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Operations of arithmetic

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The four elementary operations of arithmetic are addition, subtraction, multiplication and division.

Counting is the most basic concept of arithmetic. Counting in the most fundamental sense involves the set of numbers called the natural numbers (also called counting numbers for this reason). It is the ordered set of numbers {1, 2, 3,...}. Basic arithmetic generally takes place in this setting. When counting is done, a number is incremented from one member of the set to the next. (See Peano axioms and Counting in set theory for a higher-level discussion.)

Basic operations

Addition

Addition is the fundamental operation of arithmetic, upon which all other operations are based. Addition involves counting from one number to a number a given increment away. For example, adding five to seven begins at (7) and counts an additional five numbers (8, 9, 10, 11, 12). This final number is called the sum, while the initial number and the increment number are called addends. The addition operation is indicated by the plus sign (+). See also Addition (natural numbers).

Subtraction

Subtraction is the opposite of addition. Instead of incrementing (increasing) up from a given number, subtraction involves incrementing (increasing) down from a given number. FOR EXAMPLE, in the subtraction of five (5) from seven (7), the counter begins at seven and counts backward five more numbers, thus 7, 6, 5, 4, 3, 2. Subtraction is represented by the minus sign (-). The result of a subtraction problem is the difference.

When using natural numbers, one cannot subtract a larger number from a smaller number, as it is impossible to count below 0 (while not part of the counting numbers, zero is generally allowed as part of the whole numbers, defined as the set of natural numbers and zero). However, in more advanced math, the set of numbers known as integers (the natural numbers, their mathematical opposites and zero) are employed, allowing for negative numbers, numbers an equal distance away from zero but in the opposite direction from zero. Thus five minus seven is counted as 5, 4, 3, 2, 1, 0, -1 (said "negative one"), -2. Subtraction of a larger number from a smaller number is conceptualized also as subtracting the smaller from the larger then making the answer negative. See also Subtraction (natural numbers).

Multiplication

Multiplication is repeated addition. When one multiplies a given number by another number, they begin at zero and count up by the first number a number of times equal to the second. Thus, multiplying two by three is counted as 1, 2; 3, 4; 5, 6. The two numbers multiplied are factors and the result is the product. Multiplication is represented by the "times sign" (×), though often represented by a dot to avoid confusion with the variable x, and usually represented by an asterisk (*)in computer applications. See also Multiplication (natural numbers).

Division

Division can be considered the opposite of multiplication or repeated subtraction. Where multiplication can be considered the act of combining a number of sets a given size together, division is the breaking up of a whole into a number of equal sets or into sets of a given size. Division as the operation of repeated subtraction is subtracting a given number from a starting number as many times as possible to reach zero. Division is commonly represented by the division symbol (÷), though it is also represented by a forward slash (/), usually when working on with a computer interface where the standard division sign is a special character. The original number is the dividend, the number used for dividing into groups is the divisor, and the result is the quotient.

When a number cannot be divided evenly by another number (i.e., zero cannot be reached exactly) the difference between the last full set and zero is called the remainder (e.g., 25 ÷ 4 would have a remainder of 1). This can be expressed as a whole number remainder (as in the previous example), or by using the set of numbers known as rational numbers, numbers represented as the quotient of two integers and written as either a fraction or a decimal. See also Division (natural numbers).

Exponentiation

Where multiplication is repeated addition, exponentiation is the repeated multiplication of one number with itself. When exponentiation is employed, the number is said to be raised to a power. For example, five raised to the second power (or five to the second, or simply five squared) is saying to multiply by five twice, thus 5 \times 5 = 25. Raising a number two the third power is called cubing a number. Thus five cubed is 5 \times 5 \times 5 = 125. This operation is visually represented by a superscript number to the right of the base number, indicating the power of exponentiation. On computer interfaces and places where superscript is not possible or convenient, the convention is to indicate exponentiation by a carat (^) between the two numbers.

When a number is raised to a negative power, the number is raised to that positive power then made into a fraction of one over that number (called the reciprocal). Taking the root of a number is the operation of raising a number to a fractional power. The most common root is the square root. Finding a root is the same as finding the number which raised to the power of the root will give the original number. When an integer is the result of raising another integer to a power, it is said to be perfect with regards to that power (e.g., 64 is a "perfect square" and a "perfect cube" because 8 \times 8 = 64 and 4 \times 4 \times 4 = 64). When a number is not perfect with respect to a given power, the root of that number for that power will be an irrational number, that is, a number which cannot be represented accurately as the quotient of two integers. Thus, as 2 is not a perfect square, the square root of two (written as \sqrt{2}) will be irrational, with its decimal representation beginning 1.414....

Order of operations

The order of operations for arithmetic is as follows:

  1. Perform all exponentiation from left to right (however, multiple exponents on the same base are calculated right [top] to left [bottom])
  2. Perform all multiplication and division from left to right
  3. Perform all addition and subtraction from left to right

Parenthesis or brackets are sometimes employed to signify a deviation from that order. Operations contained within brackets are to be performed before operations outside brackets. Thus, the complete order of operations includes at the beginning:

  • Perform operations within brackets and parenthesis first, with respect to order of operations.

For example, by the order of operations, the problem 3 + 4 \times 5 - 2 equals 21, as the multiplication of four and five is performed first, with three being added to the resulting 20, followed by the subtraction of two from 21. Had the problem been evaluated from left to right without regard for order of operations, the result would have been 3 + 4 = 7, 7 \times 5 = 35, 35 - 2 = 33.

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