Onmatu- Arithmetic Article
Is the method how we add numbers a right one
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Let's assume that we're provided with two bags; one with 64 marbles in it and the other one with 53 marbles in it! Than we're asked the question—Now think of a little boy who has just learned counting, but don't know the algorithm of addition. What would he do?How many marbles are there in total in the two bags?
He would first count from first bag, and would get 64 marbles; then proceed to the second bag; he would count the first marble of the second bag as 65; and totally he would have counted 117 marbles in total!
Notice that he didn't first take 4 marbles (4 is the units digit of 64) from the first bag and 3 marbles (3 is the units digit of 53) from the second bag!
Obviously the kid's way to 'add' is the most fundamental way to add!
But what do we do? When we add 64 and 53, we first add 4 and 3, the units digit of 64 and 53, respectively! Now the question is:
Notice that he didn't first take 4 marbles (4 is the units digit of 64) from the first bag and 3 marbles (3 is the units digit of 53) from the second bag!
Obviously the kid's way to 'add' is the most fundamental way to add!
But what do we do? When we add 64 and 53, we first add 4 and 3, the units digit of 64 and 53, respectively! Now the question is:
How are we assured that if we add the units first, then the tens, then the hundreds and so on, we would get the correct sum?
Actually the ASSURANCE is a result of the fundamental axioms of the set of Real Numbers over addition, i.e., the Law of Associativity of Addition, the Law of Commutativity of Addition etc.
Note the equations below.
For simplicity, we will just point to the case of addition of 64 and 53; that will be enough as an intuitive answer the question.
We will try to show that the number that must be got as the sum of 64 and 53 is SAME as the sum of sums of units, sum of tens of the numbers.
We will try to show that the number that must be got as the sum of 64 and 53 is SAME as the sum of sums of units, sum of tens of the numbers.
Note the equations below.
Now we have, 64 + 53 = (6 tens + 5 tens) + (4 + 3). So, adding 64 and 53 is same as adding the sum of the corresponding digits with same positional value.64 + 53= 6 tens + (4 + 5 tens) + 3 [Substitution by the concept of positional value of digits in a decimal number]= {(6 tens + 4) + 5 tens}+ 3 [Associative Law]= {6 tens + (4 + 5 tens)} + 3 [Associative Law]= {6 tens + (5 tens + 4)} + 3 [Commutative Law]= {(6 tens + 5 tens) + 4} + 3 [Associative Law]= (6 tens + 5 tens) + (4 + 3) [Associative Law]
That's how the correctness of our method of addition is established!
That concludes this quest!
Happy Learning! See you again!