## Fraction Properties - I

### Transcript

In this video, we'll discuss some of the basic Fraction Properties. So we're starting out with some very basic properties here. First of all, fractions with the denominator of 1. If we take any integer and put it above 1, then we get something very simple, it just equals that integer. So, 7/1 = 7 or 135/1 = 135.

In general the, pattern is n/1 = n. Now it's very important to appreciate. That first of all, we can use this to write an integer as a fraction. So we had 4 we could write 4 as a fraction, we could write it as 4/1, we could write any integer n as the fraction n/1. Well why is this important?

This is not tested directly, the test is not gonna ask you what does 135/1 equal? That's not gonna be a question, but this is an important step in problem solving. For example, later on, when we talk about adding fractions, we talked about how to add a fraction to a number, to a whole number. Well, it turns out that is very useful in that scenario, to rewrite the whole number as a fraction.

And this is how we'll write any integer as a fraction, just by putting it /1. So 4 we can write it as a fraction, writing it as 4 / 1. That's an important problem solving step that we'll need later. Second property, fractions involving zero. This is a tricky issue. People sometimes get confused here.

So let's be clear. First of all, we cannot divide by zero ever. It breaks the mathematical law to divide by zero. So in other words to have a fraction was zero in the denominator, that actually breaks the mathematical law. That creates something that goes outside the norm, where we are able to assign mathematical meaning.

So it is in the most profound sense, something mathematically meaningless. So the test is not gonna make you deal with things that are mathematically meaningless. So don't worry, you're not gonna be asked to evaluate a fraction with zero in the denominator. You just have to remember that it's an illegal thing, that is something you have to keep an eye out for.

If that's a possibility, things no longer make sense. Now having said all that, it's perfectly allowable to divide zero by number. We can have zero in a numerator, that's perfectly fine. 0/5 = 0, 0/-3 = 0. Those are 100% valid equations, we can write the equal sign. That is well within the realm of mathematics there, 0 divided by any non zero number equals 0.

Very important, 0 can never be in the denominator. But it's perfectly fine in the numerator. Fractions of the form n/n, so what's going on here? As long as n is not equal to 0, then n / n = 1. So here, n could be an integer, but it could be any other kind of number. It could be positive, it could be negative, it could be a decimal, it could be an ugly radical, it could be any kind of thing as long as we put anything over itself.

As long as it doesn't equal zero, then anything over itself equals 1. The big idea here, is we could always multiply any expression by n/n. Because multiplying by one never changes the value of anything. And this will be hugely important, when we get to the point where we have to find common denominators. So this is a really important fundamental idea, and we'll return to this later.

Finally the issue of reciprocals, this is actually a word you need to know. The test could use the word reciprocal in a problem, so you need to recognize this word. The reciprocal of a fraction a/b, is the flipped over fraction b/a. Again, we're assuming that both a b are unequal to zero. So for example, the reciprocal of 3/5 is 5/3, the reciprocal of -7/2 is -2/7.

The reciprocal of 1/6 is 6/1, while 6/1 as we saw a moment ago just equals 6. It equals the integer 6. So the reciprocal of 1/6, the fraction 1/6 is the integer 6. So there are few important properties of reciprocals that it's important to know. First of all, the product of any fraction with its reciprocal equals 1, so they cancel out.

So 4/17 x 17/4 = 1. That's a really important idea, and we're gonna use this to solve a problem in a few minutes. Second, the reciprocal of any positive integer is simply one divided by that integer. So the reciprocal of 6 is 1/6, the reciprocal of 117 is 1/117.

In general, the pattern is the reciprocal of n is 1 / n. Now certainly this works when n is an integer, but it turns out n doesn't even have to be an integer, we can extend the pattern. And we can use that same pattern when n is not an integer. So one divided by any number, for example, one divided by a fraction equals the reciprocal that fraction.

So if I take 1 / 3/7, that's going to equal the reciprocal 3/7, which is 7/3. So that's a really deep idea there. This already suggests the rule for fraction division, which we'll discuss in an upcoming video. So if you really understand this connection well, you will understand fraction division well.

Finally, if a number is bigger than 1, then its reciprocal is smaller, between 0 and 1. If a number is between 0 and 1, then its reciprocal is larger than 1. Very important to have this fundamental number sense, that how big a number is how big its reciprocal is going to be. Very important to have a keen sense of how big and small numbers are.

So here's a practice problem, I'll recommend, pause the video here and then we'll talk about this problem. Okay, the reciprocal of a positive number, times the cube the same number equals 5. What is the number? Well, I'm gonna represent this unknown number by x. I'm gonna use a little algebra here.

So x is the number. The reciprocal is 1 / x. And of course, the cube is x cubed. So we're saying that (1 / x) * ( x cubed) =5. That's our fundamental equation. Well that x cubed, literally that means x times x times x, I'm just gonna write that out.

Now I'm gonna take advantage of the first property of reciprocals. Namely, a number times its reciprocal cancels out. A number of times its reciprocal is 1, so (1 / x) * x together, that's just gonna equal 1. So they cancel and I'm just left with x*x. So in other words, 5 = x squared.

Now, normally we'd have to consider both the positive and the negative square root, but the problem tells us that we're dealing with a positive number. So because we're giving that guarantee, we only have to consider the positive root. And that means that x = positive square root of 5. In summary, any fraction of the form n / 1 = n. So in particular, this allows us to write any integer as a fraction, and that turns out to be useful in a variety of circumstances.

Fractions with zero, we talked about these zero can be in the numerator zero divided by zero, but zero can never be in the denominator of a fraction. That's very important. We talked about the rule, n / n = 1, and again how this will be very important. As a multiplier, we can always multiply by n / n because when we're multiplying by that, we're multiplying by 1.

And finally, we talked about the properties of reciprocals.

Read full transcriptIn general the, pattern is n/1 = n. Now it's very important to appreciate. That first of all, we can use this to write an integer as a fraction. So we had 4 we could write 4 as a fraction, we could write it as 4/1, we could write any integer n as the fraction n/1. Well why is this important?

This is not tested directly, the test is not gonna ask you what does 135/1 equal? That's not gonna be a question, but this is an important step in problem solving. For example, later on, when we talk about adding fractions, we talked about how to add a fraction to a number, to a whole number. Well, it turns out that is very useful in that scenario, to rewrite the whole number as a fraction.

And this is how we'll write any integer as a fraction, just by putting it /1. So 4 we can write it as a fraction, writing it as 4 / 1. That's an important problem solving step that we'll need later. Second property, fractions involving zero. This is a tricky issue. People sometimes get confused here.

So let's be clear. First of all, we cannot divide by zero ever. It breaks the mathematical law to divide by zero. So in other words to have a fraction was zero in the denominator, that actually breaks the mathematical law. That creates something that goes outside the norm, where we are able to assign mathematical meaning.

So it is in the most profound sense, something mathematically meaningless. So the test is not gonna make you deal with things that are mathematically meaningless. So don't worry, you're not gonna be asked to evaluate a fraction with zero in the denominator. You just have to remember that it's an illegal thing, that is something you have to keep an eye out for.

If that's a possibility, things no longer make sense. Now having said all that, it's perfectly allowable to divide zero by number. We can have zero in a numerator, that's perfectly fine. 0/5 = 0, 0/-3 = 0. Those are 100% valid equations, we can write the equal sign. That is well within the realm of mathematics there, 0 divided by any non zero number equals 0.

Very important, 0 can never be in the denominator. But it's perfectly fine in the numerator. Fractions of the form n/n, so what's going on here? As long as n is not equal to 0, then n / n = 1. So here, n could be an integer, but it could be any other kind of number. It could be positive, it could be negative, it could be a decimal, it could be an ugly radical, it could be any kind of thing as long as we put anything over itself.

As long as it doesn't equal zero, then anything over itself equals 1. The big idea here, is we could always multiply any expression by n/n. Because multiplying by one never changes the value of anything. And this will be hugely important, when we get to the point where we have to find common denominators. So this is a really important fundamental idea, and we'll return to this later.

Finally the issue of reciprocals, this is actually a word you need to know. The test could use the word reciprocal in a problem, so you need to recognize this word. The reciprocal of a fraction a/b, is the flipped over fraction b/a. Again, we're assuming that both a b are unequal to zero. So for example, the reciprocal of 3/5 is 5/3, the reciprocal of -7/2 is -2/7.

The reciprocal of 1/6 is 6/1, while 6/1 as we saw a moment ago just equals 6. It equals the integer 6. So the reciprocal of 1/6, the fraction 1/6 is the integer 6. So there are few important properties of reciprocals that it's important to know. First of all, the product of any fraction with its reciprocal equals 1, so they cancel out.

So 4/17 x 17/4 = 1. That's a really important idea, and we're gonna use this to solve a problem in a few minutes. Second, the reciprocal of any positive integer is simply one divided by that integer. So the reciprocal of 6 is 1/6, the reciprocal of 117 is 1/117.

In general, the pattern is the reciprocal of n is 1 / n. Now certainly this works when n is an integer, but it turns out n doesn't even have to be an integer, we can extend the pattern. And we can use that same pattern when n is not an integer. So one divided by any number, for example, one divided by a fraction equals the reciprocal that fraction.

So if I take 1 / 3/7, that's going to equal the reciprocal 3/7, which is 7/3. So that's a really deep idea there. This already suggests the rule for fraction division, which we'll discuss in an upcoming video. So if you really understand this connection well, you will understand fraction division well.

Finally, if a number is bigger than 1, then its reciprocal is smaller, between 0 and 1. If a number is between 0 and 1, then its reciprocal is larger than 1. Very important to have this fundamental number sense, that how big a number is how big its reciprocal is going to be. Very important to have a keen sense of how big and small numbers are.

So here's a practice problem, I'll recommend, pause the video here and then we'll talk about this problem. Okay, the reciprocal of a positive number, times the cube the same number equals 5. What is the number? Well, I'm gonna represent this unknown number by x. I'm gonna use a little algebra here.

So x is the number. The reciprocal is 1 / x. And of course, the cube is x cubed. So we're saying that (1 / x) * ( x cubed) =5. That's our fundamental equation. Well that x cubed, literally that means x times x times x, I'm just gonna write that out.

Now I'm gonna take advantage of the first property of reciprocals. Namely, a number times its reciprocal cancels out. A number of times its reciprocal is 1, so (1 / x) * x together, that's just gonna equal 1. So they cancel and I'm just left with x*x. So in other words, 5 = x squared.

Now, normally we'd have to consider both the positive and the negative square root, but the problem tells us that we're dealing with a positive number. So because we're giving that guarantee, we only have to consider the positive root. And that means that x = positive square root of 5. In summary, any fraction of the form n / 1 = n. So in particular, this allows us to write any integer as a fraction, and that turns out to be useful in a variety of circumstances.

Fractions with zero, we talked about these zero can be in the numerator zero divided by zero, but zero can never be in the denominator of a fraction. That's very important. We talked about the rule, n / n = 1, and again how this will be very important. As a multiplier, we can always multiply by n / n because when we're multiplying by that, we're multiplying by 1.

And finally, we talked about the properties of reciprocals.