Elementary arithmetic

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 well-known procedures for adding and multiplying natural numbers. Other algorithms are used for subtraction and division. Mental arithmetic is elementary arithmetic performed in the head, for example to know that 100 − 37 = 63 without the use of a calculation aid, such as a sheet of paper, a slide rule, or a calculator. It is an everyday skill. Extended forms of mental calculation may involve calculating extremely large numbers, but this is a skill not usually taught at the elementary level.

Elementary arithmetic then moves on to fractions, 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. Earlier calculating tools included slide rules (for multiplication, division, logs and trig), tables of logarithms, and nomographs.

In the United States and Canada, the question of whether or not calculators should be used, and whether traditional mathematic's manual computation methods should still be taught in elementary school has provoked heated controversy as many standards-based mathematics texts deliberately omit some or most standard computation methods. 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.

In ancient times, the abacus was used to perform elementary arithmetic, and still is in many parts of Asia. A skilled user can be as fast with an abacus as with a calculator, which may require batteries.

In the 14th century Arabic numerals were introduced to Europe by Leonardo Pisano. These numerals were more efficient for performing calculations than Roman numerals, because of the positional system.

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

In decimal-counting literate cultures that use place-value written numbers, there are as many digits as fingers on the hands: the word "digit" can also mean finger (note, however, that there have been human cultures using different radices and correspondingly differently-sized digit sets, such as sexagesimal by the Babylonians and vigesimal by the pre-Columbian Mesoamericans). 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 two-digit number. This is ten sticks: I I I I I I I I I I

If a number has more than one digit, then the rightmost digit, said to be the last digit, is called the "ones-digit". The digit immediately to its left is the "tens-digit". The digit immediately to the left of the tens-digit is the "hundreds-digit". The digit immediately to the left of the hundreds-digit is the "thousands-digit".

Addition

Next, the tens-column. The tens-digit of the first number is 5, and the tens-digit 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 tens-column under the line, and write the carry digit on the hundreds-column above the first number:

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": $imes$. So the statement "five times three equals fifteen" can be written symbolically as:$5\; imes\; 3\; =\; 15.$In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. $5cdot3$. In some situations, especially in algebra, where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g. $xy$ means $x\; imes\; y$. 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".

Multiplication algorithm for a single-digit 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 multi-digit factor, then write the single-digit factor under the last digit of the multi-digit factor. Draw a horizontal line under the single-digit factor. Henceforth, the single-digit factor will be called the "multiplier" and the multi-digit factor will be called the "multiplicand".

Suppose for simplicity that the multiplicand has three digits. The first digit is the hundreds-digit, the middle digit is the tens-digit, and the last, rightmost, digit is the ones-digit. The multiplier only has a ones-digit. The ones-digits of the multiplicand and multiplier form a column: the ones-column.

Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the multiplicand and, under it, the ones-digit of the multiplier. Find the product of these two digits: write this product under the line and in the ones-column. If the product has two digits, then write down only the ones-digit of the product. Write the "carry digit" as a superscript of the yet-unwritten digit in the next column and under the line: in this case the next column is the tens-column, so write the carry digit as the superscript of the yet-unwritten tens-digit 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 tens-column. The tens-column so far contains only one digit: the tens-digit of the multiplicand (though it might contain a carry digit under the line). Find the product of the multiplier and the tens-digits of the multiplicand. Then, if there is a carry digit (superscripted, under the line and in the tens-column), add it to this product. If the resulting sum is less than ten then write it in the tens-column under the line. If the sum has two digits then write its last digit in the tens-column 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 hundreds-digit then if there is no carry digit then the multiplication algorithm has finished. If there is a carry digit (carried over from the tens-column) then write it in the hundreds-column 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 hundreds-digit... find the product of the multiplier and the hundreds-digit of the multiplicand, and to this product add the carry digit if there is one. Then write the resulting sum of the hundreds-column 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 hundreds-column and write the carry digit to its left: on the thousands-column.

Example

Say one wants to find the product of the numbers 3 and 729. Write the single-digit multiplier under the multi-digit multiplicand, with the multiplier under the ones-digit of the multiplicand, like so:

Next, the hundreds-column. The hundreds-digit of the multiplicand is 7, while the multiplier is 3. The product of three and seven is 21, and there is no previous carry-digit (carried over from the tens-column). The product 21 has two digits: write its last digit in the hundreds-column under the line, then carry its first digit over to the thousands-column. Since the multiplicand has no thousands-digit, then write this carry-digit in the thousands-column under the line (not superscripted):

Then the tens-column. The multiplicand is 789 and the tens-multiplier is 4. Perform the multiplication in the tens-row, under the previous subproduct in the ones-row, but shifted one column to the left:The answer is:$789\; imes\; 345\; =\; 272205.$

Division

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if "c" times "b" equals "a", written::$c\; imes\; b\; =\; a,$where "b" is not zero, then "a" divided by "b" equals "c", written::$frac\; ab\; =\; c$For instance, :$frac\; 63\; =\; 2$since:$2\; imes\; 3\; =\; 6,$.

In the above expression, "a" is called the dividend, "b" the divisor and "c" the quotient.

Division by zero (i.e. where the divisor is zero) is not defined.

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:$frac\; ab.$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", like this::$a/b.,$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 typographical variation, which is halfway between these two forms, uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

:frac|"a"|"b" .

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the "numerator" and "denominator"), and there is no implication that the division needs to be evaluated further.

A less common way to show division is to use the obelus (or division sign) in this manner::$a\; div\; b.$This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as 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 integers can be divided on paper using the method of long division. 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, multiply by the reciprocal (reversing the position of the top and bottom parts) of that fraction.

:$extstyle\{5\; div\; \{1\; over\; 2\}\; =\; 5\; imes\; \{2\; over\; 1\}\; =\; 5\; imes\; 2\; =\; 10\}$

:$extstyle$2 over 3} div {2 over 5} = {2 over 3} imes {5 over 2} = {10 over 6} = {5 over 3

ee also

*0
*binary arithmetic
*equals sign
*number line
*long division
*plus and minus signs
*subtraction
*unary numeral system