Beauty of MATHEMATICS:Equations
I am so fascinated by this topic that I think I am going to write maybe a series of post on the same. Last time I talked about the Fibonacci Series and Golden Ratio which you can find here.
Do read on it. Today I am just gonna talk about what is considered as one of the boring part of Mathematics(Hell bro!! Maths in entirety is itself so boring....Maybe I don't think so..) equations.
I will again try and bring in some sort of Inspiration in mathematics or equations per say. So let's start.
To start with I am going to introduce one with a simple trick which ... I guess it may even blow your mind!!
So here's some steps you need to do. You can use calculator or mentally also you can solve(I prefer mental ones). So follows these steps:
Step 1. Think of two numbers from 1 to 10.
Step 2. Add those numbers together.
Step 3. Multiply that number by 10.
Step 4. Now add the larger original number.
Step 5. Now subtract the smaller original number.
Step 6. Tell me the number you’re thinking of and I’ll tell you both
of the original numbers!
This is not any whatsapp forward.
I will come back to this later..
So now that I have your attention, lets dive into basics of equation.So what exactly is an equation? An equation is something that contains both variables and constants.
Ex: 2x+3y=5. So far so good? Okay lets move on. So why we need to learn these equations? Well firstly believe me .. the entire world is made up of equations. To start off with, at what time you will reach office if you board a slow train or the difference in time required to reach the destination if one boards fast local rather than a slower one is an example of a linear equation. Here the equation looks like below:
T= D/ S where
T: Time taken to reach destination
D: Distance between source and destination
S: Speed of the local train
So you see there are so many examples which are generalized by equations. Ahhh! interesting word generalized, this pretty much explains everything. Simple thing, if you don't wanna remember numbers, use equations...
Now lemme just actually begin something for which I am writing this post- The beauty of equations.Well one knows so many ways in which an equation can be solved like cramers rule ,Gauss elimination method etc, etc. (If you wanna know more on them, dive into this section and download the PDF ). What I am gonna take is some pretty simple concepts which I guess most of the times people fail to recognize or they simply do it mechanically.
Let's start off with linear simultaneous equations and then dive a bit deeper.
Consider the following equation:
17x+94y =119
34x+89y =238
Solve for x and y.
Now believe me most of the people will take their scientific calcis and start putting numbers to find the solution. But a good observer (like you maybe) will find the solution of above in less than 10 secs!!!. So whats the trick they use implicitly here?
Well observe the numbers carefully... one can find that in both equations,the coefficients of x- 17 and 34 and the value of constants- 119 and 238 are in a ratio(1:2). This pretty much means that y is 0, and the value of x in this case is 7 (119/7 or 238/34 is 7).
So why is y =0. This is something I guess I have to leave on you guys to think upon. As of now just remember, if one is in ratio, other is zero. So far good? Lets dive in further...
Let us say, we want to solve an equation
ax + b = px + q
ax - px = q - b
x (a - p) = q - b
x = (q - b) / (a - p)
Thus, every equation that we want to solve using this lengthy method, can be solved easily just by using the final solution that
I am so fascinated by this topic that I think I am going to write maybe a series of post on the same. Last time I talked about the Fibonacci Series and Golden Ratio which you can find here.
Do read on it. Today I am just gonna talk about what is considered as one of the boring part of Mathematics(Hell bro!! Maths in entirety is itself so boring....Maybe I don't think so..) equations.
I will again try and bring in some sort of Inspiration in mathematics or equations per say. So let's start.
To start with I am going to introduce one with a simple trick which ... I guess it may even blow your mind!!
So here's some steps you need to do. You can use calculator or mentally also you can solve(I prefer mental ones). So follows these steps:
Step 1. Think of two numbers from 1 to 10.
Step 2. Add those numbers together.
Step 3. Multiply that number by 10.
Step 4. Now add the larger original number.
Step 5. Now subtract the smaller original number.
Step 6. Tell me the number you’re thinking of and I’ll tell you both
of the original numbers!
This is not any whatsapp forward.
I will come back to this later..
So now that I have your attention, lets dive into basics of equation.So what exactly is an equation? An equation is something that contains both variables and constants.
Ex: 2x+3y=5. So far so good? Okay lets move on. So why we need to learn these equations? Well firstly believe me .. the entire world is made up of equations. To start off with, at what time you will reach office if you board a slow train or the difference in time required to reach the destination if one boards fast local rather than a slower one is an example of a linear equation. Here the equation looks like below:
T= D/ S where
T: Time taken to reach destination
D: Distance between source and destination
S: Speed of the local train
So you see there are so many examples which are generalized by equations. Ahhh! interesting word generalized, this pretty much explains everything. Simple thing, if you don't wanna remember numbers, use equations...
Now lemme just actually begin something for which I am writing this post- The beauty of equations.Well one knows so many ways in which an equation can be solved like cramers rule ,Gauss elimination method etc, etc. (If you wanna know more on them, dive into this section and download the PDF ). What I am gonna take is some pretty simple concepts which I guess most of the times people fail to recognize or they simply do it mechanically.
Let's start off with linear simultaneous equations and then dive a bit deeper.
Consider the following equation:
17x+94y =119
34x+89y =238
Solve for x and y.
Now believe me most of the people will take their scientific calcis and start putting numbers to find the solution. But a good observer (like you maybe) will find the solution of above in less than 10 secs!!!. So whats the trick they use implicitly here?
Well observe the numbers carefully... one can find that in both equations,the coefficients of x- 17 and 34 and the value of constants- 119 and 238 are in a ratio(1:2). This pretty much means that y is 0, and the value of x in this case is 7 (119/7 or 238/34 is 7).
So why is y =0. This is something I guess I have to leave on you guys to think upon. As of now just remember, if one is in ratio, other is zero. So far good? Lets dive in further...
Let us say, we want to solve an equation
ax + b = px + q
ax - px = q - b
x (a - p) = q - b
x = (q - b) / (a - p)
Thus, every equation that we want to solve using this lengthy method, can be solved easily just by using the final solution that
x = (q - b) / (a - p)
Ex: 16x+38= 11x-7
Here a=16, b=38, p=11, q=-7
x= (-7-38)/(16-11) = -9.
You can verify by substitution, 16(-9)+38=-106=11(-9)-7
Well I can go deeper as far as powers of variables are concerned, but at the moment I will restrict myself to basic discussions.
So this far we have seen two things, time to know few more. Lets go onto an application first so that you can understand what I am trying to say.
The question is find squares of any two digit number. And believe me I can do this in seconds (no bluffing!). So I am gonna say you how I and other people do it.
Lets suppose I wanna take square of 43. Heres how one must approach..
Step 1: Take nearest multiple of 10 (you can take any but that would need some practice, so as of now stick with multiple of 10)
Here it will be 40.
Step 2: Find the difference between the numbers.
In this case it will be 43-40= 3. Remember always take difference with the number to be squared and the nearest multiple of 10. (Imp follow this order only)
Step 3: Add that difference to the number to be squared.
That will make 43+3= 46.
Step 4: Take product of the results of step 1 and step 3
That means 46*40=1840. (This is simple and can be done mentally. Ignore the zero term, multiple 46 by 4 and append zero in ans)
Step 5: Add square of result obtained in step 2 to the result in step 4 and you get your square.
1840+(3*3)=1849.
Now that once you know how lets know why this worked.
Frankly speaking we all were taught this in schools when we learn about the following equation:
a2 – b2 = (a – b) (a + b)
Now don't we know this? Ofcourse we do. Now what if I add square of b on both the sides?
a2 – b2 + b2 = (a – b)(a + b) + b2
= a2
And this is what I did in here... a =43, b=3. Now I hope you will start to realize how important equations are...
Ok now that this is known, I will conclude with where I started.(There are many things to write on..hopefully I may in future if time permits me to). So at the start of this post I started by saying a question. You can scroll up to find that again.For example, if the final answer is 126, then you must have started with 9 and 3. Even if this trick is repeated a few times, it’s hard for your audience to figure out how you are doing it. Now lets ans as to how I can say.
The answer lies in....equations!! Ofcourse!!
To determine the larger number, take the last digit of the answer (6 here) and add it to the preceding number(s) (12 here), then divide by 2. Here we conclude that the larger digit is
(12+6)/2=18/2 = 9.
For the smaller digit, take the larger digit that you just computed
(9) and subtract the last digit of the answer. Here that would be
9 − 6 = 3.
One can try this on some other set of numbers and verify...
Why does it work? Suppose you start with two numbers X and Y,
where X is equal to or larger than Y. Following the original instructions and the algebra in the steps below, we see that after Step 5, you end up
with the number 10(X +Y) + (X −Y).
Step 1: X and Y
Step 2: X + Y
Step 3: 10(X + Y )
Step 4: 10(X + Y ) + X
Step 5: 10(X + Y ) + (X − Y )
Larger Number: ((X + Y ) + (X − Y ))/2 = X
Smaller Number: X − (X − Y ) = Y
Now I guess you won't tell me that equations are boring. They are boring if you make them boring. Really trust me ...no branch of any course as such is boring. One should have that curious brain fired up every time to make it interesting..
Well thats all for today. Hope you all like it..
Ex: 16x+38= 11x-7
Here a=16, b=38, p=11, q=-7
x= (-7-38)/(16-11) = -9.
You can verify by substitution, 16(-9)+38=-106=11(-9)-7
Well I can go deeper as far as powers of variables are concerned, but at the moment I will restrict myself to basic discussions.
So this far we have seen two things, time to know few more. Lets go onto an application first so that you can understand what I am trying to say.
The question is find squares of any two digit number. And believe me I can do this in seconds (no bluffing!). So I am gonna say you how I and other people do it.
Lets suppose I wanna take square of 43. Heres how one must approach..
Step 1: Take nearest multiple of 10 (you can take any but that would need some practice, so as of now stick with multiple of 10)
Here it will be 40.
Step 2: Find the difference between the numbers.
In this case it will be 43-40= 3. Remember always take difference with the number to be squared and the nearest multiple of 10. (Imp follow this order only)
Step 3: Add that difference to the number to be squared.
That will make 43+3= 46.
Step 4: Take product of the results of step 1 and step 3
That means 46*40=1840. (This is simple and can be done mentally. Ignore the zero term, multiple 46 by 4 and append zero in ans)
Step 5: Add square of result obtained in step 2 to the result in step 4 and you get your square.
1840+(3*3)=1849.
Now that once you know how lets know why this worked.
Frankly speaking we all were taught this in schools when we learn about the following equation:
a2 – b2 = (a – b) (a + b)
Now don't we know this? Ofcourse we do. Now what if I add square of b on both the sides?
a2 – b2 + b2 = (a – b)(a + b) + b2
= a2
And this is what I did in here... a =43, b=3. Now I hope you will start to realize how important equations are...
Ok now that this is known, I will conclude with where I started.(There are many things to write on..hopefully I may in future if time permits me to). So at the start of this post I started by saying a question. You can scroll up to find that again.For example, if the final answer is 126, then you must have started with 9 and 3. Even if this trick is repeated a few times, it’s hard for your audience to figure out how you are doing it. Now lets ans as to how I can say.
The answer lies in....equations!! Ofcourse!!
To determine the larger number, take the last digit of the answer (6 here) and add it to the preceding number(s) (12 here), then divide by 2. Here we conclude that the larger digit is
(12+6)/2=18/2 = 9.
For the smaller digit, take the larger digit that you just computed
(9) and subtract the last digit of the answer. Here that would be
9 − 6 = 3.
One can try this on some other set of numbers and verify...
Why does it work? Suppose you start with two numbers X and Y,
where X is equal to or larger than Y. Following the original instructions and the algebra in the steps below, we see that after Step 5, you end up
with the number 10(X +Y) + (X −Y).
Step 1: X and Y
Step 2: X + Y
Step 3: 10(X + Y )
Step 4: 10(X + Y ) + X
Step 5: 10(X + Y ) + (X − Y )
Larger Number: ((X + Y ) + (X − Y ))/2 = X
Smaller Number: X − (X − Y ) = Y
Now I guess you won't tell me that equations are boring. They are boring if you make them boring. Really trust me ...no branch of any course as such is boring. One should have that curious brain fired up every time to make it interesting..
Well thats all for today. Hope you all like it..
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