Essential Math: Types of Numbers
I have been studying math at the Open university for quite some time now and I had my reservations about studying at an academic institution again.
My main concerns were that:
I ended up enrolling and really enjoying it because I love working with numbers. The idea of sitting down in the evening, scribbling numbers down and peeking behind the curtain of the universe is a romantic one. Additionally, plenty of job listings do require a STEM degree. Especially the listings that most interest me. (Quantitative analyst, research scientist, Meta, Amazon, Google)
The duality of being someone who loves working with numbers and is left-brain-dominated whilst simultaneously having a strong emotional reason for studying math isn’t lost on me. Furthermore, I always believe we can do more as humans. If you are waiting for your sign to start something, this is it.
In this article, I aim to summarise the very first unit I learned about. It is all about number. Types of numbers, to be exact.
Let’s start by reviewing some types of numbers.
Natural Numbers.
Natural numbers, known as positive integers are the numbers we can count up with.
[1, 2, 3, 4, …]
The three dots that are after the last natural number is called an ‘ellipsis’ and is used when something is left out. You often see this in Machine Learning books or tutorials.
Integers.
When you incorporate negative numbers and the idea of 0, you now have the type of numbers called, integers. These are whole numbers that can be negative, no value or positive.
If you didn’t already know, the idea of 0 has not always existed in society. A man from India called Brahmagupt ainvented the concepts of 0 sometime in the 7th century.
The Mayans also used 0 as a placeholder in their numerical system but Brahmahgupta incorporated his teachings with equations using 0. It seems like such an obvious concept to us now. But future historians will likely say the same thing about concepts we have not yet discovered, but I digress.
Rational Numbers.
Apologies for the HORRENDOUS formatting here, LinkedIn doesn't render small images well. You can read the article as intended here.
Rational numbers are numbers that can be written in the form of:
int/int
Pay attention here, because this is where you can get confused. If I told you all of the following are rational numbers. Would you understand why?
Don’t worry, I was confused too. If we represent these numbers in a different way, they will all fit our original rule where we must have an integer over an integer.
Suddenly, it all makes sense. Remember being able to approach numbers in a different perspective will serve you well. Write it down, visualise, simplify, expand, convert and whatever else. The end goal is to simply understand the rules that bind numbers, and our universe, together.
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Real numbers.
The real numbers include all rational numbers and many other numbers too. An easy way to visualise this is to imagine a line extending into infinity. Every point on this line represents a real number.
Within this set of real numbers, there are two other groups of numbers. Rational and Irrational. We already know how a rational number can be described.
How do we differentiate irrational numbers?
You simply have to prove that they cannot be written as an integer divided by an integer. I can almost hear some of you roll your eyes at that last statement.
Another way is to find out whether the number is terminating and repeating. If the number terminates after a certain amount of digits or repeats infinitely we know it is rational.
Irrational numbers can go into infinite but are not recurring and do not terminate naturally.
Two examples include: Euler’s number and pi.
These two extend into infinity and do not terminate naturally.
The following is an example of a rational number.
1/8 = .125
Imaginary Numbers and Complex Numbers.
Why would anyone every need imaginary numbers? After all they are, as the name implies, imaginary.
The good news is whilst they do exist. Your exposure to them will be limited within Data Science. You can encounter these types of numbers by taking the square root of a negative number.
I know that can be confusing to understand at first.
Here is a simplified example:
Imagine you have 4 real coins in front of you and 3 imaginary coins in front of you.
It is quite obvious that the total sum of all coins is 7 coins. However, since the sum is made up of some imaginary components. Our sum of 7 coins is now a complex number.
As far as I know, you won’t be needing them within data science unless you end up doing some types of financial modeling.
Since you have learned about Number Theory, here is a venn diagram of all the similarities these numbers share. You won’t find a section for rational numbers because they are real numbers.
In the next tutorial, you will learn about key considerations when working with numbers. Example topics include, BIDMAS, significant digits and rounding up or down.
Even if you think you know these topics, I am sure there will be something in it for you because I felt the same way!
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