Introduction to rational and irrational numbers | Algebra I | Khan Academy

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Let’s talk a little more about the relative numbers The easiest way to know it is Any number can be represented as a ratio between two integers It is the relative number For example, any integer is a relative number 1 can be represented as 1/1 or negative 2 / negative 2 Or 10,000 / 10,000 And all of these cases are different representations For the number 1 as a ratio between two numbers Obviously, I can get an infinite number To represent the one in this way By dividing the number by itself The number minus 7 can be represented as minus 7 on 1 Or seven on negative 1, or negative 14 on positive 2 And I can keep going If minus 7 is a relative number It can be represented as a ratio between two integers. But what about the numbers that are not correct For example, let’s imagine … I don’t know … 3.75 How can we represent it as a ratio between two integers? Well, 3.75 you can rewrite it On the image 375/100, the same as 750/200 Or you could say that 3.75 It is the same 3/3/4 – so let me write it here It is the same as 4/15 4 times 3 gives 12, plus 3 gives 15, you can write this It is the same as 4/15 Or we can write it as minus 30 by minus 8 I just hit the numerator and denominator Here is minus 2 Just to be clear, it’s clearly a relative number I give you several examples of how Representing it as a ratio between two integers Now, what about periodic decimal fractions Well, let’s take perhaps the most famous example For periodic decimals Let’s say that you have … 0.3333 repeat and forever Which we can symbolize with a situation Bar top 3 This is the periodic number of 0.3 We have seen and will show later How can you convert any periodic decimal fraction To the ratio of two integers, – this is clearly 1/3 Or maybe you’ve seen things like 0.6 repeated that is 2/3 There are many, many, and many other examples of this And we will see any periodic decimal fraction Not just one who repeats a single box Even if it had a million digits repeated As long as the pattern began to repeat itself Time and time again You can always represent this as a ratio between two integers. Maybe I know what you’re thinking Sal, I mentioned a lot. You have mentioned all integers. All ending non-periodic decimal fractions are reported Also mentioned periodic decimals. What’s left Are there any numbers that are not relative? You will probably guess there are others Otherwise, when you bear people The trouble of trying to classify these numbers as relative numbers It turns out – as I imagined – That some of the most famous numbers in mathematics Not relative We call these numbers non-relativistic numbers Here I have just made a list for months Noteworthy examples Pi – the ratio between the circumference of a circle The diameter – is a non-relative number It never ends It lasts forever, and has no periodic number And the same thing does not end and has no periodic number And it has many benefits It is used in complex analysis And e appears everywhere The square root of 2 is a non-relative number So, the golden ratio is a non-relative number So these numbers are produced by nature Most of these numbers are non-relative numbers Now, you might say are these numbers not relative? These are just special types of numbers But most of them may be relative numbers And Sal just picked some special cases here But the most important thing you have to understand is that it seems strange It is strange sometimes But it is not rare They are actually too many there always A non-relative number between any two relative numbers Well, we can continue In fact, they are infinite numbers But there is at least one, so this gives you an idea Really can not say that The non-relativistic numbers are less than the relative numbers And we will prove it in the next video Give me two relative numbers, – the relative number 1 And the relative number 2– There will be at least one non-relative number between them It is a good result Because the non-relativistic numbers seem strange Another way to think about it is the square root of 2 Take the square root of a non-square whole number You will get a non-relative number Adds a non-relative number And a relative number, and we will see that later We will prove it to ourselves. The sum of a non-relative number and a relative number It will be a non-relative number. The product of a non-relative number and a relative number It will be a non-relative number. So there are many, many non-relativistic numbers ..

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