I was inspired by couple of videos I saw earlier today on kids doing mental mathematics calculations and hence decided to write a blog of slightly different nature and probably a series if this interests more folks. My intention is to remove the fear of Numbers and Maths in general and also make it more fun!
Background
I used to be fascinated by numbers from my junior school days (6th grade to be precise) and used to do mental Additions and Multiplications back then on a regular basis (Thanks to my classmate Ramesh who triggered this out of the blue noticing something that i had back then). After I enrolled into BITS, Pilani for my graduation, I started exploring mental mathematics to do n-th root of a number and logarithms - BITS was the perfect platform to explore as there was constant encouragement from my batch-mates and seniors, to do more
Now you may wonder why in the world someone has to do mental mathematics and calculate n-th root of a number when you can key in the same on a calculator or mobile these days and get the response! Yes there are enough options now and even 15 to 20 years back but performing mental maths really helped me to break-down complex problems and patterns and make you more inquisitive when you see patterns or numbers. Last but not the least, it also help remove the "fear" one has towards Maths as a subject
Enough of the preamble now and let's start with one simple example of how doing mental maths can probably help tickle our brain cells and help sharpen our analytical skills gradually as you experiment further
Example : Find the 31st root of 24 (Yes 24 ^ 1/31)!!
Must be wondering why I have taken an example of this nature.. The intent is not to teach someone how to find the root without using a calculator as it would require practice and time/effort but let's see how to break this down to simpler form and just work out the approach. The path to a solution is more important than the solution itself!
31 is a prime number and so there is no way find the 31st root as is. Hence let's look at the number nearby which has maximum factors
32 is closest to 31 and 31 can be factorized as 2x2x2x2x2
This would mean that we need to find five square roots of 24 in order to arrive at the answer. Now that sounds relatively simple compared to 31st root of a number isn't it?
Let's get into action
Step 1: Square root of 24 - 25 is the perfect square nearby and square root of that is 5. Knowing that, i would go with the law of approximation that square root of 24 would be ~ 4.9 (it doesn't matter if we need to get it very accurate as we still have more steps to go)
Step 2: Square root of 4.9... Not that easy but we know 2^2 is 4 and 2.5 ^ 2 is 6.25 (25x25). Since we have to find square root of 4.9, I would go with a number closer to 2.2 and see where we stand (22x22 is 484 and so 2.2 ^ 2 is 4.84). We are almost there. Let's go with 2.22 or 2.23 as the answer
Now drill down 3 more steps and as you go further this path, we will see that finding square root gets easier
Step 3: Square root of 2.23 is approximately 1.49 (2.25 square root is 1.5 and hence I went with 1.49)
Step 4: Square root of 1.49 is approximately 1.22 (since 144 square root is 12 and 1.44 would be 1.2)
Step 5: Square root of 1.22 is approximately 1.105 (since 121 square root is 11 and 1.21 would be 1.1)
Now what did we observe in Step 3 to 5? Most of us would remember squares upto 20 or 25 since we would have used it in some form or shape while doing Maths at school.. Invariably as you come down the chain while finding square roots, this trick would help us to arrive at the number
There is still one more step to be performed - Remember we started with 32nd root instead of 31 to ease the process and so we need to "adjust" the final answer. This would come by sheer experience but one thing we need to remember is that as n increases, 1/n decreases and approximation rule comes in handy
32nd root of 24 was approximately 1.105 and since I need 31st root, I will round it off to 1.108 or 1.109 (which is the final answer!)
In summary, while this technique may not give you near perfect answer but what you have managed to achieve is to breakdown a complex ask into smaller chunks and got your creative brain cells working.
This is especially important for students appearing in aptitude tests where they are not tested always to give answers upto 3 or 4 digits but given 3 or 4 choices, they need to identify the closest match - This is one area where practicing this comes very handy!
Have fun and try out few examples and I will back with another example in my next Blog post
Background
I used to be fascinated by numbers from my junior school days (6th grade to be precise) and used to do mental Additions and Multiplications back then on a regular basis (Thanks to my classmate Ramesh who triggered this out of the blue noticing something that i had back then). After I enrolled into BITS, Pilani for my graduation, I started exploring mental mathematics to do n-th root of a number and logarithms - BITS was the perfect platform to explore as there was constant encouragement from my batch-mates and seniors, to do more
Now you may wonder why in the world someone has to do mental mathematics and calculate n-th root of a number when you can key in the same on a calculator or mobile these days and get the response! Yes there are enough options now and even 15 to 20 years back but performing mental maths really helped me to break-down complex problems and patterns and make you more inquisitive when you see patterns or numbers. Last but not the least, it also help remove the "fear" one has towards Maths as a subject
Enough of the preamble now and let's start with one simple example of how doing mental maths can probably help tickle our brain cells and help sharpen our analytical skills gradually as you experiment further
Example : Find the 31st root of 24 (Yes 24 ^ 1/31)!!
Must be wondering why I have taken an example of this nature.. The intent is not to teach someone how to find the root without using a calculator as it would require practice and time/effort but let's see how to break this down to simpler form and just work out the approach. The path to a solution is more important than the solution itself!
31 is a prime number and so there is no way find the 31st root as is. Hence let's look at the number nearby which has maximum factors
32 is closest to 31 and 31 can be factorized as 2x2x2x2x2
This would mean that we need to find five square roots of 24 in order to arrive at the answer. Now that sounds relatively simple compared to 31st root of a number isn't it?
Let's get into action
Step 1: Square root of 24 - 25 is the perfect square nearby and square root of that is 5. Knowing that, i would go with the law of approximation that square root of 24 would be ~ 4.9 (it doesn't matter if we need to get it very accurate as we still have more steps to go)
Step 2: Square root of 4.9... Not that easy but we know 2^2 is 4 and 2.5 ^ 2 is 6.25 (25x25). Since we have to find square root of 4.9, I would go with a number closer to 2.2 and see where we stand (22x22 is 484 and so 2.2 ^ 2 is 4.84). We are almost there. Let's go with 2.22 or 2.23 as the answer
Now drill down 3 more steps and as you go further this path, we will see that finding square root gets easier
Step 3: Square root of 2.23 is approximately 1.49 (2.25 square root is 1.5 and hence I went with 1.49)
Step 4: Square root of 1.49 is approximately 1.22 (since 144 square root is 12 and 1.44 would be 1.2)
Step 5: Square root of 1.22 is approximately 1.105 (since 121 square root is 11 and 1.21 would be 1.1)
Now what did we observe in Step 3 to 5? Most of us would remember squares upto 20 or 25 since we would have used it in some form or shape while doing Maths at school.. Invariably as you come down the chain while finding square roots, this trick would help us to arrive at the number
There is still one more step to be performed - Remember we started with 32nd root instead of 31 to ease the process and so we need to "adjust" the final answer. This would come by sheer experience but one thing we need to remember is that as n increases, 1/n decreases and approximation rule comes in handy
32nd root of 24 was approximately 1.105 and since I need 31st root, I will round it off to 1.108 or 1.109 (which is the final answer!)
In summary, while this technique may not give you near perfect answer but what you have managed to achieve is to breakdown a complex ask into smaller chunks and got your creative brain cells working.
This is especially important for students appearing in aptitude tests where they are not tested always to give answers upto 3 or 4 digits but given 3 or 4 choices, they need to identify the closest match - This is one area where practicing this comes very handy!
Have fun and try out few examples and I will back with another example in my next Blog post
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