The Arithmetic Expressions Calculator will calculate:

- The value of an arithmetic expression using the PEMDAS rule.

**Arithmetic Expressions Calculator Parameters:** The numbers used are integers. For exponents (ie 4^{3}) enter 4^3 [the ^ symbol is avilable on most keyboards by holding Ctrl and pressing the number 6]. Note that the calculator computes the math within the brackets in the following order (), [], {} so { 2+1 [ 4 * (4 ÷ 6) + 2 ] - 2 }

The answer to the Arithmetic Expression is |

Arithmetic Expressions calculations |
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Arithmetic Expressions Calculator Input Values |

## Theoretical description

Numbers have been used since antiquity to represent or compare various amounts of items. We can combine numbers in various ways, where the most important combination is through the four basic operations: **addition**, **subtraction**, **multiplication** and **division**.

**Addition** occurs when the total of two different amount is to be found. The participants in addition are called **addends** and the result of addition (i.e. the total) is called **sum**. In symbols, addition is written as

a + b = c

where a and b are the addends and c is the sum. For example, in the addition 3 + 4 = 7, 3 and 4 are addends and 7 is the sum.

**Subtraction** is the inverse operation of addition. It occurs when we remove an amount from a total, to (usually) obtain a smaller amount. The biggest number is called **minuend**, the smaller number subtracted from the biggest is called **subtrahend** and the result of subtraction is called **difference**. In symbols, subtraction is written as

a - b = c

where a is the minuend, b is the subtrahend and c is the difference. For example, in the subtraction 9 - 5 = 4, 9 is the minuend, 5 is the subtrahend and 4 is the difference.

**Multiplication** is a shorter way to find the sum of a repeated addition of the same addends. Thus, instead of writing n times the addition a + a + a + + a, we simply write n × a.

The numbers involved in multiplication are called **factors** and the result of multiplication is called **product**. Thus, in the operation

n × a = b

n and a are factors and b is the product. For example, in the multiplication 7 × 3 = 21, 7 and 3 are factors and 21 is the product.

**Division** is the inverse operation of multiplication. It occurs when an item a is divided into n equal parts. The number representing the item is called **dividend**, the number that shows in how many parts the whole is divided is called **divisor** and the result of division is called **quotient**. We write

a ÷ n = b

where a is the dividend, n is the divisor and b is the quotient. For example, in the division 28 ÷ 4 = 7, 28 is the dividend, 4 is the divisor and 7 is the quotient.

Another common operation is raising a number in a certain **power**. This means multiplying n times a number a by itself. In other words, power is a recurrent multiplication by the same factor. Thus, instead of multiplying n times the number a by itself, i.e. b = a × a × a × (n times), we write

b = a^{n}

where a is called **base**, n is called **exponent** and b is called **power**. For example, in the operation 25 = 32, 2 is the base, 5 is the exponent and 32 is the power (you can test examples of this using the exponents calculator.

If all operations are involved in an arithmetic expression, then there is a hierarchy of operations, i.e. some operations have a higher order of priority than the others, and therefore, they are done first. The rule concerning this order of operation is known as the **PEMDAS Rule**, which is an acronym for Parenthesis - Exponents - Multiplication - Division - Addition - Subtraction and shows the order of operations in an arithmetic expression. According to PEMDASS Rule, we must begin from the part of the expression inside parenthesis (if any) where exponents are calculated first, then we do multiplications and divisions from left to right as they have the same order of priority, and at the end, after finishing with all the above operations, we conclude with addition and subtraction from left to right (they also have the same order of priority).

Another thing to point out here is the various types of parenthesis (otherwise known as brackets). Thus, if different types of brackets are used in an arithmetic expression, we begin with the expression inside the round brackets, as they are the innermost ones, then with the square brackets which include the round ones, and then with the curled brackets (which include the other two types).

## Using the Arithmetic Expressions Calculator

This calculator is different from the other Math calculators on iCalculator™ as there is not a specific number of inputs which are combined in an expression that potentially contains:

- Three types of brackets: round (the innermost), square (that include the round ones); and curled - the outermost (that include the other two types). There may also be some numbers outside the curled brackets.
- Five basic operations that include: a) powers (exponents) that are done first; b) division and multiplication that are done simultaneously starting from left to right after having completed the operations with exponents, and c) addition and subtraction that are done simultaneously starting from left to right after having completed divisions and multiplications.

When solving an arithmetic expression, the following must be considered:

- Operations start from the part of expression inside the curved brackets; after completing it, we focus on square ones, then the curled ones and finally the rest of operations outside brackets.
- Two consecutive operations of the same order (for example one multiplication and one division) must be carried out one by one, while when they have an addition or subtraction in-between, they can be done simultaneously.

## Example of math calculations using the PEMDAS rule

18 - 3 × 4 + {20 ÷ [10 + 2 × (3 × 2^{2} - 21 ÷ 3)]} - 3 × 2

= 18 - 3 × 4 + {20 ÷ [10 + 2 × (3 × 4 - 21 ÷ 3)]} - 3 × 2

= 18 - 3 × 4 + {20 ÷ [10 + 2 × (12-7)]} - 3 × 2

= 18 - 3 × 4 + {20 ÷ [10 + 2 × 5]} - 3 × 2

= 18 - 3 × 4 + {20 ÷ [10 + 10]} - 3 × 2

= 18 - 3 × 4 + {20 ÷ 20} - 3 × 2

= 18 - 3 × 4 + 1 - 3 × 2

= 18 - 12 + 1 - 6

= 6 + 1 - 6

= 7 - 6

## Arithmetic Math Tutorials associated with the Arithmetic Expressions Calculator

The following Math tutorials are provided within the Arithmetic section of our Free Math Tutorials. Each Arithmetic tutorial includes detailed Arithmetic formula and example of how to calculate and resolve specific Arithmetic questions and problems. At the end of each Arithmetic tutorial you will find Arithmetic revision questions with a hidden answer that reveal when clicked. This allows you to learn about Arithmetic and test your knowledge of Math by answering the revision questions on Arithmetic.

- 1.1 - Numbering Systems, a Historical View
- 1.2 - Number Sets, Positive and Negative Numbers and Number Lines
- 1.3 - Operations with Numbers and Properties of Operations
- 1.4 - Order of Operations and PEMDAS Rule
- 1.5 - Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF
- 1.6 - Divisibility Rules
- 1.7 - Decimal Number System and Other Numbering Systems

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