Grouping Symbols

Grouping Symbols include parentheses (), curly brackets {}, or square brackets [].

Grouping symbols are used to clarify which operations to do first, especially if a specific order is desired. If there is an expression to be simplified within the grouping symbols, follow the order of operations.

When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first. Terms inside a grouping symbol have to be operated upon before they can be acted upon by anything outside the grouping symbol. All the bracket types have equal weight; none is more powerful or acts differently from the others.

Students learn that when a problem contains parentheses within brackets within braces, such as 100 – {[5 x (3 + 2)] + [(8 + 4)/2]}, the first step is to simplify inside the innermost grouping symbol, the parentheses, to get 100 – {[5 x (5)] + [(12)/2]}. Next, move outward to the brackets, to get 100 / {[25] + [6]}. Finally, move outward to the braces, to get 100 – {31}, which simplifies to 69. Note that a division bar has the same effect as a set of parentheses in that it groups numbers together.

Grouping symbols in math expressions include:

  1. Parentheses ( ) – have a rounded shape
  2. Brackets: [ ]       – have a square shape
  3. Braces: { }    – have a twirled shape

All grouping symbols tell you, “Do this first!”.

Parentheses are used in math to show a part of a math expression or equation that must be solved first, before any other calculations are done. The part between the two parentheses is treated like one number; the answer replaces the expression in the larger math equation.

For complicated problems, brackets can be used to enclose sections of the problem that already include parentheses to further separate sections.

For extremely complicated problems, braces can be used to enclose sections that already include brackets and parentheses.

Fraction Line (Vinculum) —: The fraction line also acts as a grouping symbol. Everything above the line in the numerator is grouped together, and everything below the line in the denominator is grouped together.

 

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