Mathematics at every level is based on several fundamental rules. without going into details, this page outlines many of the basic mathematical rules, the knowledge of which are taken for granting when dealing with higher mathematics such as algebra, geometry and trigonomety.

 

Rules concerning numbers and their features

• Natural or counting numbers 1,2,3,4,5..
• Whole numbers 0,1,2,3...
• Integers -2, -1, 0, 1, 2...
• Negative integers -4, -3, -2, -1
• Positive Integers are natural numbers
• Rational numbers   fractions  such as 5/6 since numbers such as 4 can be written as 4/1 all integers are rational numbers.  Numbers can be written as fractions such as a/b with a being an integer and b being a natural number.  Terminating and repeating decimals are also rational numbers, since they can be written as fractions
• Irrational numbers cannot be written as fractions,  the number pi, ?,  is an example of irrational numbers
• Prime numbers a number that can be evenly divided by only itself and 1 for example The only even prime number is 2  The first ten prime numbers are 2, 3, 5, 7,11, 13, 17, 19, 23, 29
• Odd number whole number not divisible by 2: such as 1, 3, 5,7
• Even numbers divisible by 2, such as 0,2, 4, 6
• Composite number divisible by more than just 1 and itself, 4, 6, 8, 10, 12, 14, 16

 

Square, Cubes & Square Roots

Squares result from numbers multiplied by themselves 2*2=4, 4*4=16. The two number multiplied together that create a square are called the square root. The square root of some numbers cannot be easily written, for those number we simple call them the square root of the number. In the example that follows, the square root of 5 is shown. The two numbers that are multiple to get the number 5 are not who numbers and would be easy to write, so we use the number and place it under the square root symbol.

Cubes result from a number multipled by against themselves twice 3*3*3=27. The cube root of 27 would be 3.

 

Ways to show multiplicatoin

• 4x2=8
• 4*2=8
• (4) (2) =8
• 4(2)=8
• (4)2=8

 

Order of Operations

One set of rules that must be understood is Order of Operations. It is important to know which operations (addition, subtraction, multiplication, division) should take presedence under varying circumstances:

• If multiplication, division, powers. additions, parentheses and so forth are all continaued in one problem, the order of operation is

1. parentheses
2. powers and square roots
3. multiplication
4. division
   (whichever come first left to right)
   
   
5. addition
6. subtraction
   (whichever come first left to right)

 

parentheses, brackets [ ] and braces { } are used to group numbers and/or variables, parentheses first, brackets, then braces

 

Addition

• Closure all answers fall into the original set.  Even numbers added return even number, therefore they are closed, if you add to odd numbers the result is not an odd number, therefore the set of odd numbers is not closed under additional, no closure
• Commutative order does not make any difference, commutative does not for subtraction
• Associative grouping does not make any difference, does not apply to subtraction
• identity element for addition is 0, any number added to 0 gives the same number
• additive inverse is the opposite of the number thus any number added to its inverse (opposite or negative) will equal 0
  
  

Multiplication properties axioms

• Closure all answers fall into the original set.  Even and odd numbers both return a close set, that is even number multiplied by another even number  return even number, therefore they are closed set, if you multiply odd numbers by another odd number the result is also an odd number, therefore the set of odd numbers are also at closed set
• Commutative order does not  make any difference, commutative does not apply for division
• Associative grouping does not make any difference, does not apply to division
• identity element for multiplication  is 1, any number multiplied by 1 gives the same number
• multiplicative inverse is the reciprocal of the number thus any number multiplied by its reciprocal equals 1 for example 4 * 1/4 = 1, therefore 4 and 1/4 are inverses   b * 1/b = 1, therefore b and 1/b are inverses

Multiplying and dividing with zero

• zero times any number equals zero
• zero divided by any number is zero
• you cannot divide numbers by zero as dividing by zero is undefined/not permiited, that is there will be no answer.  Important fact: the answer is  not zero

Distributive Propery

A property of two operations, the distributive property is the processing distributing the number on the outside of the parentheses to each item on the inside, cannot be used with less than two operations

Exponents

An exponent is a positive or negative number placed above and to the right of a quantity. It expresses the power to which the quantity is to be raised of lowered