Overview: What Is Division?
Division is the inverse of multiplication. For example, 20/5 =4, because 4 X 5 = 20. Expressed in terms of a mathematical definition, for any whole number s, when s does not equal zero, r/s = z only if r = s X z. The number that is the answer is called the quotient, the number that is being divided by is the divisor, and the number that is being divided into is the dividend. In 20/5 =4, 4 is the quotient, 5 is the divisor, and 20 is the dividend. Solving a division problem involves multiplication and subtraction, whether it is short division (the divisor is a one-digit number), or long division (the divisor is two digits or above).
Division as Repeated Subtraction
There are two separate models of division. Divide 20 by measuring off sets of 5, and there are 4 groups of 5. Division on the abacus uses the measurement method of division, measuring the sets of the divisor and then subtracting them until the smallest possible remainder is left. (If the number is 20/5 , the answer will be 4 with no remainder.) This method can also be demonstrated using a calculator. For example 300-24=276-24=252-24=228-24=204-24=180-24=156-24=132-24=108-24=84-24=60-24=36-24=12, or 300/24 = 12 remainder 12, or 12 12/24 or 12.5.
The Division Algorithm
Whether division is long or short, the algorithm is the same. It involves multiplying the divisor by a number that is close to the dividend and subtracting the product from the dividend until the remainder is smaller than the divisor, or until there’s nothing left to subtract. For example, in 300/24, 300-240 (10 X 24) =60-48 (2 X 24) =12, so the quotient is 12 remainder 12 or 12.5. To check, multiply 12.5 X 24 =300. Alternatively, multiply 12 X 24= 288, and add 12 to equal 300.
Another Method for Long Division
Another method for long division multiplies each digit of the divisor separately, subtracts, and then adds like products, using place value. In this method 300/24 = 12.5 , because the divisor 24 equals 20 +4. Multiplying 20 X 4 by 10 gives 200 (10 X 20) + 40 (10 X 4). 300-240 = 60. The largest multiple of 20 + 4 that will divide into 60 is 48 or 40 (2 X 20) + 8 (2 X 4). The same remainder of 12/24 is still there in this problem. This method has the advantage of dividing by one digit at a time, but it is way more cumbersome than the modern method.
Why Not Divide by Zero?
All numbers multiplied by zero equal zero. Therefore, there isn’t any unique number that will give a meaningful answer. If 24 X 0 = 0 and 3 X 0 =0, 24 does not equal 3. Using the definition of division as the inverse of multiplication, dividing by zero is completely meaningless.
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