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Lesson 3: Irrational numbers

# Worked example: classifying numbers

Deciding which number set 3.4028 belongs to: integers, rationals, or irrationals? 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. • 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)
• what exactly is a irrational number • Can any number with repeating decimals be called a rational number? • When is something not a real number? • Something is not a real number exactly when it is not a real number. A better question would be: "What is a real number?". The answer to this question is more deep and subtle than it appears.

One usually defines the real numbers following this path:
Natural numbers --> Integers --> Rational Numbers --> Real numbers.

Constructing the real numbers from the rational numbers is usually done in one of two ways:
I. Defining a real number as a Dedekind cut of rational numbers.
II. Defining a real number as an equivalence class of Cauchy sequences of rational numbers.

The full constructions are rather lengthy, so I refer you to some books for the details:
I. Rudin, Walter: Principles of Mathematical Analysis, 3rd edition (1976), page 17.
II. Tao, Terence, Analysis I, 2nd edition, the first few chapters.
• What are some examples of non-real numbers? I know there are imaginary numbers but are there anything other than that? Thanks! • There is a system of numbers invented by the British combinatorial game theorist John Conway, called the surreal numbers. The surreal numbers include the real numbers and much more! If we let omega (which I'll call w) represent what we usually think of as infinity, there are many meaningfully different infinite surreal numbers such as w, w/2, sqrt(w), w-1, w+1, 2w, w^2, etc. There are also infinitesimally tiny surreal numbers that are positive but less than all positive real numbers, such as epsilon (which is 1/w), epsilon/2, 2*epsilon, sqrt(epsilon), epsilon^2, etc. The game called Red-Blue Hackenbush models surreal numbers!

Have a blessed, wonderful day!
(1 vote)
• how many rational numbers are between -5 and 5, is 1/2 and 5/1 acceptable? it seems like i could put any two of the integers together or am i missing something? 