![]() ![]() Rational NumbersĪgain, a rational number is any number that can be expressed as a fraction, where both the numerator and denominator are integers, and the denominator is not 0: Source: Rational Numbers Integers are exactly like whole numbers, but can also include negative numbers. They do not include decimals or fractions, and since they start from 0 and go up, all whole numbers are positive. Note that whole numbers are just that – whole. ![]() Whole NumbersĪ whole number is any number from 0 up to infinity: 0, 1, 2, 3, 4, 5. What Are Whole Numbers and Integers?īefore getting into integers, it would be helpful to review what whole numbers are. In this article, we'll go over what whole numbers and integers are, cover different types of rational numbers, and learn how to determine if a number is rational or not. In other words, a rational number can be expressed as p/q, where p and q are both integers and q ≠ 0. The specific numbers.A rational number is any number that can be written as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not equal to zero. You're taking the product of two irrational numbers, you don't know whether the product is going to be rational or irrational unless someone tells you To be equal to two, which is clearly a rational number. Times the square root of two, well, that's just going Same irrational number, but the square root of two Square root of two times, I think you see where this is going, times the square root of two, I'm taking the product of Same irrational number, if you square an irrational number that it's always going to be irrational. It isn't even always the case that if you multiply the Just write as pi squared, and pi squared is still What if instead I had pi times pi? Pi times pi, that you could Times the square of two, that would be one. But what if I were to multiply, and in general you could this with a lot of irrational numbers, one over square root of two The product of two irrationals became, or is, rational. ![]() To be one over pi times pi, that's just going to be pi If a was one over pi and b is pi, well, what's their product going to be? Well, their product is going Well one thing, as youĬan tell I like to use pi, pi might be my favorite irrational number. ![]() Alright, so let's think about, let's see if we can construct examples where c ends up being rational. Try to figure out some examples like we just did when we looked at sums. Pause this video and think about whether c must be rational, irrational, or whether we just don't know. Say someone tells you that both a and b are irrational. Say we have a times b is equal to c, ab is equal to c, a times b is equal to c. If you're taking the sums of two irrational numbers and people don't tell you anything else, they don't tell you which specific irrational numbers they are, you don't know whether their sum is going to be rational or irrational. This is some number right over here, but this is still going to be irrational. I would just express this as pi plus the square root of two. Or if you added pi plus the square root of two, this is still going to be irrational. Going to be equal to two pi, which is still irrational. For example, if a is pi and b is pi, well then their sum is But you could also easilyĪdd two irrational numbers and still end up withĪn irrational number. Of different combinations so that you could end up Instead of having one minus, you could have this as 1/2 minus. Orange color is irrational, what we have in thisīlue color is irrational, but the sum is going to be rational. Instead of pi you could've had square root of two plus one In general you could do this trick with any irrational number. So we were able to find one scenario in which we added two irrationals and the sum gives us a rational. But if we add these two things together, if we add pi plus one minus pi, one minus pi, well these are gonna add up to be equal to one, which is clearly going Pi, whatever this value is, this is irrational as well. What do I mean? Well what if a is equal to pi and b is equal to one minus pi? Now both of these are irrational numbers. What do I mean by that? Well, I can pick two irrational numbers where their sum actually I'm guessing that you might have struggled with this a little bitīecause the answer is that we actually don't know. To be rational or irrational? I encourage you to pause the video and try to answer that on your own. That I've given you, a and b are both irrational. Let's say that we're also told that both a and b are irrational. To add some number b and that sum is going to be equal to c. That we have some number a and to that we are going ![]()
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