数学英语 16 What are Decimals?

by Jason Marshall

Today’s article is about decimal points and numbers and how they’re related to fractions.

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Roman Numerals

When we first talked about the Fibonacci sequence a few articles ago, I mentioned that Fibonacci’s famous book introduced Europe to the 0 through 9 Arabic numeral system that we still use today. And we’re all fortunate for that because it’s very convenient to work with. To illustrate 1 this, let’s briefly 2 talk about one alternative: the Roman numeral system. In this system, numbers are represented by letters: “I” = 1, “V” = 5, “X” = 10, “L” = 50, “C” = 100, “D” = 500, and “M” = 1000. To represent a number like 3, you write “III”; or to write 15, you write “XV”. We won’t go through all the details of how the system works—you can read about it on Wikipedia if you like—but the important thing to take away is that though this system is certainly a reasonable way to write and record numbers, it’s terrible for doing arithmetic. There’s just not a good quick systematic 3 way to add a series of numbers like “XV + X + III + V”—and just imagine how hard it’d be to do division!

Decimal Integers

Herein lies the beauty of the 0 through 9 number system—also known as the decimal system. (A quick aside: The word “decimal” here is derived 4 from the same root word as “decade.” So just like there are 10 years in a decade, there are also 10 integer numerals—0 through 9—in the decimal system.) In contrast to Roman numerals, addition, subtraction 5, division, and many other things are all relatively 6 easy with decimal numbers. This system allows us to write numbers using positional notation 7—that is, where the number of ones is represented by the numeral in the first digit 8, the number of tens by the numeral just to the left of the first digit, hundreds by the next to the left, then thousands, and so on. For example, with the number 2,573, the 3 is in the ones place, the 7 is in the tens place, the 5 is in the hundreds place, and the 2 is in the thousands place. It’s a much simpler system.

Decimal Fractions

Now, here comes the leap of inspiration. Let’s try sticking to this positional notation scheme where each digit represents a number that’s ten times smaller than the one to its left, and let’s extend it out to the right—beyond the ones place. Well, that sounds great, but we’re going to have a problem: how will we know what power of ten the digit on the far right represents—we won’t just be able to assume it’s the ones column anymore. So, let’s introduce a bit of additional notation: the decimal separator—more commonly known as the decimal point in the US. Take the number 1.0 for example. The decimal point is written as a period. This is the common practice in the US, but there are many places around the world where a comma is used instead—so beware! The digit to the immediate 9 left of the decimal point is, as always, the ones column, and the first digit to the right of the decimal point is what’s called the “tenths column.” The tenths column is aptly named since it represents the number of 1/10s. And the digit to the right of that, the “hundredths column,” must therefore represent the number of 1/100s. In other words, the number 1/10 is written 0.1 and 3/10 is written 0.3. How about 11/100? That’s 0.11, which means eleven one-hundredths. That’s equivalent to 1/10 plus one 1/100.

These are called decimal fractions—they’re just like normal fractions except the denominator is always a power of ten. And this is great because it’s just a natural extension of the decimal system that we already know and love from the world of integers into the world of fractions. Pretty clever, right? But, you might be wondering why we need to have a second way to write fractions? Isn’t that redundant 10 and doesn’t it just confuse things? Well, the answer takes us back to the Roman numeral system we talked about before. Similar to the way in which arithmetic is much easier with decimal integers rather than Roman numerals, in many situations it’s much easier to solve problems using the decimal fractions we’ve been talking about rather than traditional numerator and denominator fractions. Not always, but frequently—especially when it comes to solving problems numerically on computers.

How to Add and Subtract Decimals

One more thing before we finish up for today. Since decimal fractions are really just an extension of the decimal integer system we’re accustomed to, everything you already know about working with integers also works with decimal fractions. So all the tricks we’ve talked about for speeding up addition with integers such as finding pairs of number that add to 10 and working from left-to-right instead of right-to-left still work! To test your skills, give the practice problems at the end of this article a try and then check out this week’s Math Dude “Video Extra!” episode on YouTube for an explanation.

Wrap Up

So that’s the origin and meaning of the decimal point and decimal numbers. I know this might not be earth-shattering new knowledge for many of you, but it pays to explore and understand the true meaning of things—in other words, where do these things we use every day really come from? Because once you understand that, it’s much easier to extend that knowledge into other realms. Which is exactly what we’ll start doing in next week’s article.

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!