Division (mathematics)

Division is a mathematical operation and is one of the four basic functions in mathematics, the others being multiplication, addition, and subtraction. It is the inverse of multiplication and is denoted using this symbol / or this symbol ÷. Division involves splitting a number into smaller components. This can be as simple as calculating the number of times a small number makes up a larger number, so for example 6 divided by 2 is 3, as 6 is twice the number 3. In this example, the number 6 is the dividend or numerator, 2 is the divisor or denominator and the result 3 is called the quotient.

Overview

Although the concept of division has existed for thousands of years, it was formalized by the ancient Egyptians. An example of an in-depth Egyptian mathematical text is the Rhind papyrus (also known as the Ahmes Papyrus), dated to around 1650 BC. It is an instruction manual for students in mathematics and geometry and gives area formulas and methods for mathematical functions such as division.

When the dividend divides exactly by the denominator with no remainder, as in the above example, a whole number or integer is the resulting quotient. So for example whole integers exactly divisible by 3 are the multiples of 3 (3, 6, 9, 12,...). However, in the event of it not dividing exactly, the quotient can be presented in several different ways. The simplist way is to note the remainder, that is, the amount remaining after the divisor has been computed into the dividend as a whole number. So for example 33 divided by 4 is 8 with a remainder of 1. This is conventionally how children and those new to the concept of division are first taught how to compute divisions.

A more advanced way of computing divisions is by presenting a resultant non-whole quotient as a fraction (rational number) or decimal. This denotes how many whole times and how many partial times a larger number is composed of a smaller one. So in the earlier example, 33 divided by 4 is 8.25, or 8 and a quarter, or 8 1/4. This is because 4 goes into the number 33 a total of 8 times with a remainder of 1, which when split into the divisor 4 equates to a quarter, or 0.25. In certain situations, a percentage notation may be used, in this case, 25%.

Computing Division Sums

Simple division sums like those noted so far can usually be computed by a method known as short division, which relies on the premise that the person computing the sum knows about multiplication and multiplication tables, and can therefore divide by simply performing the multiplication in reverse. Long division is used for more complex sums involving larger numbers. This involves a specific form of noting the numbers to be divided and then splitting them into smaller components, which then combine to provide the final answer. An alternative to calculating division sums by hand is to use a computer or calculator, which frequently return non-whole results in a decimal form.

Dividing by the number 1 will always return a quotient that is the same as the dividend. Division of any number by zero will always return a quotient of zero. This is because zero multiplied by any finite number always results in a product of zero. Dividing by 2 involves halving the dividend. If the dividend is a whole number, the resulting quotient will always be suffixed with the decimal .5, or the fraction 1/2. Dividing by 10 will always result in moving the decimal point one point to the left, so for example dividing 25 by 10 equals 2.5. This method applies to any exponential power of 10, so for example dividing by 100 involves moving the decimal point two points to the left so 25 divided by 100 equals 0.25, dividing by 1000 involves moving the decimal point 3 points to the left, and so on.

A simple way to divide by a fraction is to invert the fraction, or turn it upside down, and multiply. For example, dividing by 2/3 is the same as multiplying by 3/2. The resulting fraction can then be simplified if necessary by dividing one factor of the numerator and one factor of the denominator by the same number.

The rule for dividing by negative numbers is as follows. When the signs of the dividend and divisor are different (that is, one number is positive and the other negative), the resulting quotient is negative. When the signs are the same, the resulting quotient is positive. Essentially, when calculating a division, ignore the signs initially and compute based on the figures. Then look at the signs, and follow the rule: If the two signs are the same, the sign of the result is positive, otherwise it is negative. This rule may facilitate complex calculations, for example, involving negative fractions or decimals.

Bibliography

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