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Order Mathematical Operations

In mathematics and computer programming, an expression or string of symbols intended to represent a numerical value must follow commonly accepted and unambiguous rules. For example, the rule for evaluating 2 + 3 * 4 in mathematics and in most computer languages is to do the multiplication first, so the correct answer is 14. Sometimes parentheses, which have their own rules, may be used to avoid confusion, thus: 2 + (3 * 4). When a term in an expression is both preceded and followed by an operator such as minus or times. The convention needed to clarify which operator should be applied first is known as a precedence rule or, more informally, order of operation. From the earliest use of mathematics notation multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3 + 4 * 5 = 4 * 5 + 3 = 23.

The standard order of operations, or precedence, is expressed in the following chart.

• exponents and roots

• multiplication and division

This means that if a number or other symbol, or an expression grouped by one or more symbols of grouping, is preceded by one operator and followed by another, the operator higher on the list should be applied first. The commutative and associative laws of addition and multiplication allow the operators +, -, *, and / to be applied in any order that obeys this rule. The root symbol, √, requires either parentheses around the radicand or a bar over the radicand. Stacked exponents are applied from the top down.

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3/4 = 3 ÷ 4 = 3 ·¼ and 3 - 4 = 3 + (-4), that is, the sum of positive three and negative four.

Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws.

A horizontal fractional line also acts as a symbol of grouping:

For ease in reading, other grouping symbols (such as curly braces {} or square brackets [] ) are often used along with the standard round parentheses, e.g.

cs and computer programming, an expression or string of symbols intended to represent a numerical value must follow commonly accepted and unambiguous rules. For example, the rule for evaluating 2 + 3 * 4 in mathematics and in most computer languages is to do the multiplication first, so the correct answer is 14. Sometimes parentheses, which have their own rules, may be used to avoid confusion, thus: 2 + (3 * 4). When a term in an expression is both preceded and followed by an operator such as minus or times. The convention needed to clarify which operator should be applied first is known as a precedence rule or, more informally, order of operation. From the earliest use of mathematics notation multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3 + 4 * 5 = 4 * 5 + 3 = 23.

The standard order of operations, or precedence, is expressed in the following chart.

• exponents and roots

• multiplication and division

This means that if a number or other symbol, or an expression grouped by one or more symbols of grouping, is preceded by one operator and followed by another, the operator higher on the list should be applied first. The commutative and associative laws of addition and multiplication allow the operators +, -, *, and / to be applied in any order that obeys this rule. The root symbol, , requires either parentheses around the radicand or a bar over the radicand. Stacked exponents are applied from the top down.

It is helpful to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus 3/4 = 3 4 = and 3 - 4 = 3 + (-4), that is, the sum of positive three and negative four.

Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws.

A horizontal fractional line also acts as a symbol of grouping:

For ease in reading, other grouping symbols (such as curly braces {} or square brackets [] ) are often used along with the standard round parentheses, e.g.

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