From Pebbles to Pi !

I’ve always wondered how humans started using numbers. It’s pretty amazing to think that something as simple as counting or solving mathematics problems today actually started from nothing. Early humans didn’t have numbers like we do now. Instead, they used tally marks to count things like how many sheep they owned or how many belongings they had. They seem to have used simple objects like pebbles to represent tally marks. For example, they could have placed a pebble for each of the cattle that returned back after a day of grazing. This way, they could easily track if all them were back safely. Counting these pebbles was a basic but effective way to keep track of important things like livestock, which was crucial for their survival and daily life. Over time, different civilisations came up with their own number systems. 

It all seem to have started with the natural numbers—the numbers we use to count things like 1, 2, 3… and so on. But soon, people realised they needed numbers that represented nothing or something missing. This is where the whole numbers came in, which included zero. 

One of the most important breakthroughs occurred in India, where the concept of zero was introduced. It’s crazy to think that something as simple as zero, which we use all the time now, was once a huge discovery! You know, it kind of reminds me of when a teacher calls a student a "big zero" as a joke. At first, the student might feel a little down, but if he takes it positively, it could actually be the start of something great! Over time, that same student could end up doing really well in life—maybe even completely changing once they discover the world of learning, dive into their studies, and start realising how much they’re capable of. Who would have thought that starting as a "big zero" could lead to becoming a hero in their own story?

Anyways, from there, the number system evolved even further. Then came integers, which added negative numbers to the mix, allowing us to represent not just what we have, but also what we’ve lost or owe. This was important for tracking situations like debts, temperature drops below zero, or even losses in resources, helping people make sense of both positive and negative changes in their everyday lives.

After that, humans invented fractions to represent parts of things, like half a loaf of bread or a quarter of an apple. This led to the development of decimals for even more precision. While fractions helped us represent parts of a whole, decimals were introduced to make these parts easier to work with, especially in everyday situations. Decimals provide a more convenient and precise way to express numbers.

After the discovery of natural numbers, integers, fractions, and decimals, there arose a need for a more general way to express numbers that could represent both parts of a whole and quantities that couldn’t fit into the simple categories of whole numbers or decimals. This led to the development of rational numbers.

Rational numbers are numbers that can be expressed as the fraction of two integers, where the denominator is not zero. For example, ½, ¾, or even 5/3 are rational numbers because they can be written as fractions. These numbers include both proper fractions (like ½), improper fractions (like 7/3), and whole numbers (since a whole number can also be written as a fraction, like 4 = 4/1).

The need for negative fractions also emerged. For instance, if you owe ½ of a dollar to a friend, it’s represented as -½. Negative fractions are useful for situations like debts or temperatures below freezing, helping to represent things that are "less than nothing."

An interesting aspect of rational numbers is that repeating decimals are also considered fractions! For example, the decimal 0.333..., which repeats forever, is equivalent to the fraction 1/3. Similarly, 0.666... is equal to 2/3. This shows that even decimals that go on forever can still be expressed as rational numbers because they can be written as fractions of two integers.

This evolution of rational numbers allowed for a much more flexible way of describing numbers in real-world situations.

But as we humans explored numbers more, we discovered something even stranger: irrational numbers. These numbers can’t be written as fractions of two integers, and they go on forever without repeating. An example is π (pi), which is roughly 3.14159 but continues infinitely. We also have numbers like the square root of 2, which is another example of irrational numbers.

So, why do we even need irrational numbers? Well, when mathematicians tried to measure things like the circumference of a circle (using π) or the diagonal of a square (using √2), they realised that these numbers couldn’t be captured with simple fractions. These strange, never-ending decimals appeared in nature and geometry, and we needed a way to represent them. It’s fascinating to think how the number system had to evolve to account for these infinite numbers that don't fit into the neat categories of fractions or decimals. 

All these rational and irrational numbers together make up what we call real numbers. Whether we’re counting money, measuring something, or calculating distances, real numbers are everywhere!

And then comes the intriguing concept of imaginary numbers. These are numbers that cannot be represented on the number line we commonly use. The most famous of these is i, which is the square root of -1. Now, think about this for a second: When we multiply two positive numbers, we always get a positive result, right? But what happens if we multiply a positive number by a negative number? We get a negative result. But what if we multiply two negative numbers? We get a positive result! So, how can we get a negative number when we take the square of something? That’s where i comes in—it allows us to square a number and still end up with a negative result. Isn't that mind-blowing?

Imaginary numbers are still mysterious to many people, (including me) and they might make you wonder: Are these numbers really imaginary, or do they exist in some way in reality? As their name suggests, imaginary numbers seem almost impossible, but my uncle and dad who learnt from him say they do exist and they are incredibly useful in solving problems in the real world, even though they don’t seem to fit the traditional number system. 

Being a student of class 8, I feel curious to know more about them but my father says it would be too early for me to explore it in depth. I'll share my experience later once I become more familiar with them in my upcoming classes. Thanks.

- Ayaan (with guidance of my uncle and dad)

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