Elementary arithmetic

Short description: Numbers and the basic operations on them

The symbols for elementary-level math operations. From top-left going anticlockwise: addition, subtraction, multiplication, and division.

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 commonly the first branch of mathematics taught in elementary school.

Digits

Digits are used to represent the value of numbers in a numeral system. The most commonly used digits are the Arabic numerals (0 to 9 or base-ten). The Hindu-Arabic numeral system is the most commonly used numeral system, being a positional notation system used to represent numbers using these digits.

Successor function and ordering

In elementary arithmetic, the successor of a natural number (including zero) is the result of adding a value of one to that number. The predecessor of a natural number (excluding zero) 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 ([math]\displaystyle{ 0+1=1 }[/math] and [math]\displaystyle{ 11-1=10 }[/math]). Every natural number has a successor, and all natural numbers (except zero) have a predecessor.

The natural numbers have a total-ordering. If one number is greater than ([math]\displaystyle{ \gt }[/math]) another number, then the latter is less than ([math]\displaystyle{ \lt }[/math]) the former. For example, three is less than eight ([math]\displaystyle{ 3\lt 8 }[/math]), and eight is greater than three ([math]\displaystyle{ 8\gt 3 }[/math]). The natural numbers are also well-ordered.

Counting

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.

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

In more advanced mathematics, the process of counting can be thought of as constructing a one-to-one correspondence (or bijection), between the elements of a finite set and the set [math]\displaystyle{ \{1, 2, 3, ..., n\} }[/math], where [math]\displaystyle{ n }[/math] is a natural number, and the size of the set is [math]\displaystyle{ n }[/math].

Informally, two sets have the same cardinality if both of the sets' elements can be matched with one-to-one correspondence. (Example: 4 apples and 4 bananas have the same cardinality as each apple can be matched to each banana without any left-over.)

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 "[math]\displaystyle{ + }[/math]" and is performed according to the following rules:

• The sum of two numbers is equal to the number obtained by adding their individual values.
• The order in which the addends are added does not affect the sum. This is known as the commutative property of addition.
• The sum of two numbers is unique, meaning that there is only one correct answer for the sum of any given numbers.
• Addition has an 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. 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. In elementary arithmetic, students typically learn to add whole numbers, and may also learn about topics such as negative numbers and fractions.

Example

Add the numbers 653 and 274. Starting with the ones column (bolded in the figure below), the sum of three and four is seven.

	Hundreds	Tens	Ones
	6	5	3
+	2	7	4
			7

Moving on to the tens column (bolded below), the sum of 5 tens (50) and 7 tens (70) is 12 tens (120). The tens digit from 120, 2, is written under the tens column, while the hundreds digit (1) is written above the hundreds column as a carry digit.

	Hundreds	Tens	Ones
	1
	6	5	3
+	2	7	4
		2	7

In the hundreds column, the sum of 6 hundreds (600) and 2 hundreds (200) is 8 hundreds (800), but a carry digit of 1 hundred (100) is present and must also be added to the total, resulting in 9 hundreds (900).

	Hundreds	Tens	Ones
	1
	6	5	3
+	2	7	4
	9	2	7

The result:

Combine 900, 20, and 7 from their respective columns to receive a total of 927.

[math]\displaystyle{ 653 + 274 = 927 }[/math]

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 ([math]\displaystyle{ - }[/math]).

Subtraction is not commutative, which means that the order of the numbers can change the final value. [math]\displaystyle{ 3-5 }[/math] is not the same as [math]\displaystyle{ 5-3 }[/math]. In elementary arithmetic, the minuend is always larger than the subtrahend to produce a positive result.

Subtraction is also used to separate, combine (e.g., find the cardinal of a subset of the set we are interested in), 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. Reform mathematics is distinguished generally by the lack of preference for any specific technique, replaced by guiding 2nd-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. However, a method of borrowing had been known and published in prior textbooks. Crutches are the invention of William A. Brownell, who used them in a study in November 1937. In the method of borrowing, a subtraction problem such as [math]\displaystyle{ 86-39 }[/math] 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 [math]\displaystyle{ 70+16-39 }[/math]. 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".

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 making use of crutches, there is no borrowing in this method. There are also crutches that vary according to country. The method of addition involves augmenting the subtrahend, rather than reducing the minuend, as in the borrowing method. This transforms the problem into [math]\displaystyle{ (80+16)-(39+10) }[/math]. A small 1 is marked below the subtrahend digit as a reminder.

Example

Subtracting the numbers 792 and 308, starting with the ones column, 2 is smaller than 8, borrowing 10 from 90, making 90 become 80. Adding this 10 to 2, changes the problem to [math]\displaystyle{ 12-8 }[/math], which is 4.

	Hundreds	Tens	Ones
		8	12
	7	9	2
−	3	0	8
			4

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

	Hundreds	Tens	Ones
		8	12
	7	9	2
−	3	0	8
		8	4

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

	Hundreds	Tens	Ones
		8	12
	7	9	2
−	3	0	8
	4	8	4

The result:

[math]\displaystyle{ 792 - 308 = 484 }[/math]

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 "[math]\displaystyle{ 5 \times 3 = 15 }[/math]", "[math]\displaystyle{ 5 \ast 3 = 15 }[/math]", "[math]\displaystyle{ (5)(3) = 15 }[/math]", or "[math]\displaystyle{ 5 \cdot 3 = 15 }[/math]". The asterisk notation is most commonly used in computer programming languages. In algebra, the multiplication symbol may be omitted; for example, [math]\displaystyle{ xy }[/math] represents [math]\displaystyle{ x \times y }[/math].

The order in which two numbers are multiplied does not affect the result. This is known as the commutative property of multiplication. The grouping of three or more numbers in parentheses also does not affect the result. This is known as the associative property of multiplication.

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.

Example of multiplication for a single-digit factor

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:

[math]\displaystyle{ 3 \times 729 = 2187 }[/math]

Example of multiplication for multiple-digit factors

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:

[math]\displaystyle{ 789 \times 345 = 272205 }[/math]

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 [math]\displaystyle{ c \times b = a }[/math], then a divided by b equals c.

That is: [math]\displaystyle{ \frac ab = c }[/math]. For instance, [math]\displaystyle{ \frac 63 = 2 }[/math].

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:

[math]\displaystyle{ \frac ab }[/math]

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:

[math]\displaystyle{ a/b }[/math]

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 (or division sign) in this manner:

[math]\displaystyle{ a \div b. }[/math]

In some non-English-speaking cultures, "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:

[math]\displaystyle{ \textstyle{5 \div {1 \over 2} = 5 \times {2 \over 1} = 5 \times 2 = 10} }[/math]

[math]\displaystyle{ \textstyle{{2 \over 3} \div {2 \over 5} = {2 \over 3} \times {5 \over 2} = {10 \over 6} = {5 \over 3}} }[/math]

Example

Dividing 272 and 8, starting with the hundreds digit, 2 is not divisible by 8, adding 20 to 7 gets 27. The largest number that the divisor 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 (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:

[math]\displaystyle{ 272 \div 8 = 34 }[/math]

Bus stop method

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)272
     0        ( 8 ×  0 =  0)
     27       ( 2 -  0 =  2)
     24       ( 8 ×  3 = 24)
      32      (27 - 24 =  3)
      32      ( 8 ×  4 = 32)
       0      (32 - 32 =  0)

Conclusion:

[math]\displaystyle{ 272\div8=34 }[/math]

Educational standards

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. One issue has been the use of calculators versus manual computation, with some arguing that calculator use should be limited to promote mental arithmetic skills. 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 (NCTM) 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.

See also

• Early numeracy
• Elementary mathematics
• Chunking (division)
• Plus and minus signs
• Peano axioms
• Division by zero
• Real number
• Imaginary number

External links

• "A Friendly Gift on the Science of Arithmetic" – an Arabic document from the 15th century that talks about basic arithmetic.