Let's say your home is being renovated and there are 6 workers doing the renovation work. Let's say they all do the same work and everyone is working the same. We assume these for the sake of ease of understanding.
Suppose 6 workers have to spend 5 days renovating the whole house, so if 3 people leave from the beginning, how many days will it take for the remaining people to renovate the whole house?
We can calculate it in the following way:
6 workers need 5 days to work.
It takes 5*6 days for 1 worker to complete the work.
3 workers need (5*6)/3 days to complete the work.
So, it's 10 days. But what if there were 10 workers in the beginning and it would take 14 days to renovate? Are the remaining workers 30% instead of half? You must repeat these steps again.
But you can skip those lines with simple algebra. Just assume that the number of workers is x, the number of days they will spend is y, and the percentage of workers leaving is z.
Therefore, the number of days required = y/(1-(z/100)). Here you will find that if we know the percentage of workers leaving, then the number of workers is not even used in the equation.
Let's look at another example. Let's say you want to buy a new pair of shoes. But you want to wait until Singles Day, because there will be big promotions at that time. The store gave a uniform 30% discount. At present, the price of this pair of shoes is 50 yuan. So, how much does it cost when it comes to big promotions?
You will spend 50* (1-(30/100)) = 35 yuan.
But what if you want to buy a pair of shoes that cost $60 and it has a 35% discount? How will you calculate?
Let's set the price of this pair of shoes as x and the discount rate as y. With just one equation, now you can calculate the discounted price of any item. We can set the discount price to P, then we will simply get P=x*(1-(y/100)).
We can also simplify it further. Let's write 30% as 0.3 because it means the same thing. Then we get a simpler formula:
Just one formula can sum up a lot of situations. Not only is the price calculated, algebra is also used to represent physics. For example, the formula for force is:
As long as F stands for force, m stands for mass, and a stands for acceleration, this formula has always worked.
This is just the beginning. You see what equations are and how we get them. Suppose you have a lot of equations and a lot of variables. You can do something with each of them and end up calculating the value of the variable with pen and paper, but it's really tedious and not efficient at all. What if you have a thousand equations to solve, to find the values of a thousand variables? You may not be able to do it with pen or paper. We need a computer program to do this for us.
Consider the above set of equations. If we do it with pen and paper, even these three equations will take some time to solve. So, let's represent it in a different way.
We represent them all as: multiply a number by a variable, multiply a number by a variable, add... It is equal to a certain number. In addition, we must write it in a specific order (here we write it in the order of ax + by + cz = d). Now we can make these values into small tables called matrices.
The coefficient matrix is:
The variable matrix is:
The constant matrix is:
Now we can get an augmented matrix like the following and apply different methods, such as Gaussian elimination, to find the values of x, y, and z:
Even if there are thousands or even millions of equations and variables, we can solve them with a good algorithm and enough time.
Now consider the following graph:
Even this graph can be represented as a matrix. We assume that the order of the vertices is A, B, C, D, E, F, and we get the following matrix:
0 means there is no connection and 1 means there is a connection. For example, column 2 of line 1 is 1 because A and B are connected. Rows 2 and 1 are connected because B and A are connected. If this is a directed graph with an arrow pointing to B on the edge connecting A and B, then it means that only A and B are connected, and B and A are not. In this case, column 1 of row 2 would be 0.
Column 4 of row 1 is 0 because A and D are not connected. Also, column 1 in row 1 is 0 because there is no self-looping from A to A.
If each side has a weight, we can write the weight directly in the matrix instead of 1.
In this way, the matrix can represent anything from graphics to maps to vectors used in artificial intelligence. Even search engines use a matrix of keywords and websites to store whether a particular website has a certain keyword. These matrices are enormous.
These matrices can represent anything, from backlinks used in connections between neurons to different websites. Because matrices can represent connections between neurons, it is used for neural networks in machine learning.
Computing with algebra magically gets faster, and when used in computers, it becomes faster than anything imaginable. That's the real beauty of algebra, making everyday work easier, faster, and more reliable. As Lawrence Fishburne put it in the movie The Matrix, "The Matrix is everywhere, it's a fact." It's all around us." From simple searches to weather forecasts, we were surrounded by it.