Numbers, Really!
This article is a recap & extension of my discussion with my 2 elementary-grade kids in the car while driving back home from I don’t know where, a couple years back. I think the conversation started with one of them saying something like “Dad, we just learned that there are negative numbers.” I started reminiscing and must have recounted about half of the following to them on the 15 mins drive.
I learned my 123s when I was two, probably. Along with the ABCs, being able to recite 123s flawlessly used to be “a thing”, a measure of smartness. I bet it’s the same for the parents of most toddlers around the world even today.
I grew up. I learned that 123s extend beyond the venerable number 10 — the first 2 digit number! I actually vividly remember myself being excited when I was first able to narrate the numbers all the way through the first 3 digit number, 100, which I was convinced was the biggest number in the universe. I was wrong by a margin, a very wide margin. Let’s just say, I was wrong by an infinite margin. I also vividly remember my shock at learning about infinity. There is such a thing. Try dividing any Natural Number by 0; I was taught that the result is infinity.
Wait, what is a Natural Number? I learned that the set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…to infinity, was called Natural Numbers. Made sense. Discovery. Euphoria. But wait, where does 0 fit in this grand scheme of things? I was big enough to realize that this definition of Natural Numbers left out the very interesting number 0. I was taught by then that 0 is a number. I grew up in India, so it was natural (no pun intended!) that I was also taught that 0 was discovered by Aryabhata in 499 CE, a fact every Indian wears with pride ever since this revelation was widely disseminated sometime in the 20th century. Aryabhata (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.
So, I learned about Whole Numbers, which was very conveniently just the set of Natural Numbers and 0 together. Phew! Mastered an entire section of elementary school mathematics in a jiffy. But of course, that was but a start of a lifelong number journey that continues to this date.
I learned that a Natural Number can be Positive or Negative. I was taught to draw a straight line, mark its center and call it 0. Then divide the line into equal parts on either side of 0 and mark each partition on the right of 0 with +1, +2, +3,…,with the partition closest to 0 being +1 and increasing as I moved farther away from 0. And then I was taught to repeat the same exercise for each partition on the left of 0 with -1, -2, -3,…Conveniently, the ‘+’ was supposed to indicate Positive while the ‘-’ indicated Negative. So I learned Positive Numbers and Negative Numbers. 0 was once again in the middle of everything; it just can’t help itself being the center of attraction. I am glad I had learned Whole Numbers previously.
The collection of all Positive Numbers, Negative Numbers and 0 is called Integers, I was told. I was still at an age when new terminologies excited me. Each new term was a revelation. So adding Positive Integer and Negative Integer to my vocabulary was not a big deal. What did annoy me, I distinctly recollect, was the fact that I had to remember to add the ‘+’ sign in front of those Positive Numbers when showing my work in arithmetic assignments. By then, my brain had been trained to write a 2 as a 2 and not +2. Minor inconvenience, I thought, for the sake of getting good grades. I was convinced that I had mastered the entire number line. I could now go from -infinity to +infinity, stopping in between to kiss my favorite number, 0. To infinity and beyond, literally.
And then I grew up some more. Somebody had to burst my bubble. They duly did when one day they asked me what came between 0 and 1. “Nothing. There’s nothing between 0 and 1 on my number line, and my number line is pure,” were precisely my thoughts. “How about half a slice of bread for breakfast?” The bubble had burst. There is something between 0 and 1 after all. It is called half. So, I was introduced to Fractions. It turned out that there are many, many, many points between 0 and 1 on my number line. Actually, there are infinite points between 0 and 1 on my number line. A Fraction is just a ratio between an Integer and a non-0 Integer., like 1/2, 2/3, 99999999/167, etc. When the top number (called Numerator) is smaller than the bottom number (called Denominator) in a Fraction, it is called a Proper Fraction, e.g., 1/2 and 2/3. Conversely, if the bottom number is smaller in a Fraction, it is called an Improper Fraction, e.g., 99999999/167. But why this discrimination against my beloved 0? Because, dividing anything by 0 results in infinity. It’s like an automatic disqualification. Did I say it earlier that 0 is a very special number?
Fractions can be positive or negative, depending on the Integers on the top and bottom of a Fraction. If both of them are positive or negative, it becomes a Positive Fraction but becomes a Negative Fraction if the top and bottom Integers carry opposite signage, that is, one of them is positive while the other is negative.
It would have been fine if the story ended there. But one fine day, they told me that all Fractions are called Rational Numbers. So, a Rational Number is a number that can be expressed as the Fraction p/q of two Integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every Integer is a Rational Number, but the converse is not true; there are rational numbers that are not integers, e.g., 1/2 and 2/3. Just when I had mastered Fractions and Rational Numbers, came Decimal Numbers, which turned out to be just another way of representing the Fractions. So 1/2 became 0.5, 1/4 became 0.25 and 1/3 became 0.33333…
I remember how Decimal Number arithmetic was a pain to master. Addition and subtraction were relatively easy; I had to just remember to place the decimal position correctly before doing the math. I will say that my not-so-beautiful handwriting did not help at all. I would often not be able to read my own handwriting and make errors in ‘carrying’ or ‘borrowing’ digits. But that was nothing compared to the multiplication and especially, division arithmetic. Ugh! How much I hated Decimal Number division. It was all about doing the multiplication and division ignoring the decimals and then later inserting the all-important “.” at the right place. But I did master it ultimately.
But I knew something was afoot; if there are Rational Numbers, there must be something opposite of it. My logic was simple. When they introduced Positive Integers, Negative Integers followed. Proper Fractions was followed by Improper Fractions. It turned out I was correct — there are Irrational Numbers. Rational Numbers are either completely divisible, for example, 4/2(=2) or 1/2(=0.5), or if not completely divisible, their decimal expansion has a repeating number, for example, 1/3(=0.33333…) or 2/3(=0.66666…). The decimal expansion of an Irrational Number on the contrary, continues without repeating. The most famous Irrational Number is π (Pi), which can be written as 22/7 as a Fraction or as a decimal as shown below. Remember, π is an Irrational Number, which means the numbers after the decimal never repeat and never end; they go on forever. Quite irrational behavior, alright.
There are several other important Irrational Numbers in mathematics, some of which pop up more often than others, for example, e (2.71828…also known as Euler’s number is the base of natural logarithm), and φ (1.61803…also known as Golden Ratio). Together, these irrational numbers form a clique known as Mathematical Constants and are some of the most important numbers that find applications in mathematically describing the universe all around us. This is a topic of another, future essay.
Rational and Irrational Numbers are together called Real Numbers. Thankfully, I never came across Unreal Numbers but there are Imaginary Numbers and Complex Numbers. I think I was in 8th grade when I learned about these august number types. Imaginary Numbers are those that equal the product of a Real Number and the square root of -1, which is also known as the ‘unit’ Imaginary Number and commonly represented simply by the symbol i. And when you add an Imaginary Number with Real Numbers, they become Complex Numbers!
I could fit all the other numbers I had learned so far on the number line (yes, even the Mathematical Constants) but where do Imaginary Numbers go on the number line? The short answer is: They don’t. Imaginary Numbers are not that imaginary but do require a different way of thinking to visualize them. Turns out that we need to start thinking in terms of a 2-dimensional ‘Complex Plane’. Specifically, Imaginary numbers are an extension of the Real Numbers, only along the 2nd dimension. So we represent them by drawing a vertical imaginary number line through zero. These two number lines together make the complex plane.
Here is a great easy-to-read article on Imaginary and Complex Numbers that explain why Imaginary and Complex Numbers make sense and are not some bogus contraptions.
And finally, I save the best for the last: Prime Numbers. I learned about Prime Numbers in my elementary school right about the time we were being taught division, fractions, factors and multiples. There were these very interesting numbers, like 2, 3, 5, 7, 11, 13, 17, etc., that could not be divided by any number other than 1 or themselves. Formally, a Natural Number is called a Prime Number (or simply a Prime) if it is greater than 1 and cannot be written as the product of two smaller Natural Numbers. Besides being a teacher favorite for math exams and pop quizzes, it turns out that Prime Numbers are critical for many real-life applications, most famous of which is probably their use in Cryptography — the science of encoding information using secret keys. Cryptography techniques rely on the difficulty of factoring large numbers into their prime factors, which is an incredibly hard problem to solve because there are infinitely many Primes with no known simple formula that separates Prime Numbers from Non-Prime Numbers (also known as Composite Numbers.) In other words, it is not straightforward to find what is the next Prime number after 99,991; 99,991 happens to be the largest 5 digit Prime Number. What’s the next Prime Number along the number line?
Sophisticated computers have recently been used to determine the largest Prime Number. Do you want to know what it is? As of January 2020, it is 282,589,933 − 1, a number which has 24,862,048 digits!
