Algebraic Concepts - Part 0: Grouping Symbols

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• published: 30 Nov 2014
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Algebraic Concepts - Part 0: Grouping Symbols An overview of what grouping symbols are and some of the different types. Grouping symbols are used to indicate a number, numbers or a combination of numbers and variables that are to be thought of as a separate group, in carrying out an operation or procedure. For example, in the expression a plus b multiplied by d, a pair of parentheses ( ) are used to indicate that a and b are to be added to each other and d is to multiply their sum. The absolute value symbol is a grouping symbol used as a delimiter, to indicate the extent of what the absolute value is being taken of. Here, it is used to indicate that we are only taking the absolute value of negative 3, and that that result should be multiplied by negative 2. Grouping symbols include the absolute value, parentheses, square brackets, or just brackets and curly brackets or braces. Parentheses are more commonly used in math than either brackets or braces, so when a grouping of this type is mentioned, a person might use the word parentheses, even though they may mean brackets or braces. Parentheses, brackets and braces have an informal hierarchy: if parentheses are nested, one pair situated inside of another, parentheses will occur inside of brackets, which will occur inside of braces {[( )]}. If there is only one level of nesting, as in this expression, we only use parentheses inside of brackets. In this last expression, we have two levels of nesting, so all three types are used. Though this hierarchy is not universally used, it does help to distinguish which elements of an expression are grouped with which, so we can have this expression, with parentheses, brackets and braces instead of this one, with all parentheses. Radicals, such as the square root symbol use another kind of grouping symbol, called a vinculum. A vinculum is a short horizontal line or bar which when placed over or under numbers and/or variables, indicates those elements that are included in the operation. In the square root, the vinculum bar is used to indicate all digits that are to be included in the taking of the square root. In this expression, it is not clear if we mean to take the square root of 25 or to take the square root of 2, then multiply the result by 5, unless the 5 is written first, but with the vinculum, it is clear either way. The fraction line is another use of the vinculum. Using a fraction line, it is clear that the 2 and 4 should be added to each other and that 3 and 7 should be multiplied by each other, the binary operations of adding and multiplication have been physically separated by the vinculum. Other uses of the vinculum include recurring decimals, where it is used to indicate which digits repeat.