- Proof: sum & product of two rationals is rational
- Proof: product of rational & irrational is irrational
- Proof: sum of rational & irrational is irrational
- Sums and products of irrational numbers
- Worked example: rational vs. irrational expressions
- Worked example: rational vs. irrational expressions (unknowns)
- Rational vs. irrational expressions
Sums and products of irrational numbers
The sum of two irrational numbers can be rational and it can be irrational. It depends on which irrational numbers we're talking about exactly. The same goes for products for two irrational numbers. This video covers this fact with various examples.
Want to join the conversation?
- at 4.50 Sal says that pi squared is irrational .how do we know that?(14 votes)
- This is a great question and not easy to answer. Although the sum and product of rational numbers give results that are rational this is only some times true for sums and products of irrational numbers. The proof that pi^2 is irrational is a proof of contradiction, involves calculus and is detailed here : http://mathforum.org/library/drmath/view/76304.html
Sal has a video called "Sums and products of irrational numbers" here : https://www.khanacademy.org/math/algebra/rational-and-irrational-numbers/sums-and-products-of-rational-and-irrational-numbers/v/sums-and-products-of-irrational-numbers
A short text summary about the sums and products of rational and irrational can be seen here:
- Is it always the case that when we multiply two irrational numbers, the product will be EITHER an irrational number OR an integer? I am asking because I have yet to see an example where the product of two irrational numbers yields a rational non-integer.(3 votes)
- This is not always the case. A counter example would be √1.125 * √2 = 1.5(11 votes)
- how is pi+1-pi= 1 because those numbers are both irrationals. Also if pi+pi=2 pi
then what difference does it make to pi being subtracted by 1, pi has an infinite amount of numbers so you cant subtract it by 1. Even if you can wouldn't the number replace itself or something because the numbers pi have are INFINITE!!. I also want to confirm this method to so that i don't have anything else to say,let me assume that a very dumb person; ahmm (myself), assumed that if pi + pi = 2 pi but there is a minus 1 there so i can subtract 2 by 1 it would still make it pi? please dont make fun of my stupidity im only 11 so if you can pleas answer my question i will be very grateful(2 votes)
- pi + 1 - pi addition is commutable, so you can move things around as long as you keep the sign, so pi - pi + 1 is the same, and anything minus itself (even irrational numbers) is always 0, so all that is left is 1 = 1
If pi is 3.14159..., so if you subtract 1, you would have 2.14159..., but as you expected it does not mean much.(7 votes)
- at4:27, could someone please explain why 1/π is irrational, because isn't it just the ratio of two numbers?(3 votes)
- Rational numbers must be the ratio of 2 integers. Pi is not an integer. It is an irrational number. 1/Pi is also an irrational number.(5 votes)
- can you please teach me the sum of rational numbers(3 votes)
By definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero). So, adding two rationals is the same as adding two such fractions, which will result in another fraction of this same form since integers are closed under addition and multiplication. Thus, adding two rational numbers produces another rational number.
statementbox "The sum of two irrational numbers is SOMETIMES irrational."
Product of a two rational numbers is rational(2 votes)
- At2:36, he says pi-pi=c when he plugged it in for a and b, but wouldn't it be a + a then because he is using the same number for different variables?(2 votes)
- Yes your right the variables are not a and b as a and b are differet variables, it should have been a+a or 2a.(1 vote)
- Irrational minus irrational is equal to(1 vote)
- At3:10in the video, Sal tells you that the result can be rational or irrational. It depends upon what irrational numbers are being subtracted.(3 votes)
- How do you prove that the product of two irrational numbers can be rational?(1 vote)
- Not always. We cannot prove this but it can be seen through examples. For example, √2*√2=2, which is rational but √2*√3=√6, which is irrational.(2 votes)
- If pi x pi = pi squared is irrational, why isn't the sq. root of 2 x the sq. root of 2 which is the sq. root of 2 squared,not irrational?(1 vote)
- sqrt(2) * sqrt(2) = sqrt(4)
4 is a perfect square. Thus, the square root can be simplified into 2, which is a rational number.(2 votes)
- If a rational number is a number which can be expressed as a ratio of two numbers, then why is 1/pi not a rational number? But it is being expressed as a ratio isnt it?(1 vote)
- A rational number can be written as a ratio of 2 integers. 1/pi is not a ratio of 2 integers. Pi is not an integer.(2 votes)
- [Instructor] Let's say that we have some number a and to that we are going to add some number b and that sum is going to be equal to c. Let's say that we're also told that both a and b are irrational. Irrational. So based on the information that I've given you, a and b are both irrational. Is their sum, c, is that going to be rational or irrational? I encourage you to pause the video and try to answer that on your own. I'm guessing that you might have struggled with this a little bit because the answer is that we actually don't know. It depends on what irrational numbers a and b actually are. What do I mean by that? Well, I can pick two irrational numbers where their sum actually is going to be rational. 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. Pi is irrational and one minus pi, whatever this value is, this is irrational as well. 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 to be a rational number. So we were able to find one scenario in which we added two irrationals and the sum gives us a rational. In general you could do this trick with any irrational number. Instead of pi you could've had square root of two plus one minus the square root of two. Both of these, what we have in this orange color is irrational, what we have in this blue color is irrational, but the sum is going to be rational. And you could do this, instead of having one minus, you could have this as 1/2 minus. You could have done it a bunch of different combinations so that you could end up with a sum that is rational. But you could also easily add two irrational numbers and still end up with an irrational number. For example, if a is pi and b is pi, well then their sum is going to be equal to two pi, which is still irrational. Or if you added pi plus the square root of two, this is still going to be irrational. In fact, mathematically I would just express this as pi plus the square root of two. This is some number right over here, but this is still going to be irrational. So the big takeaway is 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. Now let's think about products. Similar exercise, let's say we have a times b is equal to c, ab is equal to c, a times b is equal to c. And once again, let's say someone tells you that both a and b are irrational. Pause this video and think about whether c must be rational, irrational, or whether we just don't know. Try to figure out some examples like we just did when we looked at sums. Alright, so let's think about, let's see if we can construct examples where c ends up being rational. Well one thing, as you can tell I like to use pi, pi might be my favorite irrational number. If a was one over pi and b is pi, well, what's their product going to be? Well, their product is going to be one over pi times pi, that's just going to be pi over pi, which is equal to one. Here we got a situation where the product of two irrationals became, or is, rational. 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 times the square of two, that would be one. What if instead I had pi times pi? Pi times pi, that you could just write as pi squared, and pi squared is still going to be irrational. This is irrational, irrational. It isn't even always the case that if you multiply the same irrational number, if you square an irrational number that it's always going to be irrational. For example, if I have 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 two irrational numbers. In fact, they're the same irrational number, but the square root of two times the square root of two, well, that's just going to be equal to two, which is clearly a rational number. So once again, when 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 the specific numbers. Whether you're taking the product or the sum of irrational numbers, in order to know whether the resulting number is irrational or rational, you need to know something about what you're taking the sum or the product of.