Here are some of the basic rules and vocabulary of Algebra.

1.  Operations – There are four basic operations.

The four basic operations are addition, subtraction, multiplication and division.  [ +, -, x. / ]

To subtract we add the opposite and to divide we multiply by the reciprocal.  So in reality, there are only two operations to learn – addition and multiplication.

2.  Opposite – Another name for the opposite of a number is the additive inverse.      \left [ a\mid -a \right ]

a + (-a)= 0

remove the brackets

a -a = 0

3.  Reciprocal – Another name for the reciprocal of a number is the multiplicative inverse.  \left [ a\mid \frac{1}{a}\right ]

a*\frac{1}{a}= 1

4.  Fractions - A fraction has both a denominator and a numerator. 

\frac{a}{b}=\frac{Numerator}{Denominator}

 

 5.  Properties of Operations

  • Commutative property of addtion and multiplication.

I like to think of the word order when I think of this property.  Almost every day we commute to school or to work.  We leave home and go to school in the morning.  At the end of the day we leave school and go home.  The order changes but the route remains relatively the same.  When we add or multiply the result is the same no matter what order the terms are in.

a+b=b+a 

a*b=b*a

  • Associative property of addition and multiplication.

What group or association do you belong to?  Are you a member of the Red Cross or the Red Crescent Society?  In this property the groups are changed but the end result is the same.

\left (a+b \right )+c=a + \left ( b+c \right )

\left (a*b \right )*c=a * \left ( b*c \right )

  • Distributive property of multiplication over addition.

In the property the number in front of a group of addition is distributed over all the terms of the addition.  In other words multiply each term inside the brackets by the number in front of the bracket.

a*\left ( b+c \right )= a*b + a*c

  • Identity Property of Addition.

The quantity zero [0] is the identity element of addition.  When you add 0 to any number you end up the identical number you started with.

a+0=a

  • Identity Property of Multiplication.

The quantity one [1] is the identity element of multiplication.  When you multiply any number by 1 you end up with the identical number.

a*1=a

  • Inverse Property of Addition.

The additive inverse of a number is called the opposite of the number.  If you add a number and its additive inverse the result is equal to 0.

a+\left ( -a \right )=0

  • Inverse Property of Multiplication.

The multiplicative inverse of a number is called the reciprocal of the number.  If you multiply a number by its multiplicative inverse the result is equal to 1.

a*\frac{1}{a}= 1

6.  Rules for Adding Terms

  • Signs the same –> Add and keep the sign

\left ( +,+ \right )\overset{add}{\rightarrow}+

\left ( -,-\right )\overset{add}{\rightarrow}-

  • Signs different –> Subract and keep the sign of the larger number

\left ( \left ( + \right ),-\right )\overset{subract}{\rightarrow}+

\left ( \left ( - \right ),+\right )\overset{subract}{\rightarrow}-

 7.  Rules for Multiplication

  • Signs the same –> multiply and the result is positive

\left ( +*+ \right )=+

\left ( -*-\right )=+

  • Signs different –> multiply and the result is negative

\left ( +*-\right )=-

\left ( -*+\right )=- 

8.  More Vocabulary

  • factors – a*b=c , a and b are factors of c

  • prime numbers — numbers that only have two factors, (1 and itself)

\left \{ 2,3,5,7,11,...\left. \right \} \right.

  • composite numbers — numbers that are not prime.  They have more than 2 factors.

24=1*24=1*\left ( 2*12 \right )=1*2*\left ( 2*6 \right )=1*2*2*\left ( 2*3 \right )=1*2*2*2*3

 

Filed under: Pre-requisites for Pre-Calculus

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