Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. Most people learn elementary arithmetic in elementary school.
Elementary arithmetic starts with the natural numbers and the Arabic numerals used to represent them. It requires the memorization of addition tables and multiplication tables for adding and multiplying pairs of digits. Knowing these tables, a person can perform certain wellknown procedures for adding and multiplying natural numbers. Other algorithms are used for subtraction and division.
Though starting out with natural numbers, elementary arithmetic then moves on to fractions, then decimals and negative numbers, which can be represented on a number line.
Nowadays people routinely use electronic calculators, cash registers, and computers to perform their elementary arithmetic for them. Before that, people used slide rules, tables of logarithms, nomographs, mechanical calculators, or they employed calculating prodigies. Otherwise, they just performed calculations by hand on paper, following the wellknown rules of arithmetic.
In ancient times, the instrument to perform arithmetical calculations was the abacus. In the 14th century Arabic numerals were introduced to Europe by Leonardo Pisano. These numerals were more efficient for performing calculations that Roman numerals, because of the positional system, which also made multiplication by hand more efficient than the use of the abacus.
The digits
0 , zero, represents absence of objects to be counted.
1 , one. This is one stick: I
2 , two. This is two sticks: I I
3 , three. This is three sticks: I I I
4 , four. This is four sticks: I I I I
5 , five. This is five sticks: I I I I I
6 , six. This is six sticks: I I I I I I
7 , seven. This is seven sticks: I I I I I I I
8 , eight. This is eight sticks: I I I I I I I I
9 , nine. This is nine sticks: I I I I I I I I I
There are as many digits as fingers on the hands: the word "digit" can also mean finger. But if counting the digits on both hands, the first digit would be one and the last digit would not be counted as "zero" but as "ten": 10 , made up of the digits one and zero. The number 10 is the first twodigit number.
If a number has more than one digit, then the rightmost digit, said to be the last digit, is called the "onesdigit". The digit immediately to its left is the "tensdigit". The digit immediately to the left of the tensdigit is the "hundredsdigit". The digit immediately to the left of the hundredsdigit is the "thousandsdigit".
Addition table
+ 
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 
What does it mean to add two natural numbers? Suppose you have two bags, one bag holding five apples and a second bag holding three apples. Grabbing a third, empty bag, move all the apples from the first and second bags into the third bag. The third bag now holds eight apples. This illustrates 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 which 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.
Symbolically, addition is represented by the "plus sign": . So the statement "three plus five equals eight" can be written symbolically as . The order in which two numbers are added does not matter, so . This is the commutative property of addition.
To add 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 sum of the two digits. Some pairs of digits add up to twodigit numbers, with the tensdigit always being a 1. In the addition algorithm the tensdigit of the sum of a pair of digits is called the "carry digit".
Addition algorithm
For simplicity, consider only numbers with three digits or less. Two add a pair of numbers (written in Arabic numerals), write the second number under the first one, so that digits line up in columns: the rightmost column will contain the onesdigit of the second number under the onesdigit of the first number. This rightmost column is the onescolumn. The column immediately to its left is the tenscolumn. The tenscolumn will have the tensdigit of the second number (if it has one) under the tensdigit of the first number (if it has one). The column immediately to the left of the tenscolumn is the hundredscolumn. The hundredscolumn will line up the hundredsdigit of the second number (if there is one) under the hundredsdigit of the first number (if there is one).
After the second number has been written down under the first one so that digits line up in their correct columns, draw a line under the second (bottom) number. Start with the onescolumn: the onescolumn should contain a pair of digits: the onesdigit of the first number and, under it, the onesdigit of the second number. Find the sum of these two digits: write this sum under the line and in the onescolumn. If the sum has two digits, then write down only the onesdigit of the sum. Write the "carry digit" above the top digit of the next column: in this case the next column is the tenscolumn, so write a 1 above the tensdigit of the first number.
If both first and second number each have only one digit then their sum is given in the addition table, and the addition algorithm is unnecessary.
Then comes the tenscolumn. The tenscolumn might contain two digits: the tensdigit of the first number and the tensdigit of the second number. If one of the numbers has a missing tensdigit then the tensdigit for this number can be considered to be a zero. Add the tensdigits of the two numbers. Then, if there is a carry digit, add it to this sum. If the sum was 18 then adding the carry digit to it will yield 19. If the sum of the tensdigits (plus carry digit, if there is one) is less than ten then write it in the tenscolumn under the line. If the sum has two digits then write its last digit in the tenscolumn under the line, and carry its first digit (which should be a one) over to the next column: in this case the hundreds column.
If none of the two numbers has a hundredsdigit then if there is no carry digit then the addition algorithm has finished. If there is a carry digit (carried over from the tenscolumn) then write it in the hundredscolumn under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the sum of the two numbers.
If at least one of the numbers has a hundredsdigit then if one of the numbers has a missing hundredsdigit then write a zero digit in its place. Add the two hundredsdigits, and to their sum add the carry digit if there is one. Then write the sum of the hundredscolumn under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundredscolumn and write the carry digit to its left: on the thousandscolumn.
Example
Say one wants to find the sum of the numbers 653 and 274. Write the second number under the first one, with digits aligned in columns, like so:
Then draw a line under the second number and start with the onescolumn. The onesdigit of the first number is 3 and of the second number is 4. The sum of three and four is seven, so write a seven in the onescolumn under the line:
Next, the tenscolumn. The tensdigit of the first number is 5, and the tensdigit of the second number is 7, and five plus seven is twelve: 12, which has two digits, so write its last digit, 2, in the tenscolumn under the line, and write the carry digit on the hundredscolumn above the first number:
Next, the hundredscolumn. The hundredsdigit of the first number is 6, while the hundredsdigit of the second number is 2. The sum of six and two is eight, but there is a carry digit, which added to eight is equal to nine. Write the nine under the line in the hundredscolumn:
No digits (and no columns) have been left unadded, so the algorithm finishes, and
 653 + 274 = 927.
Successorship and size
The result of the addition of one to a number is the successor of that number. Examples:
the successor of zero is one,
the successor of one is two,
the successor of two is three,
the successor of ten is eleven.
Every natural number has a successor.
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. Every natural number except zero has a predecessor.
If a number is the successor of another number, then the first number is said to be larger than the other number. If a number is larger than another number, and if the other number is larger than a third number, then the first number is also larger than the third number. Example: five is larger than four, and four is larger than three, therefore five is larger than three. But six is larger than five, therefore six is also larger than three. But seven is larger than six, therefore seven is also larger than three... therefore eight is larger than three... therefore nine is larger than three, etc.
If two nonzero natural numbers are added together, then their sum is larger than either one of them. Example: three plus five equals eight, therefore eight is larger than three (8>3) and eight is larger than five (8>5). The symbol for "larger than" is >.
If a number is larger than another one, then the other is smaller than the first one. Examples: three is smaller than eight (3<8) and five is smaller than eight (5<8). The symbol for smaller than is <. A number cannot be at the same time larger and smaller than another number. Neither can a number be at the same time larger than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true:
 the first number is larger than the second one,
 the first number is equal to the second one,
 the first number is smaller 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 to be 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.
The process of counting a group is the following:
Step 1: Let "the count" be equal to zero. "The count" is a variable quantity, which though beginning with a value of zero, will soon have its value changed several times.
Step 2: Find at least one object in the group which has not been labeled with a natural number. If no such object can be found (if they have all been labeled) then the counting is finished. Otherwise choose one of the unlabeled objects.
Step 3: Increase the count by one. That is, replace the value of the count by its successor.
Step 4: Assign the new value of the count, as a label, to the unlabeled object chosen in Step 2.
Step 5: Go back to Step 2.
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 labeled, so as to be able to identify unlabeled objects necessary for Step 2. 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.
Multiplication table
x 
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.
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 should be added onto itself; 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
In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. . In some situations, especially in algebra, where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g xy means . 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 twodigit numbers. In the multiplication algorithm the tensdigit of the product of a pair of digits is called the "carry digit".
Multiplication algorithm for a singledigit factor
Consider a multiplication where one of the factors has only one digit, whereas the other factor has an arbitrary quantity of digits. Write down the multidigit factor, then write the singledigit factor under the last digit of the multidigit factor. Draw a horizontal line under the singledigit factor. Henceforth, the singledigit factor will be called the "multiplier" and the multidigit factor will be called the "multiplicand".
Suppose for simplicity that the multiplicand has three digits. The first digit is the hundredsdigit, the middle digit is the tensdigit, and the last, rightmost, digit is the onesdigit. The multiplier only has a onesdigit. The onesdigits of the multiplicand and multiplier form a column: the onescolumn.
Start with the onescolumn: the onescolumn should contain a pair of digits: the onesdigit of the multiplicand and, under it, the onesdigit of the multiplier. Find the product of these two digits: write this product under the line and in the onescolumn. If the product has two digits, then write down only the onesdigit of the product. Write the "carry digit" as a superscript of the yetunwritten digit in the next column and under the line: in this case the next column is the tenscolumn, so write the carry digit as the superscript of the yetunwritten tensdigit of the product (under the line).
If both first and second number each have only one digit then their product is given in the multiplication table, and the multiplication algorithm is unnecessary.
Then comes the tenscolumn. The tenscolumn so far contains only one digit: the tensdigit of the multiplicand (though it might contain a carry digit under the line). Find the product of the multiplier and the tensdigits of the multiplicand. Then, if there is a carry digit (superscripted, under the line and in the tenscolumn), add it to this product. If the resulting sum is less than ten then write it in the tenscolumn under the line. If the sum has two digits then write its last digit in the tenscolumn under the line, and carry its first digit over to the next column: in this case the hundreds column.
If the multiplicand does not have a hundredsdigit then if there is no carry digit then the multiplication algorithm has finished. If there is a carry digit (carried over from the tenscolumn) then write it in the hundredscolumn under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the product of the two numbers.
If the multiplicand has a hundredsdigit... find the product of the multiplier and the hundredsdigit of the multiplicand, and to this product add the carry digit if there is one. Then write the resulting sum of the hundredscolumn under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundredscolumn and write the carry digit to its left: on the thousandscolumn.
Example
Say one wants to find the product of the numbers 3 and 729. Write the singledigit multiplier under the multidigit multiplicand, with the multiplier under the onesdigit of the multiplicand, like so:
Then draw a line under the multiplier and start with the onescolumn. The onesdigit of the multiplicand is 9 and the multiplier is 3. The product of three and nine is 27, so write a seven in the onescolumn under the line, and write the carrydigit 2 as a superscript of the yetunwritten tensdigit of the product under the line:
Next, the tenscolumn. The tensdigit of the multiplicand is 2, the multiplier is 3, and three times two is six. Add the carrydigit, 2, to the product 6 to obtain 8. Eight has only one digit: no carrydigit, so write in the tenscolumn under the line:
Next, the hundredscolumn. The hundredsdigit of the multiplicand is 7, while the multiplier is 3. The product of three and seven is 21, and there is no previous carrydigit (carried over from the tenscolumn). The product 21 has two digits: write its last digit in the hundredscolumn under the line, then carry its first digit over to the thousandscolumn. Since the multiplicand has no thousandsdigit, then write this carrydigit in the thousandscolumn under the line (not superscripted):

7 
2 
9 
_ 
_ 
_ 
3 
2 
1 
8^{2} 
7 
No digits of the multiplicand have been left unmultiplied, so the algorithm finishes, and
.
Multiplication algorithm for multidigit factors
Given a pair of factors, each one having two or more digits, write both factors down, one under the other one, so that digits line up in columns.
For simplicity consider a pair of threedigits numbers. Write the last digit of the second number under the last digit of the first number, forming the onescolumn. Immediately to the left of the onescolumn will be the hundredscolumn: the top of this column will have the second digit of the first number, and below it will be the second digit of the second number. Immediately to the left of the hundredscolumn will be the hundredscolumn: the top of this column will have the first digit of the first number and below it will be the first digit of the second number. After having written down both factors, draw a line under the second factor.
The multiplication will consist of two parts. The first part will consist of several multiplications involving onedigit multipliers. The operation of each one of such multiplications was already described in the previous multiplication algorithm, so this algorithm will not describe each one individually, but will only describe how the several multiplications with onedigit multipliers shall be coördinated. The second part will add up all the subproducts of the first part, and the resulting sum will be the product.
First part. Let the first factor be called the multiplicand. Let each digit of the second factor be called a multiplier. Let the onesdigit of the second factor be called the "onesmultiplier". Let the tensdigit of the second factor be called the "tensmultiplier". Let the hundredsdigit of the second factor be called the "hundredsmultiplier".
Start with the onescolumn. Find the product of the onesmultiplier and the multiplicand and write it down in a row under the line, aligning the digits of the product in the previouslydefined columns. If the product has four digits, then the first digit will be the beginning of the thousandscolumn. Let this product be called the "onesrow".
Then the tenscolumn. Find the product of the tensmultiplier and the multiplicand and write it down in a row — call it the "tensrow" — under the onesrow, but shifted one column to the left. That is, the onesdigit of the tensrow will be in the tenscolumn of the onesrow; the tensdigit of the tensrow will be under the hundredsdigit of the onesrow; the hundredsdigit of the tensrow will be under the thousandsdigit of the onesrow. If the tensrow has four digits, then the first digit will be the beginning of the tenthousandscolumn.
Next, the hundredscolumn. Find the product of the hundredsmultiplier and the multiplicand and write it down in a row — call it the "hundredsrow" — under the tensrow, but shifted one more column to the left. That is, the onesdigit of the hundredsrow will be in the hundredscolumn; the tensdigit of the hundredsrow will be in the thousandscolumn; the hundredsdigit of the hundredsrow will be in the tenthousandscolumn. If the hundredsrow has four digits, then the first digit will be the beginning of the hundredthousandscolumn.
After having down the onesrow, tensrow, and hundredsrow, draw a horizontal line under the hundredsrow. The multiplications are over.
Second part. Now the multiplication has a pair of lines. The first one under the pair of factors, and the second one under the three rows of subproducts. Under the second line there will be six columns, which from right to left are the following: onescolumn, tenscolumn, hundredscolumn, thousandscolumn, tenthousandscolumn, and hundredthousandscolumn.
Between the first and second lines, the onescolumn will contain only one digit, located in the onesrow: it is the onesdigit of the onesrow. Copy this digit by rewriting it in the onescolumn under the second line.
Between the first and second lines, the tenscolumn will contain a pair of digits located in the onesrow and the tensrow: the tensdigit of the onesrow and the onesdigit of the tensrow. Add these digits up and if the sum has just one digit then write this digit in the tenscolumn under the second line. If the sum has two digits then the first digit is a carrydigit: write the last digit down in the tenscolumn under the second line and carry the first digit over to the hundredscolumn, writing it as a superscript to the yetunwritten hundredsdigit under the second line.
Between the first and second lines, the hundredscolumn will contain three digits: the hundredsdigit of the onesrow, the tensdigit of the tensrow, and the onesdigit of the hundredsrow. Find the sum of these three digits, then if there is a carrydigit from the tenscolumn (written in superscript under the second line in the hundredscolumn) then add this carrydigit as well. If the resulting sum has one digit then write it down under the second line in the hundredscolumn; if it has two digits then write the last digit down under the line in the hundredscolumn, and carry over the first digit to the thousandscolumn, writing it as a superscript to the yetunwritten thousandsdigit under the line.
Between the first and second lines, the thousandscolumn will contain either two or three digits: the hundredsdigit of the tensrow, the tensdigit of the hundredsrow, and (possibly) the thousandsdigit of the onesrow. Find the sum of these digits, then if there is a carrydigit from the hundredscolumn (written in superscript under the second line in the thousandscolumn) then add this carrydigit as well. If the resulting sum has one digit then write it down under the second line in the thousandscolumn; if it has two digits then write the last digit down under the line in the thousandscolumn, and carry the first digit over to the tenthousandscolumn, writing it as a superscript to the yetunwritten tenthousandsdigit under the line.
Between the first and second lines, the tenthousandscolumn will contain either one or two digits: the hundredsdigit of the hundredscolumn and (possibly) the thousandsdigit of the tenscolumn. Find the sum of these digits (if the one in the tensrow is missing think of it as a zero), and if there is a carrydigit from the thousandscolumn (written in superscript under the second line in the tenthousandscolumn) then add this carrydigit as well. If the resulting sum has one digit then write it down under the second line in the tenthousandscolumn; if it has two digits then write the last digit down under the line in the tenthousandscolumn, and carry the first digit over to the hundredthousandscolumn, writing it as a superscript to the yetunwritten tenthousands digit under the line. However, if the hundredsrow has no thousandsdigit then do not write this carrydigit as a superscript, but in normal size, in the position of the hundredthousandsdigit under the second line, and the multiplication algorithm is over.
If the hundredsrow does have a thousandsdigit, then add to it the carrydigit from the previous row (if there is no carrydigit then think of it as a zero) and write the singledigit sum in the hundredthousandscolumn under the second line.
The number under the second line is the soughtafter product of the pair of factors above the first line.
Example
Let our objective be to find the product of 789 and 345. Write the 345 under the 789 in three columns, and draw a horizontal line under them:
First part. Start with the onescolumn. The multiplicand is 789 and the onesmultiplier is 5. Perform the multiplication in a row under the line:

7 
8 
9 

3 
4 
5 
3 
9^{4} 
4^{4} 
5 
Then the tenscolumn. The multiplicand is 789 and the tensmultiplier is 4. Perform the multiplication in the tensrow, under the previous subproduct in the onesrow, but shifted one column to the left:


7 
8 
9 


3 
4 
5 

3 
9^{4} 
4^{4} 
5 
3 
1^{3} 
5^{3} 
6 
Next, the hundredscolumn. The multiplicand is once again 789, and the hundredsmultiplier is 3. Perform the multiplication in the hundredsrow, under the previous subproduct in the tensrow, but shifted one (more) column to the left. Then draw a horizontal line under the hundredsrow:



7 
8 
9 



3 
4 
5 


3 
9^{4} 
4^{4} 
5 

3 
1^{3} 
5^{3} 
6 
2 
3^{2} 
6^{2} 
7 


Second part. Now add the subproducts between the first and second lines, but ignoring any superscripted carrydigits located between the first and second lines.



7 
8 
9 



3 
4 
5 


3 
9^{4} 
4^{4} 
5 

3 
1^{3} 
5^{3} 
6 
2 
3^{2} 
6^{2} 
7 


2 
7^{1} 
2^{2} 
2^{1} 
0 
5 
The answer is
See also
Last updated: 05132005 23:06:58
Last updated: 10292005 02:13:46