- To multiply by the reciprocal, and used to prove recirocals are real. [a(1/a)=1, and to prove that if a is real, then 1/a is real)
- identity of multiplication
- To multiply by 1. [a(1)=a]
- subtraction property of equality
- Subtracting the same quantity to both sides of an equation. (If a=b, then a-c=b-c)
- reflexive
- Looks exactily the same on both sides. (a=a)
- division property of equality
- Dividing the same quantity to both sides of an equation. (If a=b, then a/c=b/c)
- reciprocal of a product
- Seperating one fraction into two. [1/ab=1/a(1/b)]
- commutative of multiplication
- Changes the order of an eqation that uses multiplication and or division. [(3)4=(4)3]
- associative of multiplication
- in an equation using multiplication and division, the order will remain the same but parenthesis groupings will change. [2(3)4=(2)3(4)]
- opposites of a sum
- Like the distributive property, but used when only distributing a negative through. [-(a+3)=-a-3]
- addition property of equality
- Adding the same quantity to both sides of an equation. (If a=b, then a+c=b+c)
- cancellation
- To multiply a negative by a negative. [-(-a)=a]