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**Elementary arithmetic** is a branch of mathematics involving basic numerical operations, namely addition, subtraction, multiplication, and division. Due to the low level of abstraction, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally known as the first branch of mathematics that is taught in schools.^{[1]}^{[2]}

Main article: Numerical digit |

Digits are used to represent the value of numbers in a numeral system. The most commonly used digits^{[3]} are the Arabic numerals 0 through 9 and count with Decimal (base 10). The Hindu-Arabic numeral system is the most commonly used numeral system, being a positional notation system used to represent numbers using these digits.^{[4]} However, other systems are used, such as Kaktovik numerals, commonly used in the Eskimo-Aleut languages of Alaska, Canada, and Greenland (base 20), binary used by computers (base 2),^{[5]} and the Telefol language (base 27). Other non-positional number systems are occasionally used, such as the Roman numerals.^{[6]} A disadvantage of non-positional systems is that there is generally a highest expressible number.

In elementary arithmetic, the *successor* of a natural number (including zero) is the next natural number and is the result of adding a value of one to that number. The *predecessor* of a natural number (excluding zero) is the previous natural number and is the result obtained by subtracting a value of one from that number. For example, the successor of zero is one, and the predecessor of eleven is ten (** and ****). Every natural number has a successor, and every natural number except the first (zero or 1) has a predecessor.**^{[7]}

The natural numbers have a total ordering, meaning that the value of any two natural numbers can be compared to each other.^{[further explanation needed]} If one number is greater than () another number, then the latter is less than () the former. For example, three is less than eight (), thus eight is greater than three (). The natural numbers are also well-ordered, meaning that any subset of the natural numbers has a least element.

Main article: Counting § Counting in mathematics |

Counting involves assigning a natural number to each object in a set, starting with one for the first object and increasing by one for each subsequent object. The number of objects in the set is the count, which is equal to the highest natural number assigned to an object in the set. This count is also known as the cardinality of the set containing such objects.

Counting can also be the process of tallying, the process of drawing a mark for each object in a set.

Informally, two sets have the same cardinality if both of the sets' elements can be matched with one-to-one correspondence. As an example, a set of 4 apples and another of 4 bananas have the same cardinality, because each apple can be assigned a banana with no fruit remaining.

Main article: Addition |

Addition is a mathematical operation that combines two or more numbers, called addends or summands, to produce a combined number, called the sum. The addition of two numbers is expressed using the plus sign ().^{[8]} It is performed according to the following rules:

- The sum of two numbers is equal to the number obtained by adding their individual values.
^{[9]} - The order in which the addends are added does not affect the sum. This is known as the commutative property of addition. For example, (a + b) and (b + a) will produce the same output.
^{[10]}^{[9]} - The sum of two numbers is unique, meaning that there is only one correct answer for the sum of any given numbers.
^{[9]} - Addition's inverse operation, called subtraction, which can be used to find the difference between two or more numbers.

Addition is used in a variety of contexts, including comparing quantities, joining quantities, and measuring.^{[11]} When the sum of a pair of digits results in a two-digit number, the "tens" digit is referred to as the "carry digit" in the addition algorithm.^{[12]} In elementary arithmetic, students typically learn to add whole numbers and may also learn about topics such as negative numbers and fractions.

Main article: Subtraction |

Subtraction is used to evaluate the difference between two numbers, where the minuend is the number being subtracted from, and the subtrahend is the number being subtracted. It is represented using the minus sign (). The minus sign is also used to notate negative numbers, and these can be considered to be the numbers subtracted from 0.

Subtraction is not commutative, which means that the order of the numbers can change the final value; is not the same as . In elementary arithmetic, the minuend is always larger than the subtrahend to produce a positive result. However, the absolute values of and are the same ().

Subtraction is also used to separate, combine (e.g., find the size of a subset of a specific set), and find quantities in other contexts. For example, "Tom has 8 apples. He gives away 3 apples. How many is he left with?" represents separation, while "Tom has 8 apples. Three of the apples are green, and the rest are red. How many are red?" represents combination. In some cases, subtraction can also be used to find the total number of objects in a group, as in "Tom had some apples. Jane gave him 3 more apples, so now he has 8 apples. How many did he start with?"

There are several methods to accomplish subtraction. The traditional mathematics method teaches elementary school students to subtract using methods suitable for hand calculation.^{[13]} Reform mathematics is distinguished generally by the lack of preference for any specific technique, replaced by guiding 2^{nd}-grade students to invent their own methods of computation, such as using properties of negative numbers in the case of TERC.

American schools currently teach a method of subtraction using borrowing,^{[14]} which had been known and published in textbooks prior to the method's wider adoption in American curricula. In the method of borrowing, a subtraction problem such as can be solved by borrowing a 10 from the tens place to add to the ones place in order to facilitate the subtraction. For example, subtracting 9 from 6 involves borrowing a 10 from the tens place, making the problem into . This is indicated by crossing out the 8, writing a 7 above it, and writing a 1 above the 6. These markings are called "crutches", which were invented by William A. Brownell, who used them in a study in November 1937.^{[15]}

The Austrian method, also known as the additions method, is taught in certain European countries and employed by some American people from previous generations. In contrast to the previous method, no borrowing is used, although there are crutches that vary according to certain countries.^{[16]}^{[17]} The method of addition involves augmenting the subtrahend, rather than reducing the minuend, as in the borrowing method. This transforms the previous problem into . A small 1 is marked below the subtrahend digit as a reminder.

Subtracting the numbers 792 and 308, starting with the ones column, 2 is smaller than 8. Using the borrowing method, 10 is borrowed from 90, reducing 90 to 80. Adding this 10 to 2 changes the problem to , which is 4.

Hundreds | Tens | Ones | |

8 |
^{1}2 | ||

7 | |||

− | 3 | 0 | 8 |

4 |

In the tens column, the difference between 80 and 0 is 80.

Hundreds | Tens |
Ones | |

8 |
^{1}2
| ||

7 | |||

− | 3 | 0 | 8 |

8 | 4 |

In the hundreds column, the difference between 700 and 300 is 400.

Hundreds |
Tens | Ones | |

8 |
^{1}2
| ||

7 | |||

− | 3 | 0 | 8 |

4 | 8 | 4 |

The result:

Main article: Multiplication |

Multiplication is a mathematical operation of repeated addition. When two numbers are multiplied, the resulting value is a product. The numbers being multiplied are called multiplicands and multipliers and are altogether known as factors. For example, if there are five bags, each containing three apples, and the apples from all five bags are placed into an empty bag, the empty bag will contain 15 apples. This can be expressed as "five times three equals fifteen", "five times three is fifteen" or "fifteen is the product of five and three".

Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅). Therefore, the statement "five times three equals fifteen" can be written as "", "", "", or "". The multiplication sign is the most used symbol for multiplication^{[citation needed]}, while the asterisk notation is most commonly used in computer programming languages. In algebra, the multiplication symbol may be omitted; for example, represents .

In elementary arithmetic, multiplication satisfies the following properties^{[a]}:

- Commutativity. Switching the order in a product does not change the result: .
- Associativity. Rearranging the order of parentheses in a product does not change the result: .
- Distributivity. Multiplication
*distributes*over addition: . - Identity. Any number multiplied by 1 is itself,
*i.e.*1 is the**multiplicative identity**: . - Zero. Any number multiplied by 0 is 0,
*i.e.*0 is the**zero**or**absorbing element**: .

In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit". To multiply a pair of digits using a table, one must locate the intersection of the row of the first digit and the column of the second digit, which will contain the product of the two digits. Most pairs of digits, when multiplied, result in two-digit numbers.

Multiplying 729 and 3, starting on the ones column, the product of 9 and 3 is 27. 7 is written under the ones column and 2 is written above the tens column as a carry digit.

Hundreds | Tens | Ones | |

2 |
|||

7 | 2 | 9 | |

× | 3 | ||

7 |

The product of 2 and 3 is 6, and the carry digit adds 2 to 6, so 8 is written under the tens column.

Hundreds | Tens |
Ones | |

7 | 2 | 9 | |

× | 3 | ||

8 | 7 |

The product of 7 and 3 is 21, and since this is the last digit, 2 will not be written as a carry digit, but instead beside 1.

Hundreds |
Tens | Ones | |

7 | 2 | 9 | |

× | 3 | ||

2 | 1 | 8 | 7 |

The result:

Multiplying 789 and 345, starting with the ones column, the product of 789 and 5 is 3945.

7 | 8 | 9 | |

× | 3 | 4 | 5 |

3 | 9 | 4 | 5 |

4 is in the tens digit. The multiplier is 40, not 4. The product of 789 and 40 is 31560.

7 | 8 | 9 | ||

× | 3 | 4 | 5 | |

3 | 9 | 4 | 5 | |

3 | 1 | 5 | 6 | 0 |

3 is in the hundreds digit. The multiplier is 300. The product of 789 and 300 is 236700.

7 | 8 | 9 | |||

× | 3 | 4 | 5 | ||

3 | 9 | 4 | 5 | ||

3 | 1 | 5 | 6 | 0 | |

2 | 3 | 6 | 7 | 0 | 0 |

Adding all the products,

7 | 8 | 9 | ||||

× | 3 | 4 | 5 | |||

3 | 9 | 4 | 5 | |||

3 | 1 | 5 | 6 | 0 | ||

+ | 2 | 3 | 6 | 7 | 0 | 0 |

2 | 7 | 2 | 2 | 0 | 5 |

The result:

Main articles: Division (mathematics) and Long division |

Division is an arithmetic operation that is the inverse of multiplication.

Specifically, given a number *a* and a non-zero number *b*, if another number *c* times *b* equals *a*, that is , then *a* divided by *b* equals *c*.

That is: . For instance, .

The number *a* is called the dividend, *b* the divisor, and *c* the quotient. Division by zero is considered impossible at an elementary arithmetic level, and is generally disregarded.

Division can be shown by placing the *dividend* over the *divisor* with a horizontal line, also called a vinculum, between them. For example, *a* divided by *b* is written as:

This can be read verbally as "*a* divided by *b*" or "*a* over *b*".

Another way to express division all on one line is to write the *dividend*, then a slash, then the *divisor*, as follows:

This is the usual way to specify division in most computer programming languages.

A handwritten or typographical variation uses a solidus (fraction slash) but elevates the dividend and lowers the divisor:

*a*⁄*b*

Any of these forms can be used to display a fraction. A *common fraction* is a division expression where both dividend and divisor are numbers (although typically called the *numerator* and *denominator*), and there is no implication that the division needs to be evaluated further.

A more basic way to show division is to use the obelus (÷) in this manner:

In some non-English-speaking cultures^{[which?]}, "*a* divided by *b*" is written *a* : *b*. However, in English usage, the colon is restricted to the concept of ratios ("*a* is to *b*").

Two numbers can be divided on paper using the method of long division. An abbreviated version of long division, short division, can be used for smaller divisors as well.

A less systematic method involves the concept of chunking, involving subtracting more multiples from the partial remainder at each stage.

To divide by a fraction, one can simply multiply by the reciprocal (reversing the position of the top and bottom parts) of that fraction. For example:

Dividing 272 and 8, starting with the hundreds digit, 2 is not divisible by 8. Add 20 and 7 to get 27. The largest number that the divisor of 8 can be multiplied by without exceeding 27 is 3, so the digit 3 is written under the tens column to start constructing the quotient. Subtracting 24 (the product of 3 and 8) from 27 gives 3 as remainder.

2 | 7 | 2 | |

÷ | 8 | ||

3 |

8 is necessarily bigger than the remainder 3. Going to the ones digit to continue the division, the number is 2. Adding 30 and 2 gets 32, which is divisible by 8, and the quotient of 32 and 8 is 4. 4 is written under the ones column.

2 | 7 | 2 | |

÷ | 8 | ||

3 | 4 |

The result:

Another method of dividing taught in some schools is the bus stop method, sometimes notated as

result(divisor) dividend

The steps here are shown below, using the same example as above:

034(Explanations) 8|2720( 8 × 0 = 0) 27 ( 2 - 0 = 2)24( 8 × 3 = 24) 32 (27 - 24 = 3)32( 8 × 4 = 32) 0 (32 - 32 = 0)

Conclusion:

Elementary arithmetic is typically taught at the primary or secondary school levels and is governed by local educational standards. In the United States and Canada, there has been debate about the content and methods used to teach elementary arithmetic.^{[18]}^{[19]} One issue has been the use of calculators versus manual computation, with some arguing that, to promote mental arithmetic skills, calculator usage should be limited. Another debate has centered on the distinction between traditional and reform mathematics, with traditional methods often focusing more on basic computation skills and reform methods placing a greater emphasis on higher-level mathematical concepts such as algebra, statistics, and problem-solving.

In the United States, the 1989 National Council of Teachers of Mathematics standards led to a shift in elementary school curricula that de-emphasized or omitted certain topics traditionally considered to be part of elementary arithmetic, in favor of a greater focus on college-level concepts such as algebra and statistics. This shift has been controversial, with some arguing that it has resulted in a lack of emphasis on basic computation skills that are important for success in later math classes.