Elementary arithmetic

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

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]}

Digits

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, which count with base 20. The Telefol language is known for using base 27, and computers are known for using binary (base 2). ^[5] Other non-positional number systems are occasionally used, such as Roman numerals. ^[6] A disadvantage of non-positional systems is that there is generally a highest expressible number.

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 (${\displaystyle 0+1=1}$ and ${\displaystyle 11-1=10}$). 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. If one number is greater than (${\displaystyle >}$) another number, then the latter is less than (${\displaystyle <}$) the former. For example, if three is less than eight (${\displaystyle 3<8}$), then eight is greater than three (${\displaystyle 8>3}$). The natural numbers are also well-ordered.

Counting

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. Some sets, known as non-enumerable sets, cannot be ordered. For example, while it is possible to enumerate (order) the rational numbers, the real numbers are non-enumerable.

Counting can also be the process of tallying using tally marks, 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, 4 apples and 4 bananas have the same cardinality, as each apple can be matched to each banana with no fruit remaining.

Addition

Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, the addition of 59 + 27 gives the result 86.

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 (${\displaystyle +}$). ^[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.

Subtraction

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 (${\displaystyle -}$). The minus sign is also used to notate negative numbers, and these can be considered to be the number subtracted from 0.

Subtraction is not commutative, which means that the order of the numbers can change the final value. ${\displaystyle 3-5}$ is not the same as ${\displaystyle 5-3}$. In elementary arithmetic, the minuend is always larger than the subtrahend to produce a positive result. However, the absolute values of ${\displaystyle a-b}$ and ${\displaystyle b-a}$ are the same (${\displaystyle |a-b|=|b-a|}$).

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 ${\displaystyle 86-39}$ 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 ${\displaystyle 70+16-39}$. 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 ${\displaystyle (80+16)-(39+10)}$. 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. Using the borrowing method, 10 is borrowed from 90, reducing 90 to 80. Adding this 10 to 2 changes the problem to ${\displaystyle 12-8}$, 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:

${\displaystyle 792-308=484}$

Multiplication

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 "${\displaystyle 5\times 3=15}$", "${\displaystyle 5\ast 3=15}$", "${\displaystyle (5)(3)=15}$", or "${\displaystyle 5\cdot 3=15}$". 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, ${\displaystyle xy}$ represents ${\displaystyle x\times y}$.

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:

${\displaystyle 3\times 729=2187}$

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:

${\displaystyle 789\times 345=272205}$

Division

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 ${\displaystyle c\times b=a}$, then a divided by b equals c.

That is: ${\displaystyle {\frac {a}{b}}=c}$. For instance, ${\displaystyle {\frac {6}{3}}=2}$.

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:

${\displaystyle {\frac {a}{b}}}$

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:

${\displaystyle a/b}$

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:

${\displaystyle a\div b.}$

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:

${\displaystyle \textstyle {5\div {1 \over 2}=5\times {2 \over 1}=5\times 2=10}}$

${\displaystyle \textstyle {{2 \over 3}\div {2 \over 5}={2 \over 3}\times {5 \over 2}={10 \over 6}={5 \over 3}}}$

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:

${\displaystyle 272\div 8=34}$

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:

${\displaystyle 272\div 8=34}$

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. ^{[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.

See also

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

References

1. Mitchelmore, Michael C.; White, Paul (2012). Seel, Norbert M. (ed.). Abstraction in Mathematics Learning. Boston, MA: Springer US. pp. 31–33. doi:10.1007/978-1-4419-1428-6_516. ISBN   978-1-4419-1428-6.
2. Björklund, Camilla; Marton, Ference; Kullberg, Angelika (2021). "What is to be learnt? Critical aspects of elementary arithmetic skills". Educational Studies in Mathematics. 107 (2): 261–284. doi: 10.1007/s10649-021-10045-0 . ISSN   0013-1954.
3. "numeral system | mathematics | Britannica". www.britannica.com. Paragraph 2, sentence 4. Archived from the original on 2023-08-10. Retrieved 2022-11-24.
4. Tillinghast-Raby, Amory. "A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut". Scientific American. Archived from the original on 19 July 2023. Retrieved 24 July 2023.
5. "Computer Language". Encyclopedia.com. August 1, 2016. Retrieved April 11, 2024.
6. ITL Education Solutions Limited (2011). Introduction to Computer Science. Pearson Education India. p. 28. ISBN   978-81-317-6030-7.
7. Madden, Daniel J.; Aubrey, Jason A. (2017). An Introduction to Proof through Real Analysis. John Wiley & Sons. p. 3. ISBN   9781119314721.
8. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. p. 87. ISBN   978-1-118-48700-6.
9. Hall, F. M. (1972). An Introduction to Abstract Algebra. Cambridge University Press. p. 171. ISBN   978-0-521-08484-0.
10. Rosen, Kenneth (2013). Discrete Maths and Its Applications Global Edition. McGraw Hill. ISBN   978-0-07-131501-2. See the Appendix I.
11. "What is addition?". Splashlearn.com. April 11, 2024. Retrieved April 11, 2024.
12. Resnick, L. B.; Ford, W. W. (2012). Psychology of Mathematics for Instruction. Routledge. p. 110. ISBN   978-1-136-55759-0.
13. "Everyday Mathematics4 at Home". Everyday Mathematics Online. Retrieved December 26, 2022.
14. "Subtraction Algorithms - Department of Mathematics at UTSA". mathresearch.utsa.edu. Retrieved 2024-04-01.
15. Ross, Susan. "Subtraction in the United States: An Historical Perspective" (PDF). Microsoft Word - Issue 2 -9/23/. Archived from the original (PDF) on August 11, 2017. Retrieved June 25, 2019.
16. Klapper, Paul (1916). "The Teaching of Arithmetic: A Manual for Teachers. pp. 177" . Retrieved 2016-03-11.
17. Smith, David Eugene (1913). "The Teaching of Arithmetic. pp. 77" . Retrieved 2016-03-11.
18. "Debate about Teaching style of Maths". edmontonjournal.com.
19. Gollom, Mark (April 10, 2016). "Educators debate whether some math basics are 'a dead issue in the year 2016'".

External links

• "A Friendly Gift on the Science of Arithmetic" ^{[ dead link ]} – an Arabic document from the 15th century that talks about basic arithmetic.