Elementary arithmetic

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The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction. Arithmetic symbols.svg
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 addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first branch of mathematics taught in schools. [1] [2]

Contents

Numeral systems

In numeral systems, digits are characters used to represent the value of numbers. An example of a numeral system is the predominantly used Indo-Arabic numeral system (0 to 9), which uses a decimal positional notation. [3] Other numeral systems include the Kaktovik system (often used in the Eskimo-Aleut languages of Alaska, Canada, and Greenland), and is a vigesimal positional notation system. [4] Regardless of the numeral system used, the results of arithmetic operations are unaffected.

Successor function and ordering

In elementary arithmetic, the successor of a natural number (including zero) is the next natural number and is the result of adding one to that number. The predecessor of a natural number (excluding zero) is the previous natural number and is the result of subtracting 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 0 has a predecessor. [5]

The natural numbers have a total ordering. 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.

Counting

Counting assigns a natural number to each object in a set, starting with 1 for the first object and increasing by 1 for each subsequent object. The number of objects in the set is the count. This is also known as the cardinality of the set.

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

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, 59 + 27 = 86. Addition with carry.png
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, 59 + 27 = 86.

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 with the plus sign (). [6] It is performed according to these rules:

When the sum of a pair of digits results in a two-digit number, the "tens" digit is referred to as the "carry digit". [9] In elementary arithmetic, students typically learn to add whole numbers and may also learn about topics such as negative numbers and fractions.

Subtraction

Subtraction evaluates 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.

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.

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.

There are several methods to accomplish subtraction. The traditional mathematics method subtracts using methods suitable for hand calculation. [10] Reform mathematics is distinguished generally by the lack of preference for any specific technique, replaced by guiding students to invent their own methods of computation.

American schools teach a method of subtraction using borrowing. [11] A subtraction problem such as is solved by borrowing a 10 from the tens place to add to the ones place in order to facilitate the subtraction. 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. [12]

The Austrian method, also known as the additions method, is taught in certain European countries[ which? ]. In contrast to the previous method, no borrowing is used, although there are crutches that vary according to certain countries. [13] [14] The method of addition involves augmenting the subtrahend. This transforms the previous problem into . 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. This changes the problem to .

HundredsTensOnes
812
792
308
4

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

HundredsTensOnes
812
792
308
84

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

HundredsTensOnes
812
792
308
484

The result:

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 multiplicands, multipliers, or factors. Multiplication 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 (⋅). The statement "five times three equals fifteen" can be written as "", "", "", or "".

In elementary arithmetic, multiplication satisfies the following properties [lower-alpha 1] :

In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit".

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.

HundredsTensOnes
2
729
×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.

HundredsTensOnes
2
729
×3
87

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.

HundredsTensOnes
2
729
×3
2187

The result:

Example of multiplication for multiple-digit factors

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

789
×345
3945

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

789
×345
3945
31560

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

789
×345
3945
31560
236700

Adding all the products,

789
×345
3945
31560
+236700
272205

The result:

Division

Division is an arithmetic operation, and the inverse of multiplication. Given that , ,

Division can be written as , , or ab. This can be read verbally as "a divided by b" or "a over b".

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

In an equation , a is the dividend, b the divisor, and c the quotient. Division by zero is considered impossible at an elementary arithmetic level.

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

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

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 it is written under the tens column. Subtracting 24 (the product of 3 and 8) from 27 gives 3 as the remainder.

272
÷8
3

Going to the ones digit, the number is 2. Adding 30 (the remainder, 3, times 10) and 2 gets 32. The quotient of 32 and 8 is 4, which is written under the ones column.

272
÷8
34

The result:

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         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)

The result:

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. [15] [16]

See also

Notes

  1. While elementary arithmetic mainly operates under the set of natural numbers (sometimes including 0), multiplication under other number sets can satisfy more or less properties than those listed here, such as having an inverse element in the rational numbers and beyond, or lacking commutativity in the quaternions and higher order number sets.

Related Research Articles

<span class="mw-page-title-main">Arithmetic</span> Branch of elementary mathematics

Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.

<span class="mw-page-title-main">Multiplication</span> Arithmetical operation

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

<span class="mw-page-title-main">Division (mathematics)</span> Arithmetic operation

Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient.

<span class="mw-page-title-main">Multiplication table</span> Mathematical table

In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.

A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the decimal numeral system.

<span class="mw-page-title-main">Addition</span> Arithmetic operation

Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5".

Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ1 + φ0 = φ2. For instance, 11φ = 100φ.

<span class="mw-page-title-main">Subtraction</span> One of the four basic arithmetic operations

Subtraction is one of the four arithmetic operations along with addition, multiplication and division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are 5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the difference of 5 and 2 is 3; that is, 5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.

Brahmagupta was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text.

Two's complement is the most common method of representing signed integers on computers, and more generally, fixed point binary values. Two's complement uses the binary digit with the greatest value as the sign to indicate whether the binary number is positive or negative; when the most significant bit is 1 the number is signed as negative and when the most significant bit is 0 the number is signed as positive. As a result, non-negative numbers are represented as themselves: 6 is 0110, zero is 0000, and -6 is 1010. Note that while the number of binary bits is fixed throughout a computation it is otherwise arbitrary.

In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps.

<span class="mw-page-title-main">Method of complements</span> Method of subtraction

In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement. Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement is also valuable in number theory, such as in Midy's theorem.

<span class="mw-page-title-main">Mental calculation</span> Arithmetical calculations using only the human brain

Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies or devices such as a calculator. People may use mental calculation when computing tools are not available, when it is faster than other means of calculation, or even in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems. People with unusually high ability to perform mental calculations are called mental calculators or lightning calculators.

The Trachtenberg system is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp.

Location arithmetic is the additive (non-positional) binary numeral systems, which John Napier explored as a computation technique in his treatise Rabdology (1617), both symbolically and on a chessboard-like grid.

In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of the standard algorithm to add numbers together by starting with the rightmost digits and working to the left. For example, when 6 and 7 are added to make 13, the "3" is written to the same column and the "1" is carried to the left. When used in subtraction the operation is called a borrow.

In elementary arithmetic, a standard algorithm or method is a specific method of computation which is conventionally taught for solving particular mathematical problems. These methods vary somewhat by nation and time, but generally include exchanging, regrouping, long division, and long multiplication using a standard notation, and standard formulas for average, area, and volume. Similar methods also exist for procedures such as square root and even more sophisticated functions, but have fallen out of the general mathematics curriculum in favor of calculators. As to standard algorithms in elementary mathematics, Fischer et al. (2019) state that advanced students use standard algorithms more effectively than peers who use these algorithms unreasoningly. That said, standard algorithms, such as addition, subtraction, as well as those mentioned above, represent central components of elementary math.

<span class="mw-page-title-main">Karatsuba algorithm</span> Algorithm for integer multiplication

The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a divide-and-conquer algorithm that reduces the multiplication of two n-digit numbers to three multiplications of n/2-digit numbers and, by repeating this reduction, to at most single-digit multiplications. It is therefore asymptotically faster than the traditional algorithm, which performs single-digit products.

<span class="mw-page-title-main">Rod calculus</span>

Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.

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