Elementary arithmetic

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Short description: Numbers and the basic operations on them
The symbols for each basic elementary operations. Starting from the top-left going clockwise, is addition, division, multiplication, and subtraction.

In arithmetic, the elementary operations are addition, subtraction, multiplication, and division which are operated on fractions and negative numbers, which can be represented on a number line.

Digits

Main page: Numerical digit

Digits are the entire set of symbols used to represent numbers. In a numeral system, each digit represents a unique value, although the meaning of certain symbols used for digits vary between cultures.

In modern usage, the Arabic numerals (0-9) are the most common set of symbols, and the most frequently used form of these digits is the Western style.

A numeral system defines the value of all numbers that contain more than one digit, most often by addition of the value for adjacent digits. The Hindu–Arabic numeral system includes positional notation to determine the value for any numeral. In this type of system, the increase in value for an additional digit includes one or more multiplications with the radix value and the result is added to the value of an adjacent digit. With Arabic numerals, the radix value of ten produces a value of twenty-one (equal to 2×10 + 1) for the numeral "21". An additional multiplication with the radix value occurs for each additional digit, so the numeral "201" represents a value of two-hundred-and-one (equal to 2×100 + 0×10 + 1).

The elementary level of study typically includes understanding the value of individual whole numbers using Arabic numerals with a maximum of seven digits, and performing the four elementary operations using Arabic numerals with a maximum of four digits each.

Addition

Main page: Addition
Addition Table From 0 to 9
+ 0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18

When two numbers are added together, the result is called a sum. The two numbers being added together are called addends.

Adding two natural numbers

We have 2 bags, one bag holding five apples and a second bag holding three apples. With the third empty bag, the apples from bag 1 and bag 2 are put into bag 3. The third bag now holds eight apples. This illustrates that the combination of three apples and five apples is eight apples, or more generally: "three plus five is eight" or "three plus five equals eight" or "eight is the sum of three and five". Numbers are abstract, and the addition of a group of three things to a group of five things will yield a group of eight things. Addition is a regrouping: two sets of objects that were counted separately are put into a single group and counted together: the count of the new group is the "sum" of the separate counts of the two original groups.

This operation of combining is only one of several possible meanings that the mathematical operation of addition can have. Other meanings for addition include:

  • comparing ("Tom has 5 apples. Jane has 3 more apples than Tom. How many apples does Jane have?")
  • joining ("Tom has 5 apples. Jane gives him 3 more apples. How many apples does Tom have now?")
  • measuring ("Tom's desk is 3 feet (0.91 meters) wide. Jane's is also 3 feet (0.91 meters) wide. How wide will the desks be when put together?")
  • separating ("Tom had some apples. He gave 3 to Jane. Now he has 5. How many did he start with?").

Symbolically, addition is represented by the "plus sign": +. So the statement "three plus five equals eight" can be written symbolically as 3 + 5 = 8. The order in which two numbers are added does not matter, so 3 + 5 = 5 + 3 = 8. This is the commutative property of addition.

Some pairs of digits add up to two-digit numbers, with the tens-digit always being a 1. In the addition algorithm, the tens-digit of the sum of a pair of digits is called the "carry digit".

Example

Using the numbers 653 and 274, starting with the ones column, we find that the sum of three and four is seven.

6 5 3
+ 2 7 4
7

Next, the tens-column. The sum of 5 and 7 is 12, which has two digits. The last digit of 12 is written under the tens-column, while the first digit is written above the hundreds-column as a carry digit.

1
6 5 3
+ 2 7 4
2 7

Next, the hundreds-column. The sum of 6 and 2 is 8, but the carry digit is present, which added to eight is equal to nine.

1
6 5 3
+ 2 7 4
9 2 7

There are no other digits to add, so the algorithm is finished, yielding the following equation as a result:

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

Successorship and size

The result from adding a one to any whole number (i.e. the set of all natural numbers and zero) is the successor of that number, and the result from the subtraction of one from any whole number (excluding zero) is the predecessor of that number. For example, the successor of zero is one and the predecessor of eleven is ten. Every natural number has a successor, and all except zero has a predecessor.

The predecessor of the successor of a number is the number itself. For example, five is the successor of four therefore four is the predecessor of five.

If a number is the successor of another number, then the first number is said to be greater than the other number. If a number is greater than another number, and if the other number is greater than a third number, then the first number is also greater than the third number. For example, five is greater than four, and four is greater than three, therefore five is greater than three. But six is greater than five, therefore six is also greater than three, and so on.

If two non-zero natural numbers are added together, then their sum is greater than either one of them. Example: three plus five equals eight, therefore eight is greater than three (8 > 3) and eight is greater than five (8 > 5). The symbol for "greater than" is >.

If a number is greater than another number, then the other is less than (<) the first one. Examples: three is less than eight (3 < 8) and five is less than eight (5 < 8). Given a pair of natural numbers, one and only one of the following cases must be true:

  • the first number is greater than the second one,
  • the first number is equal to the second one,
  • the first number is less than the second one.

Counting

To count a group of objects means to assign a natural number to each one of the objects, as if it were a label for that object. Such that a natural number is never assigned to an object unless its predecessor was already assigned to another object, with the exception that zero is not assigned to any object. The smallest natural number assigned is one, and the largest natural number assigned depends on the size of the group. It is called the count and it is equal to the number of objects in that group. Counting can also be seen as the process of tallying using tally marks.

We have 7 apples, and in order to count them, we must assign the natural number 1 to an apple, and then increment by 1 for each apple we count, until we have no other apples. When the counting is finished, the last value of the count will be the final count. This count is equal to the number of objects in the group.

Often, when counting objects, one does not keep track of what numerical label corresponds to which object: one only keeps track of the subgroup of objects which have already been labelled, so as to be able to identify unlabeled objects. However, if one is counting persons, then one can ask the persons who are being counted to each keep track of the number which the person's self has been assigned. After the count has finished it is possible to ask the group of persons to file up in a line, in order of increasing numerical label. What the persons would do during the process of lining up would be something like this: each pair of persons who are unsure of their positions in the line ask each other what their numbers are: the person whose number is smaller should stand on the left side and the one with the larger number on the right side of the other person. Thus, pairs of persons compare their numbers and their positions, and commute their positions as necessary, and through repetition of such conditional commutations they become ordered.

In higher mathematics, the process of counting can be also likened to the construction of a one-to-one correspondence (a.k.a. bijection) between the elements of a set and the set {1, ..., n} (where n is a natural number). Once such a correspondence is established, the first set is then said to be of size n.

Subtraction

Main page: Subtraction

Subtraction is the mathematical operation which describes a reduced quantity. The result of this operation is the difference between two numbers, the minuend and the subtrahend. As with addition, subtraction can have a number of interpretations, such as:

  • separating ("Tom has 8 apples. He gives away 3 apples. How many does he have left?")
  • finding the difference ("Tom has 8 apples. Jane has 3 fewer apples than Tom. How many does Jane have?")
  • combining ("Tom has 8 apples. Three of the apples are green and the rest are red. How many are red?")
  • and sometimes joining ("Tom had some apples. Jane gave him 3 more apples, so now he has 8 apples. How many did he start with?").

As with addition, there are other possible interpretations, such as motion.

Symbolically, the minus sign ("−") represents the subtraction operation. So, the statement "five minus three equals two" is also written as 5 − 3 = 2. In elementary arithmetic, subtraction uses smaller positive numbers for all values to produce simpler solutions.

Unlike addition, subtraction is not commutative, so the order of numbers in the operation might change the result. Therefore, each number is provided with a different distinguishing name. The first number (5 in the previous example) is formally defined as the minuend and the second number (3 in the previous example) as the subtrahend. The value of the minuend is larger than the value of the subtrahend so that the result is a positive number, but a smaller value of the minuend will result in negative numbers.

There are several methods to accomplish subtraction. The method which in the United States of America is referred to as traditional mathematics teaches elementary school students to subtract using methods suitable for hand calculation.[1] The particular method used varies from country to country, and within a country, different methods are in fashion at different times. 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 and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks prior, apparently the crutches are the invention of William A. Browell, who used them in a study in November 1937.[2] This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

Students in some European countries are taught, and some older Americans employ, a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid the memory) which vary according to country.[3][4]

In the method of borrowing, a subtraction such as 86 − 39 will accomplish the ones-place subtraction of 9 from 6 by borrowing a 10 from 80 and adding it to the 6. The problem is thus transformed into (70 + 16) − 39, effectively. This is indicated by striking through the 8, writing a small 7 above it, and writing a small 1 above the 6. These markings are called crutches. The 9 is then subtracted from 16, leaving 7, and the 30 from the 70, leaving 40, or 47 as the result.

In the additions method, a 10 is borrowed to make the 6 into 16, in preparation for the subtraction of 9, just as in the borrowing method. However, the 10 is not taken by reducing the minuend, rather one augments the subtrahend. Effectively, the problem is transformed into (80 + 16) − (39 + 10). Typically a crutch of a small one is marked just below the subtrahend digit as a reminder. Then the operations proceed: 9 from 16 is 7; and 40 (that is, 30 + 10) from 80 is 40, or 47 as the result.

The additions method seems to be taught in two variations, which differ only in presentation. Continuing the example of 86 − 39, the first variation attempts to subtract 9 from 6, and then 9 from 16, borrowing a 10 by marking near the digit of the subtrahend in the next column. The second variation attempts to find a digit which, when added to 9, gives 6, and recognizing that is not possible, gives 16, and carrying the 10 of the 16 as a one marking near the same digit as in the first method. The markings are the same; it is just a matter of preference as to how one explains its appearance.

As a final caution, the borrowing method can become complicated in cases such as 100 − 87, where a borrow cannot be made immediately, and must be obtained by reaching across several columns. In this case, the minuend is effectively rewritten as 90 + 10, by taking a 100 from the hundreds, making ten 10s from it, and immediately borrowing that down to nine 10s in the tens column and finally placing a 10 in the ones column.

Example

Let us find the difference between the numbers 792 and 308. Starting with the ones-column, the number 2 is smaller than 8, so we must borrow 10 from 90, making it 80. We add this 10 to 2, which changes the problem to 12 - 8, which is 4.

8 12
7 9 2
3 0 8
4

Next is the tens-column. Since we took 10 from 90, it is now 80, which means we must find the difference of 80 and 0, which is just 80.

8 12
7 9 2
3 0 8
8 4

Next is the hundreds-column. The difference of 700 and 300 is 400.

8 12
7 9 2
3 0 8
4 8 4

The algorithm is completed and yields the final result:

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

Multiplication

Main page: Multiplication
Multiplication Table From 0 to 9
× 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
9 0 9 18 27 36 45 54 63 72 81

When two numbers are multiplied together, the result is called a product. The two numbers being multiplied together are called factors, with multiplicand and multiplier also used.

What does it mean to multiply two natural numbers?

Suppose there are five red bags, each one containing three apples. Now grabbing an empty green bag, move all the apples from all five red bags into the green bag. Now the green bag will have fifteen apples.
Thus the product of five and three is fifteen.
This can also be stated as "five times three is fifteen" or "five times three equals fifteen" or "fifteen is the product of five and three". Multiplication can be seen to be a form of repeated addition: the first factor indicates how many times the second factor occurs in repeated addition; the final sum being the product.

Symbolically, multiplication is represented by the multiplication sign: ×. So the statement "five times three equals fifteen" can be written symbolically as

[math]\displaystyle{ 5 \times 3 = 15. }[/math]

In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. 5 ⋅ 3. In some situations, especially in algebra, where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g. xy means x × y. The order in which two numbers are multiplied does not matter, so that, for example, three times four equals four times three. This is the commutative property of multiplication.

To multiply a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the product of the two digits. Most pairs of digits produce two-digit numbers. In the multiplication algorithm the tens-digit of the product of a pair of digits is called the "carry digit".

Example of a multiplication algorithm for a single-digit factor

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

2
7 2 9
× 3
7

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

7 2 9
× 3
8 7

Next, the hundreds-column. 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 the 1.

7 2 9
× 3
2 1 8 7

No digits of the multiplicand have been left unmultiplied, so the algorithm finishes, yielding the following equation as a result:

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

Example of a multiplication algorithm for multiple-digit factors

Let our objective be to find the product of two numbers, 789 and 345.

7 8 9
3 4 5

First part, Starting with the ones-column, the product of 789 and 5 is 3945.

7 8 9
× 3 4 5
3 9 4 5

Then the tens-column. We are using the multiplier 4, which is in the tens-digit. This means that we are using the multiplier 40, not 4. We must add a 0 at the end of the answer because of this. The product of 789 and 40 is 31560.

4
7 8 9
× 3 4 5
3 9 4 5
3 1 5 6 0

Next, the hundreds-column. Since we are using the multiplier 3 and that is in the hundreds-digit, that means it is the multiplier 300, and so 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

Second part, Now we have all of our products. To find the total product of 789 and 345, we must find the sum of all of our 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 answer is

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

Division

Main pages: Division (mathematics) and Long division

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which 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]

since

[math]\displaystyle{ 2 \times 3 = 6 }[/math].

In the above expression, a is called the dividend, b the divisor and c the quotient. Division by zero — where the divisor is zero — is said to be either meaningless or undefined in elementary arithmetic.

Division notation

Division is most often 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 out loud as "a divided by b" or "a over b". A 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 since it can easily be typed as a simple sequence of characters.

A handwritten or typographical variation — which is halfway between these two forms — uses a solidus (fraction slash) but elevates the dividend and lowers the divisor, as follows:

ab

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]

This form is infrequent except in basic arithmetic. The obelus is also used alone to represent the division operation itself, for instance, as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

With a knowledge of multiplication tables, 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 — but which leads to a more holistic understanding of division in general — involves the concept of chunking. By allowing one to subtract more multiples from the partial remainder at each stage, more free-form methods can be developed as well.

Alternatively, if the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones' place as far as desired. If the divisor has a decimal fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

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

Let us find the quotient of 272 and 8. Starting with the hundreds-digit, 2 is not divisible by 8. So we must go to 7, and add 20 to 7, to get 27. In order to divide 27 and 8, we must subtract the dividend by the closest factor of the divisor, that is, a factor of the divisor that cannot be increased without being bigger than the dividend. The closest factor of 27 from the divisor is 24, so the difference between 27 and 24 is 3. 3 should be written under the hundreds-column because 3 is the closest factor of 8 to 27 without being bigger than 27.

2 7 2
÷ 8
3

Using the difference of 27 and 24, 3. We can try to divide 3 by 8, but 8 is bigger than 3, so we must go to the ones-digit, 2. We add 3 to 2, and get 32 which is divisible by 8 and so the quotient of 32 and 8 is 4. 4 is written under the ones-column.

2 7 2
÷ 8
3 4

There is no other digits, and we can check that 34 is truly the answer, by multiplying the quotient with the divisor, 8, to get 272. So thus the algorithm is complete, yielding the result:

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

Educational standards

Local standards usually define the educational methods and content included in the elementary level of instruction. In the United States and Canada, controversial subjects include the amount of calculator usage compared to manual computation and the broader debate between traditional mathematics and reform mathematics.[5]

In the United States, the 1989 NCTM standards led to curricula which de-emphasized or omitted much of what was considered to be elementary arithmetic in elementary school and replaced it with emphasis on topics traditionally studied in college such as algebra, statistics and problem solving, and non-standard computation methods unfamiliar to most adults.

Tools

The abacus is an early mechanical device for performing elementary arithmetic, which is still used in many parts of Asia. Modern calculating tools that perform elementary arithmetic operations include cash registers, electronic calculators, and computers.

Generalizations

Elementary arithmetic usually deals with the real numbers; the set of numbers is given algebraic structure by the two operations (addition and multiplication) and their inverses (subtraction and division). In general, a set of objects together with notions of addition, subtraction, multiplication, and division which behave in expected ways (obeying the associative and distributive properties for example) is called a field (mathematics). In general, fields can look and behave differently from the real numbers, but the basic rules of arithmetic still hold. For example, modular integer arithmetic modulo a prime number is a field. Relaxing the rules of arithmetic allows the creation of numerous other algebraic objects such as division rings and integral domains.

See also

References

External links