Lesson RULES OF ALGEBRAIC AND ARITHMETIC OPERATIONS

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This lesson covers a review of the basic rules of algebraic and arithmetic operations.

REFERENCES

<a href = "http://www.mathgoodies.com/lessons/vol7/order_operations.html" target = "_blank">http://www.mathgoodies.com/lessons/vol7/order_operations.html</a>
<a href = "http://www.themathpage.com/alg/algebraic-expressions.htm" target = "_blank">http://www.themathpage.com/alg/algebraic-expressions.htm</a>
<a href = "http://whyslopes.com/etc/ThreeSkillsForAlgebra/ch18.html" target = "_blank">http://whyslopes.com/etc/ThreeSkillsForAlgebra/ch18.html</a>
<a href = "http://www.themathpage.com/alg/algebraic-expressions.htm" target = "_blank">http://www.themathpage.com/alg/algebraic-expressions.htm</a>
<a href = "http://www.themathpage.com/Alg/rules-of-algebra.htm" target = "_blank">http://www.themathpage.com/Alg/rules-of-algebra.htm</a>
<a href = "http://wiki.answers.com/Q/What_are_the_basic_rules_of_algebra" target = "_blank">http://wiki.answers.com/Q/What_are_the_basic_rules_of_algebra</a>
<a href = "http://www.themathpage.com/aPreCalc/algebraPre.htm" target = "_blank">http://www.themathpage.com/aPreCalc/algebraPre.htm</a>
<a href = "http://mentorproducts.com/laws.html" target = "_blank">http://mentorproducts.com/laws.html</a>
<a href = "http://mathforum.org/dr.math/faq/faq.property.glossary.html" target = "_blank">http://mathforum.org/dr.math/faq/faq.property.glossary.html</a>

DIFFERENCE BETWEEN ARITHMETIC AND ALGEBRA

Definition of algebra from Deb at <a href = "http://math.about.com/cs/algebra/g/algebradef.htm" target = "_blank">http://math.about.com/cs/algebra/g/algebradef.htm</a>

A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc. Moving from Arithmetic to Algebra will look something like this: Arithmetic: 3 + 4 = 3 + 4 in Algebra it would look like: x + y = y + x. [end]

The actual operations are the same, but in some cases you would favor one method over another even though both methods provide the same answer.
Example:
5 * (3 + 2 + 3) = 5 * (8) = 40
you would simply add the numbers together and then multiply.
x * (y + z + t) = (x*y) + (x*z) + (x*t)
with the x and y and z and t, since you don't know what the numbers are, an intermediate step might be to do what is shown.  You could do the same thing with the numbers and get the same answer but it was not necessary since the numbers were in a form that favored the alternate method.
just to show you:
5 * (3 + 2 + 3) = (5*3) + (5*2) + (5*3) = 15 + 10 + 15 = 40

REAL NUMBERS

real numbers are not imaginary numbers. They do not include the letter i which stands for the square root of -1, nor do they include undefined numbers such as infinity.

real numbers cover:
1.  integers
2.  rational numbers
3.  irrational numbers

INTEGERS

integers are numbers without a fractional part.
examples of integers are:
1,2,3, etc.

RATIONAL NUMBERS

rational numbers are numbers that can be expressed as a ratio of 2 integers.
examples of rational numbers are:
1,2,3, 1/2, 1/3, 1/4, 5/6, etc.

integers are part of the rational number set since all integers are the ratio of that number divided by 1.
example:  5 = 5/1 = a number that is expressed as the ratio of 2 integers, namely 5 and 1.

if the decimal equivalent of the fractional part stops, then the number is a rational number.

an example would be 5/8 which has a decimal equivalent of .625

if the decimal equivalent of the fractional part does not stop, but has a repeating pattern, then it is more then likely a rational number even if you can't find the ratio of the integers that created it.

an example would be 1/3 which has a decimal equivalent of .33333333...
another example would be 5/11 which has a decimal equivalent of .45454545...

IRRATIONAL NUMBERS

irrational numbers are numbers that cannot be expressed as a ratio of 2 integers.

if the decimal equivalent of the fractional part does not have a repeating pattern, the number is more than likely an irrational number.

{{{pi}}} = 3.141592654... is an example of an irrational number.
{{{e}}} = 2.7182818284590452353602875... is another example of an irrational number.

Note that {{{e}}} looked like it was repeating (18281828) but ended up not repeating after carrying it out more decimal places.

While the fractional part of a real number can be endless, the capabilities of the calculator used dictates how many decimal places the operations are actually carried out for.

In this lesson, the letters of the alphabetic are used as symbols to represent unknown numbers.  The letters a, b, c, d, e, f, � are mostly used.  Some smattering of x, y, z, � might also be used where required to distinguish from the a, b, c, �, etc.

These symbols are called variables if the number they represent can be different depending on the circumstances in which they are used.

These symbols are called constants if the number they represent remains the same every time.  The greek letter {{{pi}}} represents the constant used in translating from a radius of a circle to a circumference of a circle, among other functions.  The small letter {{{e}}} usually written in script, represents the base of the natural logarithm called "ln", among other functions.  The fractional part of both of these are endless and do not contain any repeating patterns.

ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

The order of algebraic and arithmetic operations  is briefly mentioned here, but will be covered in more detail in the lesson titled ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS.

ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

calculate the results of an expression within the most inner set of parentheses in the equation.

replace the most inner set of parentheses and the expression within the set of parentheses with the results of the calculation.

calculate the results of an expression within the next most inner set of parentheses in the equation.

replace the next most inner set of parentheses and the expression within the set of parentheses with the results of the calculation.

continue working your way out until there are no more sets of parentheses to process.

within each set of parentheses the calculations are done in the following order:

process all powers and roots first going from left to right.
process all multiplications and divisions next going from left to right.
process all addition and subtraction next going from left to right.

once all sets of parentheses have been processed and you are left with an equation that has no more sets of parentheses, then make one last pass through the equation using the same order of algebraic and arithmetic operations that was used within each set of parentheses.

Note:
parentheses enclosing a number without any operation being performed are not considered as being an inner set of parentheses for purposes of ordering the calculations to be performed.  entries would be something like (59) or (-59) or (-x).  You can clear the parentheses if it makes sense to do so, or you can leave it to be processed later on.
Example:
+ (-59) could be cleared and replaced with - 59 since these are equivalent.  This clears the parentheses.

Parentheses containing expressions involving an operation are what you are looking for.  Those would contain something like {{{x - 5}}} or {{{5*(x^2+3x-2)}}}.

Examples of this procedure will be seen in more detail in the lesson on ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS

COMMUTATIVE LAW OF ADDITION

{{{a+b}}} = {{{b+a}}}

This law means that you can add a string of numbers together and get the same answer regardless of the order in which you add them.

Example:
if we let:
a = 5
b = 7
then the equation representing the law becomes:
{{{5+7}}} = {{{7+5}}}
since:
{{{5+7}}} = {{{12}}} and:
{{{7+5}}} = {{{12}}} then:
both forms of the equation are equivalent.

COMMUTATIVE LAW OF MULTIPLICATION

{{{a*b}}} = {{{b*a}}}

This law means that you can multiply a string of numbers together and get the same answer regardless of the order in which you multiply them.

Example:
If we let:
a = 5
b = 7
then the equation representing the law becomes:
{{{5*7}}} = {{{7*5}}}
since:
{{{5*7}}} = {{{35}}} and:
{{{7*5}}} = {{{35}}} then:
both forms of the equation are equivalent.

ASSOCIATIVE LAW OF ADDITION

{{{a+(b + c)}}} = {{{(a+b)+c}}}

The numbers within the parentheses are processed first based on the ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS.

This law means that you can add a string of numbers together and get the same answer regardless of how you combine those numbers into different groups.

Example:
If we let:
a = 5
b = 7
c = 9
then the equation representing the law becomes:
{{{5+(7+9)}}} = {{{(5+7)+9}}}
since:
{{{5+(7+9)}}} = {{{5+16}}} = {{{21}}} and:
{{{(5+7)+9}}} = {{{12+9}}} = {{{21}}} then:
both forms of the equation are equivalent.

ASSOCIATIVE LAW OF MULTIPLICATION

{{{a*(b*c) = (a*b)*c}}}

The numbers within the parentheses are processed first.

This law means that you can multiply a string of numbers together and get the same answer regardless of how you combine those numbers into different groups.

Example:
if you let:
a = 5
b = 6
c = 7
then the equation representing the law becomes:
{{{5*(6*7) = (5*6)*7}}}
since:
{{{5*(6*7)}}} = {{{5*42}}} = {{{210}}} and:
{{{(5*6)*7}}} = {{{30*7}}} = {{{210}}} then:
both forms of the equation are equivalent.

FIRST DISTRIBUTIVE LAW

{{{a*(b+c) = a*b + a*c}}}

This law means that if you multiply a group of numbers being added together by a number, then you have to multiply every member of the group by the same number.

Example:
If we let:
a = 7
b = 8
c = 9
then the equation representing the law becomes:
{{{7*(8+9)}}} = {{{(7*8) + (7*9)}}}
since:
{{{7*(8+9)}}} = {{{7*17}}} = {{{119}}} and:
{{{(7*8) + (7*9)}}} = {{{56+63}}} = {{{119}}} then:
both forms of the equation are equivalent.

SECOND DISTRIBUTIVE LAW

{{{(a+b)*(c+d) = ((a*c)+(a*d))+((b*c)+(b*d))}}}
Example:
If we let:
a = 4
b = 5
c = 6
d = 7
then the equation representing the law becomes:
{{{(4+5)*(6+7) = ((4*6)+(4*7))+((5*6)+(5*7))}}}
since:
{{{(4+5)*(6+7)}}} = {{{9*13}}} = {{{117}}} and:
{{{(4*6)+(4*7)+(5*6)+(5*7)}}} = {{{24+28+30+35}}} = {{{117}}} then:
both forms of the equation are equivalent.

GENERALIZATION OF THE LAWS

Most of these laws can be generalized fairly easily.

example from commutative law of multiplication:
{{{a*b = b*a}}} can be generalized to {{{a*b*c*d*e}}} = {{{e*d*c*b*a}}} = {{{c*b*d*a*e}}}, etc.
if you let:
a=1
b=2
c=3
d=4
e=5
then:
{{{1*2*3*4*5}}} = {{{120}}} and:
{{{5*4*3*2*1}}} = {{{120}}} and:
{{{3*2*4*1*5}}} = {{{120}}} so they are all equivalent.

example from associative law of multiplication:
{{{a*(b*c) = (a*b)*c}}} can be generalized to {{{a*(b*c*d*e)}}} = {{{(a*b*c)*(d*e)}}}, etc.
if you let:
a=1
a=2
c=3
d=4
e=5
then:
{{{1*(2*3*4*5)}}} = {{{1*(120)}}} = {{{120}}} and:
{{{(1*2*3)*(4*5)}}} = {{{(6)*(20)}}} = {{{120}}} so they are both equivalent.

example from first distributive law:
{{{a*(b+c) = (a*b) + (a*c)}}} can be generalized to {{{a*(b+c+d+e)}}} = {{{(a*b)+(a*c)+(a*d)+(a*e)}}}, etc.
if you let:
a=1
b=2
c=3
d=4
e=5
then:
{{{1*(2+3+4+5)}}} = {{{1*(14)}}} = {{{14}}} and:
{{{(1*2)+(1*3)+(1*4)+(1*5)}}} = {{{2+3+4+5}}} = {{{14}}} so they are both equivalent.

example from second distributive law:
{{{(a+b)*(c+d) = ((a*c)+(a*d))+((b*c)+(b*d))}}} can be generalized to:
{{{(a+b+c)*(d+e+f)}}} = {{{(a*d)+(a*e)+(a*f)}}} + {{{(b*d)+(b*e)+(b*f)}}} + {{{(c*d)+(c*e)+(c*f)}}}
The a in the first group multiplies each member of the second group and the results are summed.
The b in the first group multiplies each member of the second group and the results are summed.
The c in the first group multiplies each member of the second group and the results are summed.
The overall result is the sum of all 3 sums.

example:
if you let:
a=1
b=2
c=3
d=4
e=5
f=6
then:
{{{((1+2+3)*(4+5+6))}}} = {{{((6)*(15))}}} = {{{(90)}}} and:
{{{((1*4)+(1*5)+(1*6))}}} + {{{((2*4)+(2*5)+(2*6))}}} + {{{((3*4)+(3*5)+(3*6))}}} = {{{(4+5+6)}}} + {{{(8+10+12)}}} + {{{(12+15+18)}}} = {{{(15)}}} + {{{(30)}}} + {{{(45)}}} = {{{(90)}}} so they are both equivalent.

If you are in doubt as to whether the laws can be generalized or not, just extend the logic to whatever you think is still covered by the law and test it out.  Care in making these tests is necessary because the logic can sometimes fool you.  If you are not 100% sure, then confirm with some other knowledgeable resource.

OPERATORS AND OPERANDS

Addition operator is shown as +.
Subtraction operator is shown as -.

THE ADDITION OF A POSITIVE OPERAND IS THE SAME AS THE SUBTRACTION OF A NEGATIVE OPERAND

Example:
+ (500) = - (-500) = 500

THE ADDITION OF A NEGATIVE OPERAND IS THE SAME AS THE SUBTRACTION OF A POSITIVE OPERAND

Example:
+ (-500) = - (500) = (-500)

The parentheses are not always there unless they absolutely have to be.
The sign of the operand is not always there unless it absolutely has to be.

+ 500 means you are adding a positive 500 and the parentheses are not necessary.
+ (-500) means you are adding a negative 500 and the parentheses are necessary.
- 500 means you are subtracting a positive 500 and the parentheses are not necessary.
- (-500) means you are subtracting a negative 500 and the parentheses are necessary.

THE SIZE OF ANY OPERAND IS ALWAYS POSITIVE REGARDLESS OF THE SIGN OF THE OPERAND.

This is useful when comparing numbers of opposite signs.

Example 1:
Size of (-1000) is 1000
Size of (1000) is 1000
Size of (-500) is 500
Size of (500) is 500

When comparing sizes of operands, the signs are ignored.

Where applicable, the sign of the result is determined by the sign of the larger operand.

Where applicable, the sign of the operand is determined by the combination of the sign of the operand and the operator.
Example:
- (+500) is equivalent to (-500).
+ (-500) is equivalent to (-500).
- (-500) is equivalent to (+500)

LAW OF SIGNS

THE SIGN OF THE RESULT OF A MINUS TIMES A MINUS IS A PLUS

Examples:
{{{(-25)*(-5)}}} = {{{125}}}
{{{(-5)*(-25)}}} = {{{125}}}

THE SIGN OF THE RESULT OF A PLUS TIMES A PLUS IS A PLUS

Examples:
{{{(25)*(5)}}} = {{{125}}}
{{{(5)*(25)}}} = {{{125}}}

THE SIGN OF THE RESULT OF A MINUS TIMES A PLUS IS A MINUS

Example:
{{{(-25)*(5)}}} = {{{-125}}}
{{{(-5)*(25)}}} = {{{-125}}}

THE SIGN OF THE RESULT OF A PLUS TIMES A MINUS IS A MINUS.

Example:
{{{(25)*(-5)}}} = {{{-125}}}
{{{(5)*(-25)}}} = {{{-125}}}

THE SIGN OF THE RESULT OF A MINUS DIVIDED BY A MINUS IS A PLUS.

Examples:
{{{(-25)/(-5)}}} = {{{(5)}}}
{{{(-5)/(-25)}}} = {{{(1/5)}}}

THE SIGN OF THE RESULT OF A PLUS DIVIDED BY A PLUS IS A PLUS.

Example:
{{{(25)/(5)}}} = {{{(5)}}}
{{{(5)/(25)}}} = {{{(1/5)}}}

THE SIGN OF THE RESULT OF A MINUS DIVIDED BY A PLUS IS A MINUS

Examples:
{{{(-25)/(5)}}} = {{{-125}}}
{{{(-5)/(25)}}} = {{{-(1/5)}}}

THE SIGN OF THE RESULT OF A PLUS DIVIDED BY A MINUS IS A MINUS

{{{(25)/(-5)}}} = {{{-125}}}
{{{(5)/(-25)}}} = {{{-(1/5)}}}

THE SIGN OF THE RESULT OF A PLUS ADDED TO A PLUS IS A PLUS

Example:
{{{(500) + (700)}}} = {{{(1200)}}}
The two numbers are added together and the sign remains the same.

THE SIGN OF THE RESULT OF A MINUS ADDED TO A MINUS IS A MINUS

Example:
{{{(-500) + (-700)}}} = {{{(-1200)}}}
The two numbers are added together and the sign remains the same.

THE SIGN OF THE RESULT OF A MINUS ADDED TO A PLUS IS DETERMINED BY THE SIZE OF THE OPERANDS

Example 1:
{{{(1000) + (-500)}}} = {{{(500)}}}
The size of (1000) is larger than the size of (-500) so the sign of the result is positive.

Example 2:
{{{(500) + (-1000)}}} = {{{(-500)}}}
The size of (-1000) is larger than the size of (500) so the sign of the result is negative.

THE SIGN OF THE RESULT OF A PLUS ADDED TO A MINUS IS DETERMINED BY THE SIZE OF THE OPERANDS

Example 1:
{{{(-500) + (1000)}}}
= {{{(500)}}}
The size of (1000) is larger than the size of (-500) so the sign of the result is positive.

Example 2:
{{{(-1000) + (500)}}} = {{{(-500)}}}
The size of (-1000) is larger than the size of (500) so the sign of the result is negative.

THE SIGN OF THE RESULT OF A PLUS SUBTRACTED FROM A PLUS IS DETERMINED BY THE SIZE OF THE OPERANDS

Subtracting a positive number is the same as adding a negative number so the signs of the numbers become opposite resulting in the rule.

Example 1:
{{{(1000) - (500)}}} = {{{(500)}}}
The size of (1000) is greater than the size of - (500) so the sign of the result is positive.
Note that - (500) is the same as + (-500).

Example 2:
{{{(500) - (1000)}}} = {{{(-500)}}}
The size of - (1000) is greater than the size of (500) so the sign of the result is negative.
Note that - (1000) is the same as + (-1000).

THE SIGN OF THE RESULT OF A MINUS SUBTRACTED FROM A MINUS IS DETERMINED BY THE SIZE OF THE OPERANDS

Subtracting a negative number is the same as adding a positive number so the signs of the numbers become opposite resulting in the rule.

Example 1:
{{{(-1000) - (-500)}}} = {{{(-500)}}}
The size of (-1000) is greater than the size of - (-500) so the sign of the result is negative.
Note that - (-500) is the same as + (500).

Example 2:
{{{(-500) - (-1000)}}} = {{{(500)}}}
The size of - (-1000) is greater than the size of (-500) so the sign of the result is positive.
Note that - (-1000) is the same as + (1000).

THE SIGN OF THE RESULT OF A MINUS SUBTRACTED FROM A PLUS IS ALWAYS PLUS

Subtracting a negative number is the same as adding a positive number so the signs of the numbers become the same resulting in the rule.

Example 1:
{{{(1000) - (-500)}}} = {{{(1500)}}}
Note that - (-500) is the same as + (500).

Example 2:
{{{(500) - (-1000)}}} = {{{(1500)}}}
Note that - (-1000) is the same as + (1000).

THE SIGN OF THE RESULT OF A PLUS SUBTRACTED FROM A MINUS IS ALWAYS MINUS

Subtracting a positive number is the same as adding a negative number so the signs of the numbers become the same resulting in the rule.

Example 1:
{{{(-1000) - (500)}}} = {{{(-1500)}}}
Note that - (500) is the same as + (-500).

Example 2:
{{{(-500) - (1000)}}} = {{{(-1500)}}}
Note that - (1000) is the same as + (-1000).

This lesson did not cover the rules of exponents and radicals.
Those are covered in the lesson on EXPONENTS and the lesson on RADICALS
A complex example of the application of the LAWS OF ALGEBRAIC AND ARITHMETIC OPERATIONS can be found in the lesson on the ORDER OF ALGEBRAIC AND ARITHMETIC OPERATIONS.