Division (mathematics)
In mathematics, the word "division" means the operation which is the opposite of multiplication. The symbol for division can be a slash, a line, or the division sign ([math]\displaystyle{ \div }[/math]), as in:
- [math]\displaystyle{ 6/3\, }[/math] or [math]\displaystyle{ \frac 63 }[/math] or [math]\displaystyle{ 6 \div 3. }[/math]
Each, of those three, means "6 divided by 3" giving 2 as the answer. The first number is the dividend (6), and the second number is the divisor (3). The result (or answer) is the quotient. Whole numbers, any left-over amount is called the "remainder" (such as 14/4 gives 3 with the remainer as 2, as the number Template:Convert/numdisp, same as Template:Convert/numdisp or 3.5).
The numbers can be very big, such as with two hundred: 200/5 = 40, or with 7 billion: 7,000,000,000 / 1000 = 7,000,000 (equals 7 million).
With multiplication
If c times b equals a, written as:
- [math]\displaystyle{ c \times b = a }[/math]
where b is not zero, then a divided by b equals c, written as:
- [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.
- [math]\displaystyle{ \frac x0 = ? }[/math]
...is not defined.
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
- [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, like this:
- [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 typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor:
- 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). A fraction is an accepted way of writing numbers. It is not always expected that the result of the division is written in decimals.
A less common way to show division is to use the obelus (or division sign) in this manner:
- [math]\displaystyle{ a \div b. }[/math]
But in elementary arithmetic this form is used rather often. 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 (with a:b said as "a is to b").
Related pages
- Divisor, another meaning as a number which evenly divides an amount
- Division by two
- Remainder