تبدیل، تقسیم و دهدهی

سرفصل: بخش ریاضی / سرفصل: حساب و فراکسیون / درس 10

تبدیل، تقسیم و دهدهی

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Conversions: Fractions and Decimals

Conversions between fractions and decimals. Fractions form a group that mathematicians call the rational numbers. Rational because they are ratios. We write a fraction, a rational number, as a decimal. One of two things happens, it either terminates or repeats.

Well those are fancy words, what does that mean? Fractions that form terminating decimals. And of course terminating simply means, coming to an end. So very simply, we write one-fourth as a decimal, it’s 0.25. After two places, the decimal just comes to an end. There’s no more decimal after that.

There are just two decimal places, that’s it. That’s a terminating decimal. Another example, three-eighths, that’s 0.375, so that’s also a terminating decimal. It’s a decimal that just comes to an end. That’s not an approximation, that’s the whole thing right there.

So these are terminating decimals. If it doesn’t terminate, if it goes on forever, then it repeats. Fractions that form repeating decimals. So for example, with one-third, that’s just 0.333 repeating so it’s the same digit repeating over and over again. Technically, to write it correctly, we’d have to write an infinite number of threes, that would be inconvenient.

Instead, we write it with a bar over the last three, and that bar just means continue to repeat this decimal forever. Similarly, one-seventh. This is not a single digit repeating, this is a set of digits repeating, 142857, 142857, 142857. That whole pattern repeats like mathematical wallpaper forever, and the way we indicate that is by writing the long bar over that entire set of repeating digits.

So once again, fractions, when written in decimals, they either are terminating decimals, that means they come to an end, or they are repeating decimals, decimals that repeat a particular pattern. There a few fraction to decimal conversions you should have memorized for efficiency on the test. Many of these are probably already familiar.

Basically, you should be familiar with the decimal form of almost every fraction with a single digit denominator. Now the really easy ones, one half is 0.5, one-fourth is 0.25, three-fourths is 0.75. You may have these memorized already. If you don’t have these memorized, please memorize these. These show up everywhere on the test.

These are very important. Another very popular one, one-third, is 0.3333 repeating. We’ve met this already. If we multiply this by two, we get two-thirds is 0.666 repeating, and of course we round that off, the last 6 becomes a 7. One-fifth is 0.2.

Suppose I just multiply both sides by two, then I would get two-fifths is 0.4. Or I could multiply both sides by three, I would get three-fifths is 0.6, or multiply by four and I get four-fifths is 0.8. Very important to realize that these are just multiples of one another. One-sixth is .1666 repeating.

And of course, if we round that off, we get a final seven. Now, we don’t have to worry about two-sixths because two out of six is one-third. Three-sixths is one half, four-sixths is two-thirds. But the one that does actually show up as a new decimal, five-sixths, is 0.8333 repeating.

That’s a good one to know. Now one-seventh, I’m putting this in a slightly different color, one-seventh. I’m gonna say you probably don’t need to memorize this six-digit pattern repeating over and over again. If you want to memorize this six-digit pattern, go ahead, be my guest. That’s wonderful.

But you don’t need to have it memorized. I would say if you just memorize the first three digits, 0.143. So, that’s just rounded off to the thousandth place. If you know that, that’s an excellent approximation for one-seventh, and that would tell you virtually everything that you would need to know about one-seventh on the test.

One eighth is 0.125. Multiply both sides by three, we get three-eighths is 0.375. Start out with one-eighth, multiply both sides by 5, we get five-eighths is 0.625. Multiply by seven, we get seven-eighths is 0.875. Notice that I wasn’t worried about even numerators, because if I have an even numerator over eight, it would cancel and it would be something else in lowest terms, something that we’ve already handled.

One-ninth, that’s a bunch of ones repeating. Now this is particularly convenient because it means I can multiply by any digit and it would be, just be that digit repeating forever. So for example, if I multiply both sides by seven, seven-ninths is a bunch of sevens repeating. Any digit repeating is that digit over nine.

One-tenth is 0.1. One-hundredth is 0.01. One-thousandth is 0.001. These should be familiar already. But we can use these to get some other fractions. So for example, one-twentieth.

Well, one way to think about that, I’m just gonna multiply numerator and denominator by five. And the reason I’m doing that is because I want to get 100 in the denominator. Five over 100, which of course is 0.05. That’s one way to think about the decimal form of one-twentieth. Another way to think about fractions with denominators that are multiples of ten.

So, one-twentieth. I could write that as one half times one-tenth. Well one half, I can write that as a decimal. That’s 0.5 and one-tenth, of course, is ten to the negative one. Well, if i multiply these, multiplying by ten to the negative one, as we found two videos ago, this slides the decimal place over one place to the left and I get 0.05.

Again, one-fortieth. Well, this is 0.25 times ten to the negative one. Again slide the decimal place over one place to the left. I get 0.025 and that’s the decimal for one-fortieth. One-six hundredth, well of course this one sixth times one hundredth. The one-sixth I know is a decimal.

And one hundredth is ten to the negative two, that means slide the decimal place two places to the left. And so then this is 0.001666 repeating. That is the decimal form of one over 600. With these tips in mind, you should be able to find the decimal equivalent of almost any fraction given on the test.

Finally, I will make a few remarks that aren’t directly tested but may help to clear up any misconceptions. All the decimals in this lesson, so far, either terminated or repeated but that’s not true of all possible decimals. When fractions, aka rational numbers, are written as decimals, they always terminate or repeat.

That part is true. The decimals that neither terminate nor repeat are called the irrational numbers, the numbers that can’t possibly be written as a ratio. These numbers go on forever in a non-repeating pattern. Familiar examples include pi and square-roots of numbers that aren’t perfect squares.

So for example, here are some famous irrational numbers. Of course, pi, the square-root of two, the famous mystical number, the Golden Ratio. All these go on forever and there’s no repeating pattern. None of these can be written as a fraction of integer over integer. In this lesson, we studied the common fraction-to-decimal equivalents, and discussed how to generate more using powers of ten.

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