- Math Symbols -
The Most Valuable and Important Grouping Symbols Used:
"Nested Groupings (Nested Brackets)"
Background - why math symbols are used. . .
Symbols are a concise way of giving lengthy instructions related to
numbers and logic.
Math Symbols are an invention, not a discovery. They are a
communication tool. Symbols are used to eliminate
the need to write long, plain language instructions to describe
calculations and other processes.
The most valuable, most frequently used Symbols in mathematics . . .
The most important, most frequently used Grouping symbols
are listed below.
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.
Evaluate the following expression:
{[3(15 + 5) + 27]/3}
Note the arrangement of the nested brackets:
Will be evaluated first:
Most Inner Brackets - Parentheses (shown in blue):
{[3(15 + 5) + 27]/3}
Will be evaluated second:
Middle Brackets - Square Bracket Symbols (shown in brown):
{[3(15 + 5) + 27]/3}
Will be evaluated third (last) :
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 parentheses
has been replaced by the number 20: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 now represent the innermost
bracket grouping. There are no
additional bracket groupings to
evaluate inside the square brackets.
"E"
- no exponents to evaluate inside the
square brackets
"M"
3 x 20
60
"D"
- no division to evaluate inside
the square brackets
"A" [60+27]
[87]
87
"S"
- no subtraction to evaluate inside
the square brackets
The expression within the square brackets
has been 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 now represent the innermost
bracket grouping. There are no
additional bracket groupings to
evaluate inside the curly brackets.
"E"
- no exponents to evaluate inside the
curly brackets
"M"
- no multiplication to evaluate inside
the curly brackets
"D"
{87/3}
{29}
29
"A"
- no addition to evaluate inside the
curly brackets
"S"
- no subtraction to evaluate inside
the curly brackets
The expression within the curly brackets
has been replaced by the number 29:
{87/3} =
{29} =
29
- FINAL ANSWER:
{[3(15 + 5) + 27]/3} = 29
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