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Lesson 3: Irrational numbers

# Worked example: classifying numbers

How do repeating decimals fit into the number sets? We'll explore if a number like 3.4028 repeating is a real number, a rational number, or both. We'll use multiplication and subtraction, plus a bit of algebra, to convince ourselves of whether or not there is a fraction representation of the number. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• With sets, what is a union and an intersection?
• Think union meaning getting together. We have marriage that is talked about as a union (getting a couple together and saying they are linked as partners). You have unions that are people that get together to bargin in a workplace. Or you can have unions of sets where you take two sets and put them together.

Think intersection as what two things have in common. An intersection in a city is a place where two streets cross and a place where you are on both streets at once. An intersection of functions on a plane are the points which are part of both (or all) functions. An intersection with sets are the points that are in common between the sets.
• I'm a bit confused about doing arithmetic with numbers that go on forever. Let me show you:
If x = 0.999... then
10x = 9.999...
we then get
9x = 10x - x = 9.999... - 0.999... = 9
9x = 9
and
x = 9x/9 = 9/9 = 1
x = 1
But x is defined as 0.999...
Can someone explain this? Am I missing something here?
• Okay, so the number in the video has a .28 repeating forever. But are there numbers that have something like .28 repeating a hundred times, and then they terminate? It seems impossible, since once you get into the pattern the second time while dividing, I don't see any way for it to get out of the pattern.

But I don't know for sure. What's the most times a number can repeat itself without it having infinite repetitions? Is the answer that it can't repeat at all unless its infinite? Is there a different answer if it's a single digit that repeats instead of multiple?
• It's simpler to see for a single repeating decimal.
For .3 repeating, that would be the same as 3/9 (3 divided by 9).
If you want it to repeat a certain number of times:
.3 = 3/10
.33 = 33/100
.333 = 333/1000
and so on.

To get .28 repeating several times without going forever, you could look at it as a pattern. You could punch this into a calculator to see the pattern.

.28 = 28/100
.2828 = 2828/10000
.282828 = 282828/1000000
and so on.

.28 repeating would be the same as 28/99.

So, to answer your question, yes it's definitely possible to have a number that repeats a certain number of times without repeating forever.
• The product of two irrational numbers is rational or irrational? And why? Thank you.
• It might be either, depending on the irrational numbers involved. As an example, √2, √3 and √8 are all irrational. The product of the first two is √2 * √3 = √6, which is also irrational. But √2 * √8 = √16 = 4, which is clearly rational (as all integers are).
• Can any number with repeating decimals be called a rational number?
• Yes, all repeating decimals are rational numbers. Decimals that terminate (end) are rational numbers as well.

Have a blessed, wonderful day!
• So does this mean that when numbers repeat they are rational number? Because that would make e a rational number.
• You probably saw that e = 2.718281828 ... The fact that 1828 repeats here is pure coincidence (1 in 10,000 probability of that happening!). If you carry e out to further decimal places you will see that the 1828 repeats only once, and after that, it behaves much like pi does (ie, non-repeating). e = 2.7182818284590452353602874713526 ...
(I picked that up from wikipedia)
• i thought pie was 3.14
• It is most commonly known as 3.14 so that it's easier to work with when it comes to geometry or other forms of math because no one can really multiply or divide something that never ends, so it depends on how far your teacher wants you to use the digits, but it's most commonly known as about 3.14
• If there are 'real' numbers...are there 'fake' numbers?
• Good question! In the real number system, the square roots of negative numbers do not exist.

So mathematicians invented the imaginary unit i and defined it as the square root of -1, to make it possible to take square roots of negative numbers.

This produces an extension of the real number system, called the complex number system. Numbers in the form a + bi, where a and b are real numbers, are called complex numbers (which includes real numbers when b = 0).

Have a blessed, wonderful New Year!