Let’s continue on our journey of learning the language of the Universe so we can converse with it! This article will talk all about the numbers that make up the Universe, which range from very big to teeny tiny. For instance, did you know that our Solar System is 4.5 billion years old, or put another way, 4,500,000,000 years old? Or that the average distance between our planet Earth and Jupiter is 715 million kilometers, or 715,000,000 km? Or that the mass of an electron in an atom is 9.1 x 10ˉ³¹ kg (0.0000000000000000000000000000091 kg) while the mass of the Sun is 1.989 x 10³¹ kg (19890000000000000000000000000000 kg)? All of those numbers have one thing in common — lots of zeros. So now that we know what a number means in a computer program and what numbers mean in a mathematical sense, it’s time to learn how to express the numbers of the Universe! Now, this is probably a pretty basic refresher for those of you up-to-date with your math skills, but it’s been a hot minute since I’ve taken a math class, so away we go!
Let’s start with the very basics- the place value of the number itself. Each part of a number gets its own name. There is the ones place, the tens places, the hundreds place, the thousands place, and so on. And it works in reverse, too! For numbers with decimals, there is the tenths place, the hundredths place, the thousandths place, and so on (the decimal does not have a oneths place, it just starts at the tenths place). Now, you might have noticed something unique about these numbers. Perhaps it might be easier to see if I convert the word to a number: the 1s place, the 10s place, the 100s place, the 1000s place. See it now?
Take 1 and multiply by 10, what do you get? 10! Take 10 and multiply by 10 and you get? 100! Take 100 and multiply by 10 and you get 1000! See the pattern now? This is known as the base-10 number system, or decimal system. Now, this is an important concept to understand. Because these numbering systems also apply to computers, but not this exact numbering system. Computers use what is known as a base-2 number system. However, we will come to that further down the article. For now, just recognize that our number system has a base of 10 and uses the numbers 0-9. This is a different way of thinking than the previous article. While the previous article covered number sets, this article covers number systems.
Let’s visualize place value using that distance from Jupiter to Earth:
The place value for 715,000,000. Image credit: Alexandra Nachman
When taking that 715,000,000 km average distance and putting it into a place value table, it is easy to see just what each value represents. Fun fact: this is also how we give numbers names (and write them out with letters)! We usually pronounce a number beginning with the leftmost digit. So with 7 being in the hundred millions place, we would call the number seven hundred fifteen million. Basically how you would say 715, but with a million added on at the end. Now, there are also other ways to write numbers as well such as 715 million or 7.15x10⁸ . But working out how to say numbers based on their place value is a nifty way to spell things out!
Now, we are very familiar with this base ten system, having used it our whole lives. However, there is another number system we must understand, at least, if you want to understand how a computer works. And that is the base-2 number system, or binary system, I mentioned earlier. The base two number system only uses two numbers, a 1 and a 0. This is because a computer only uses 1s and 0s to operate. These 1s and 0s are known as bits. A 1 allows electricity to flow while a 0 does not hence why 1s and 0s are used. For programming languages, unless you plan to go into a career that works with machine or assembly languages, that’s all you really need to know about the binary system. However, this will most likely get covered in a future article.
But what about all those really large or really small numbers with lots of zeros that we talked about earlier? What does a number like 9.1 x 10ˉ³¹ or 1.989 x 10³¹ mean? Writing 9.1 x 10³¹ kg is the same as writing 0.0000000000000000000000000000091 kg and writing 1.989 x 10³¹ kg is the same as writing 19890000000000000000000000000000 kg. Writing the numbers as 9.1 x 10ˉ³¹ kg and 1.989 x 10³¹ kg is just a much simpler way of writing the number. It gets rid of all of those pesky zeros and makes it much easier to read and interpret. This method of expressing numbers is known as scientific notation.
Quick segue into exponents- that’s what the -31 and 31 to the upper right of the 10 are. When a number is raised to an exponent, it means to multiply the number by itself that many times. 10² is 10 x 10 which equals 100. 10³ is 10 x 10 x 10 which equals 1000. If 10² is 10 x 10 then 10¹ is just 10. But what is 10⁰? The answer is 1. Any nonzero number that is raised to the power of 0 is always 1. What? How does that make sense? Well, unfortunately the answer to that is a bit too in depth for this article, so I encourage you to Google it. Just know that anything to the power of 0 will always be 1. Alright, back to the normal exponents. Can you see the pattern for larger and larger exponents? 10⁴ is 10 x 10 x 10 x 10 = 10000, 10⁵ is 10 x 10 x 10 x 10 x 10 = 100000, and so on. This is true for any number, whether you are finding 2³ (2 x 2 x 2 = 8) or 4⁵ (4 x 4 x 4 x 4 x 4 = 1024). So when we have a number like 10³¹, we write 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1000000000000000000000000000000. And that is no fun to write out all of those numbers and zeros. Especially if we miscount our numbers! So we make it shorthand by writing 10³¹.
Whew! That was a lot to learn about exponents! Now back to scientific notation. Scientific notation is usually written with a number that is in the ones place and everything else trailing after it. This number must can only go up to 9 (we can technically write numbers "incorrectly" such as 500 x 10⁴ to represent 5,000,000 but the preferred method is to write it so that the number is 9 or lower. So 5,000,000 would be properly represented as 5 x 10⁶). So, let’s work with a smaller number to convert to scientific notation. Say that 715,000,000 km average distance from Earth to Jupiter we keep talking about. Imagine that there is always a decimal at the end of a number, even one that might not be typed out. So 715,000,000 is really 715,000,000.0 even though we often don't type out the decimal. Then take the decimal point and move it to the left 8 times. This places the decimal in 7.15. Then add x10 and add the number 8 as the exponent. So it would be written as 7.15x10⁸ . Let’s visualize this:
This image shows how to convert a positive number into scientific notation. Image credit: Alexandra Nachman
This works for all positive numbers. Even those very large ones with lots of zeros. Just don’t make any errors counting the number of zeros! Now, you might be thinking- what about those negative numbers? Well, the opposite is true. Instead of moving the decimal to the left, you move the decimal to the right. See here:
This image shows how to convert a positive number into scientific notation. Image credit: Alexandra Nachman
And instead of writing a positive exponent, the exponent will be negative. So positive numbers get a positive exponent and negative numbers get a negative exponent. That’s easy enough to remember! So 715,000,000.0 is the same as 7.15 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 is the same as 7.15x10⁸ (don’t believe me? Get out a calculator and try multiplying it yourself!). And that is how we turn very large numbers or very small numbers with lots of zeros into something we can easily read and understand. So let’s recap what we learned!
There are two important number systems to know and understand (there are more than just these two, but let’s keep it easy for now). There is the base-10 number system, which consists of all the numbers we use everyday- from 0 to 9. Then there is the base-2 number system, which consists of just two numbers, 1 and 0, that a computer uses to operate (I will do a whole separate article on binary numbers as well-so keep an eye out!). Each number has a place value ranging from the ones, tens, hundreds, and so on as it gets larger and to the tenths, hundredths, thousandths, and so on as it gets smaller. Because numbers concerning our Universe can be very large or very small, with lots and lots of zeros, we often use scientific notation to write these numbers. Phew. That was a lot to learn about number systems and how to express very large and very small numbers!
Now one side note before I end this article- as I am working my way up the math and computer ladder and slowly building up concepts from both to understand how to converse with the Universe, there are some things that I will not cover. These are things such as addition, subtraction, multiplication, division, rounding numbers, and so on that are usually covered when starting with math basics. They will only be covered in relation to computer programming- how to represent these calculations as well as how numbers are rounded and can be expressed in a program. But for anyone looking to know more about basic math, Khan Academy has excellent lessons on how to do all of that for those of you who might want to teach younger folks (or those who are older- I don’t judge). I hope you enjoyed learning all about number systems. We are now one step closer to being able to converse with the Universe!
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