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- Math Symbols -
The Most Valuable and Important Grouping Symbols Used:

"Nested Groupings (Nested Brackets)"

Nested Groupings (nested Brackets)

Math Symbols: . . . why math symbols are used . . .

Symbols are a concise way of giving lengthy instructions related to numbers and logic.


Symbols are a communication tool. Symbols are used to eliminate the need to write long, plain language instructions to describe calculations and other processes.

For example, a single symbol stands for the entire process for addition. The familiar plus sign eliminates the need for a long written explanation of what addition means and how to accomplish it.

The same symbols are used worldwide . . .

The symbols used in mathematics are universal.

The same math symbols are used throughout the civilized world. In most cases each symbol gives the same clear, precise meaning to every reader, regardless of the language they speak.

The most valuable, most frequently used Symbols in mathematics . . .

The most important, most frequently used Grouping symbols are listed below.


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Nested Groupings (nested Brackets) Nested Groupings (nested Brackets) -


If there are three levels of grouping in a nested expression, generally parentheses are used in the inner most groupings. Square brackets are used in the next higher level grouping, and braces are used in the most outer groupings.

All mathematical expressions must be evaluated in PEMDAS order to arrive at the correct answer.

If there are several nested parentheses, begin by evaluating the innermost parentheses first - before you do anything else. After evaluating the innermost parentheses, evaluate the next innermost parentheses, and so on - until there are no parentheses left.

If there are several non-nested parentheses, evaluate every one of them separately - before you do anything else.



Nested Groupings (nested Brackets) - Example 1 Math - Nested Groupings (nested Brackets) Evaluate the following expression:

{[3(15 + 5) + 27]/3}


Note the arrangement of the nested brackets:

Will be evaluated first :

Expression inside Inner Brackets - Parentheses (shown in blue):

{[3 ( 15 + 5 ) + 27]/3}


Will be evaluated second :

Unevaluated portion of expression inside Middle Brackets - Square Bracket Symbols (shown in brown):

{ [ 3(15 + 5) + 27 ] /3}


Will be evaluated third :

Unevaluated portion of expression inside Outer Brackets - Curly Bracket Symbols (shown in purple):

{ [3(15 + 5) + 27]/3 }


first : evaluation of the expression inside the inner most brackets . . .


"PEMDAS" Order
(while reading the mathematical
expression left to right)


{[3 ( 15 + 5 ) + 27]/3}



"P" - The parentheses are the math symbols which represent the innermost bracket grouping. There are no additional bracket groupings to evaluate inside the parentheses.


"E" - no exponents to evaluate inside the parentheses.


"M" - no multiplication to evaluate inside the parentheses.


"D" - no division to evaluate inside the parentheses.


"A" - ( 15+5 )

( 20 )

20


"S" - no subtraction to evaluate inside the parentheses.


The expression within the inner most parentheses can be replaced by the number 20:


{[3 ( 15 + 5 ) + 27]/3}

{[3 ( 20 ) + 27]/3}

{[3 x 20 + 27]/3}


second : evaluation of the expression inside the middle brackets . . .


"PEMDAS" Order
(while reading the mathematical
expression left to right)


{ [ 3 x 20 + 27 ] /3}



"P" - The square brackets are the math symbols which represent the middle bracket grouping.


"E" - no exponents to evaluate inside the middle bracket grouping.


"M" - 3 x 20

60


"D" - no division to evaluate inside the middle backet grouping.


"A" - [ 60+27 ]

[ 87 ]

87


"S" - no subtraction to evaluate inside the parentheses.


The expression within the square brackets can be replaced by the number 87:


{ [ 3 x 20 + 27 ] /3}


{ [ 60 + 27 ] /3}


{ [ 87 ] /3}


{ 87 /3}


third : evaluation of the expression inside the outer brackets . . .


"PEMDAS" Order
(while reading the mathematical
expression left to right)


{ 87/3 }



"P" - The curly brackets are the math symbols which represent the outer bracket grouping.


"E" - no exponents to evaluate inside the outer bracket grouping.


"M" - no multiplication to evaluate inside the outer backet grouping.


"D" - { 87/3 }

{ 29 }

29


"A" - no addition to evaluate within the outer backet grouping.


"S" - no subtraction to evaluate inside the parentheses.


The expression within the outer brackets can be replaced by the number 29:


{ 87/3 }

{ 29 }

29

Final Answer - Nested Brackets - Example 1 - FINAL ANSWER: {[3(15 + 5) + 27]/3} = 29