数学英语 19 What are Natural Numbers?

by Jason Marshall

Do you remember your very first encounter with numbers? No, I don’t either. But I imagine it might have involved someone pointing to each of my fingers in turn, and with great fanfare 1, counting aloud: “One, Two, Three, Four, Five!” And although I wouldn’t have known it at the time, I was learning about the natural numbers and integers. Not familiar with those terms? Well, they both refer to numbers without fractional parts. Oh, not quite sure about what a fractional part is? No problem. Since fractions are the subset of… Oh. What’s a subset? Or a set, for that matter? All good questions. And to ensure that everybody starts out on an equal footing, we’re going to spend the next few episodes going over some math fundamentals. We’ll start at the beginning and take things nice and slowly—step-by-step. So if you aren’t familiar with the terms I threw around before (and I certainly won’t think less of you if you aren’t), this is your chance to introduce yourself to them. Or, if you do already know them, see this as an opportunity to freshen up your memory just in case things might have gotten a little foggy. Either way, by the time we’re finished, you’ll be well-acquainted with these terms and many more too. So sit back, relax, and get ready for math basic training.

The Beginning of Math

Let’s kick things off by traveling back in time. Way, way back… Nobody knows exactly when numbers first came into use. Bone fragments and other knick-knacks with little notches 3 in them have been found and dated to have been made about 30,000 years ago. People have speculated that the marks on these bones were used to help people count or tally 4 things—perhaps something like keeping track of the number of days that have passed since the last full moon. The age of these artifacts is rather remarkable 5 when you consider that the oldest known human-made pottery 6 dates back to something like 18,000 years ago—that’s about 12,000 years after somebody started doing primitive 7 math with their “tally-bones.” So, in one form or another, basic math has been around for a very long time!

Where Did Numbers Come From?

However, more sophisticated mathematical systems—like the one you’re used to using—weren’t developed until much, much later…about 5,000 years ago, in fact, by the Egyptians. The base-10 (also called decimal) number system that you and I use every day requires only ten digits 9 to represent all the numbers that most people need to worry about in their daily lives. (By the way, it’s no coincidence that digit 8 is also the word for finger. People have been counting digits on their, well, digits nearly forever.) These ten digits are the same ones that you’ve known about since you first started having your fingers counted as a child: 1, 2, 3, 4, 5, 6, 7, 8, 9. But wait—that’s only nine digits, right? What’s the tenth? 10? Nope, that’s actually two digits used together—one…and…yes...also the elusive 10 tenth digit—zero.

All About Zero

Now, zero is an interesting character. It’s had a long and complex history of people getting all riled-up over its meaning and even its very existence. Picture ancient Greeks shouting: “How can nothing be something?” This was all good times for mathematicians 11 and philosophers, no doubt, but eventually people figured out that zero was just plain useful...and necessary. For instance, how many dollars do you have left after giving away your last one? Zero! It’s hard to imagine life without it.

How We Count

Okay, so we’ve established that we all use a base-10 number system to count from 0 to 9. But then what? How do we keep counting? Of course, you’ve known for years that the number after 9 is 10. But have you ever stopped to notice that the number ‘10’ is actually a combination of the digits 1 and 0—it’s not an altogether different symbol. What’s going on here? It’s quite clever really. The value of 10 is made up of one-10 and zero-1s. Similarly, the value of 11 is made up of one-10 and one-1; and the value of, say, 123 is made up of one-100, two-10s, and three-1s. In other words, the farthest digit to the right represents how many 1s, the next digit to the left represents how many 10s, the next one how many 100s, and so on. Each successive place to the left represents one larger multiple of ten. Not only is it clever, it’s also incredibly efficient compared to the ancient system of carving 12 marks on scraps 13 of bone.

To illustrate 14 exactly what I mean by being more efficient, think for a few moments about how the ancient notch 2-carving people counted. It’s easy enough at first, sure, you just keep making those little notches. But what happens after you’ve made hundreds of notches, and the time comes for you to figure out exactly how many you have in total. What do you do? Well, you have no choice but to count every single one of them. There aren’t any shortcuts 15. Now, imagine that your tally-bone carving buddies 16 all brought over their tally-bones, and you had to figure out the total number of notches that everyone has carved altogether. How would that work? Not well. You have to lay all the bones out and count every single notch individually. It’s a very long and tiring process, and one that is severely 17 error prone 18 since your eyes are almost certain to bug-out long before you obtain your total. Not good.

Now, jump forward 30,000 years. Imagine you’re given the task of counting the people coming through the turnstiles at a baseball stadium. Wisely, you station one person at each turnstile, each of whom keeps track of the number of people coming through their entrance. After the first person passes, they write down 1. After the second, they cross out the 1 and write 2. Next they cross out 2 and write 3…and so on, and on, and on…until after a while, they’ve counted lots and lots of people who’ve passed through their turnstile. Now, if you asked them how many people they’d counted, they’d be able to tell you right away because they wouldn’t have to waste time counting notches. Furthermore, if you got all the turnstile workers together to figure out the total attendance at the game, all you’d have to do is gather each person’s individual total and do one addition problem—a far more efficient and accurate technique than counting notches.

What are Natural Numbers?

Alright, jargon 19 alert time. The numbers we’ve been talking about are known as the “natural numbers.” 0, 1, 2, 3, 4, 5, and so on, up to as big a number as you can think of, and then even bigger. (Is zero really a natural number? See "Can a Math Problem Have More Than One Right Answer?") Begin with zero and add one to the result, and you get the next natural number; repeat the process to get the next; repeat the process to get the next; repeat the process to get… Okay, you get the idea. There are an infinite number of them. Give me the biggest one you can think of, and I’ll add one to it and give you an even bigger one. You can add, subtract, multiply, and divide them to solve lots of problems. Doing arithmetic like this is something I’m going to assume you’re relatively 20 comfortable with, although we do have a few episodes coming up discussing how to do these types of calculations more efficiently 21. You also use natural numbers to order things—1st, 2nd, 3rd, and so on. Natural numbers are truly the foundation for the rest of math.

Natural Numbers Test-Taking Tip

One more thing. If, by chance, you’re preparing to take a test like the SAT, you’ll be seeing lots of problems based entirely 22 on doing arithmetic with natural numbers. The math isn’t all that complicated, but sometimes the problems can be presented in ways that make them seem tougher than they are. Just to give your brain a little workout, here’s an example of something you might see: If two books are checked-out from the library every minute, and one is returned every five minutes, how many fewer books are in the library after 20 minutes? It sounds like a mouthful, I know. But how do you go about solving it? Well, first off, if you find yourself overcome by feelings of despair, here’s a quick and dirty tip for you: After hearing a word problem like this, the first thing you should do is take a deep breath, regain 23 your confidence, and re-read the problem…s l o w l y…making sure you understand exactly what is being asked.

So, let’s slow down and read it again: If two books are checked-out from the library every minute, that’s two checked-out each minute, and one is returned every five minutes…that’s one returned every five minutes, then how many fewer books are in the library after 20 minutes?” If it seems hard, don’t worry—stick with me and things will get easier. And if it seems easy, then pat yourself on the back and relax knowing that you’re well prepared for the more difficult subjects we’re going to cover once we finish up basic training. In the meantime, think about the problem, give it a shot, and check out the next episode to see if you get the right answer. We’ll also be continuing our basic training talking about integers.

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