What is 18 - 9 equal to ?
As funny as this meme is, this made me realize (after discussing with a friend) that you know the answer to this is 9 because it is burned in your memory from all the countless addition/subtraction problems you have solved in your school. Or you counted this on your fingers, which is rather unlikely if you’ve finished primary schooling.
In fact, if you want to add any two numbers, you need to have memorized all the additions between the numbers from 0 to 9, unless you’re going to count on your fingers every time you see them. Let me illustrate why:
As you see in the above illustration, when you’re adding 84 to 53, you break it down into additions of 3 and 4 and 5 and 8. (Some people break it down into a 50 + 80 and then a 3 + 4. Even then, it is essentially a 5+8 when you’re adding 50 and 80.) Your brain instantly replies 3+4 is 7 and 5+8 is 13. Now, just stop and ask a question to yourself. Why do you say 3+4 is 7? How do you know 3+4 is 7? That is because you were taught in your school that 3+4 is 7 indeed by telling you to count on your fingers. After you counted it on your fingers, you logically concluded that 3+4 is 7. From then on, you have practiced an awful lot of additions that the answers to 3+4 and 5+8 have been etched into your brain’s memory. The next time you see a 3+4 you just fetch the answer from your memory. The same is true for all the additions between 0 and 9.
Similarly, if you want to subtract any two numbers, you need to have memorized all the subtractions between the numbers from 0 to18, unless you’re going to count on your fingers here too.
Why is it 18 and not 9 for the subtraction ? It is obvious why we have the numbers from 0 to 9. But why do we have numbers from 10 to 18 as well ? Consider a case where you’ve to take a carry when you’re subtracting. Let us consider 28 - 9 as in the below illustration. Since 8 is less than 9, you take a carry (from the next place’s digit which is 2) and this temporarily creates a new number in the minuend(18) that you can subtract from. You can have a maximum of nine such new numbers that you have to subtract from (10 to 18). That is why you have an extra set of 9 numbers involved when you’re subtracting.
Many people usually convert the original subtraction into other equivalent forms to their convenience. Let us consider 27 - 13. Some people would start from 13 and keep adding numbers to 13 until they reach 27. Then count how much was added to 13 in total. I will write down two ways of doing it.
→ 13 + 10 = 23 ; adding 10 to get 23
→ 23 + 4 = 27 ; adding 4 to get 27
→ 10 + 4 = 14 ; finding in total how much was added to 13
→ 27 - 13 = 14 ; finally arriving at the result
— — — — — — — — — — — — — — — — — — — — — — — —
→ 13 + 7 = 20; adding 7 to get 20
→ 20 + 7 = 27; adding 7 to get 27
→ 7 + 7 = 14 ; finding in total how much was added to 13
→ 27–13 = 14 ; finally arriving at the result
Of course, these are not the only ways subtract. In math, you can do the same thing in a lot of different ways :). But, no matter to what form you convert the subtraction/addition, those forms too rely on your memory to be solved. If you convert into a convenient form, you will probably need to memorize a lesser set of numbers and not until 18. But it will be memory that you shall rely on.
I had never realized until now, that addition and subtraction, in its simplest form, were based on the memory of your brain. I always thought I was purely applying logic whenever I was solving them. This doesn’t mean that it is purely based on the memory of your brain. Nobody counted the addition of 552 and 2723 on their fingers.You must credit the logical reasoning ability of the human brain for extending the addition/subtraction to numbers that are bigger than the numbers in the set that we have memorized.
The point I want to make is that, for addition/subtraction it is not that apparent that you’re subconsciously using your memory. We can say the same thing for multiplication/division[See the clip from the movie ‘Ma and Pa Kettle’]. But hey, multiplication and division in the simplest form is nothing but addition/subtraction. :)
We use the memory of our brain for a lot of things and it is an invaluable ability we possess. Having knowledge of established facts and practices is definitely important, so that you don’t have to reinvent the wheel. At the same time, the ability to logically extend and improvise them is equally important. Well, isn’t that what progress is ?
