What is Repeated Subtraction?
Repeated Subtraction is a process that subtracts the same number of objects from a set, also known as division. The Repeated Division Method is when the same number is continuously subtracted from another greater number until the remainder is zero or less than the number subtracted.
What is Subtraction?
Subtraction is an arithmetic operation that is used to subtract objects from a list. The outcome of subtraction is referred to as a difference. Although mainly concerned with natural arithmetic numbers, subtraction can also reflect the elimination or reduction of physical and abstract quantities using various kinds of artifacts, including negative numbers, ratios, irrational numbers, vectors, decimals, functions, and matrices.
What is Division?
The division is one of the four fundamental operations of algebra, the way that numbers are added to produce new numbers. Other operations include addition, subtraction, and multiplication. A division sign, a symbol consisting of a short horizontal line with a dot above and a dot below, is sometimes used to denote a mathematical division. This use, although common in English-speaking countries, is neither universal nor recommended: the ISO 80000-2 standard for mathematical notation only recommends the solidus/or fraction bar for division, or the colon for ratios; it specifies that this symbol "should not be used" for the division. The division is a method of separating a set of items into equal parts. It is one of the four simple arithmetic operations that give a fair outcome of sharing. The key purpose of the division is to see how many comparable groups or how many of each group share equally. In other words, we can say that repeated subtraction is called division.
Division as Repeated Subtraction
The division is only counting how many times you can deduct from another number (divisor) (dividend). The division as subtraction is repeated till the remainder becomes lesser than the divisor. The number of times you can subtract is called the quotient, and any number less than the remaining divisor is called the remainder. Let us take a look at the Repeated Subtraction Method.
Steps for Repeated Subtraction:
Divide using repeated subtraction i.e, subtract the divisor from the dividend.
Repeat step 1 until you have a number lesser than the divisor or zero.
The answer is the number of times the step is repeated.
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Suppose you've got 15 bits of candy to be given out to 3 people. This is the dilemma of division 15/3. You want each person to get the same amount of candy, so you plan to give five candy pieces at a time (one piece each) before you run out or you don't have enough to give each of the three people another piece.
Notice that any time you hand out 3 pieces of candy, you have to deduct 3 out of the total leftover, and each person gets one piece of candy.
(15 – 3) = 12 Each person now has 1 piece of candy, and there are 12 more to pass out.
(12 – 3) = 9 Each person now has 2 pieces of candy, and there are 9 more to pass out.
(9 – 3) = 6 Each person now has 3 pieces of candy, and there are 6 more to pass out.
(6 – 3) = 3 Each person now has 4 pieces of candy, and there are 3 more to pass out.
(3 – 3) = 0 Each person now has 5 pieces of candy, and there are 0 to pass out.
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If there are 25 balls and we make up a bunch of five balls each.
Here, the number 5 was consistently subtracted five times. We may say the number 5 has been subtracted 5 times out of 25. So, we can write this subtraction as 25 ÷ 5 = 5.
Similarly, to solve the problem of division by repetitive subtraction, we repeatedly group and deduct the same number again and again to find the response.
Square Root as Repeated Subtraction
One of the important applications of repeated subtraction is finding the square root of a number. Each squared natural number can be written as the sum of consecutive odd natural numbers starting from zero. So to find the square root, we begin subtraction from 1 and proceed until it reaches 0. This process of performing repeated division will help us to get the value of the square root of a number. The number of steps that can hit zero is the square root.
Let us take an example to understand the topic better. Let us find the square root of 25.
We will start by subtracting 25 by odd natural numbers,
25 - 1 = 24
24 - 3 = 21
21 - 5 = 16
16 - 7 = 9
9 - 9 = 0
This process happened 5 times, therefore the square root of 25 is 5.
Theorem
Euclid’s Division Lemma states that provided the dividend and the divisor, there will be a special pair of quotients and the remainder, satisfying the equation.
Dividend = Divisor × Quotient + Remainder
This is valid for any two positive integers and is referred to as Euclid’s Division Lemma.
It says that:
Provided the positive integers m and n, there are two unique integers q and r, which satisfy
m = n × q + r,
In which 0 ≤ r < n.
This lemma is useful to find a large number of HCFs because it is difficult to divide them into variables.
Solved Examples
1. Find the value of 38 ÷ 7using the repeated subtraction method.
Ans:
We start by subtracting 7 from 38,
38 - 7 = 31
The division is repeated until we attain a value lesser than 7,
31 - 7 = 24
24 - 7 = 17
17 - 7 = 10
10 - 7 = 3
The quotient and the remainder are 5 and 3 respectively.
2. Find the value of 169 being divided by 13 using the repeated subtraction method.
Ans:
Use repeated subtraction to divide169 ÷ 13. Let us start by subtracting 13 from 169 until we get a value lesser than 13.
169 - 13 = 156
156 - 13 = 143
143 - 13 = 130
130 - 13 = 117
117 - 13 = 104
104 - 13 = 91
91 - 13 = 78
78 - 13 = 65
65 - 13 = 52
52 - 13 = 39
39 - 13 = 26
26 - 13 = 13
13 - 13 = 0
The quotient and the remainder are 13 and 0 respectively.
Fun Facts
When you divide anything by 1, the answer will always be the original number. This implies that if the divisor is 1, the quotient will always be equal to the dividend, for example, 10 ÷ 1 = 10.
The remainder is often smaller than the divisor in a division.
In algebra, division by zero is a division where the divider is zero. Such a division can be formally expressed as a/0 where a is a dividend. In ordinary arithmetic, the expression has no sense, as there is no number which, when multiplied by 0, gives a, and hence the division by zero is undefined.
FAQs on Repeated Subtraction
1. What is the Division Method?
Ans: In algebra, long division is a method used to separate large numbers into classes or sections. Long division tends to divide the dilemma of division into a series of simpler moves. Just like all division problems, a big number, which is a dividend, is divided by another number, which is called a divider, to give a result called a quotient and sometimes a remainder.
The method starts by splitting or discovering how many times the left digit of the dividend can be separated by the divisor.
The consequence or answer from step 1, which becomes the first digit of the quotient, is then multiplied by the divisor and written in the first digit of the dividend.
The subtraction shall be rendered on the first digit of the dividend and the remainder shall be in writing. The next digit of the dividend is reduced and the process is repeated until all the digits of the dividend are reduced and the remainder is found.
2. What are the Terminologies Used for Subtraction?
Ans: Subtraction is the opposite additional operation. If you subtract, you take the value of the smaller number from the value of the greater number. You often deduct because you want to see how many you need to apply to the smaller number so that it becomes the same as the greater number. The concepts of subtraction are called minuend and subtrahend, the product is called the difference.
The minute is the first number, the number from which you take something, and it must be the greater number.
The subtrahend is the number that is subtracted and the lower number must be subtracted.
The difference is the result of the subtraction.
3. How to Find Square Root in the Process of Repeated Subtraction?
Ans: The square root of a number is a value that gives the initial number when multiplied by itself. Suppose that x is the square root of y, then it is defined as x = √y, or we can express the same equation as x2 = y. If the Natural Number is a square number, it must be the sum of successive odd numbers beginning from 1. As explained above, a square number is the sum of consecutive odd numbers beginning from 1.
You can therefore calculate the square root of a number by continuously subtracting successive odd numbers (starting from 1) from the given square number before you get 0.
