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The fascinating irrational numbers
A teaching guideline/lesson plan giving you more insight to irrational numbers

Most children learn about Pi and square roots somewhere during the middle school.  They may hear said 'irrational number' and some even remember the phrase, but very few really understand what it means.  Well, irrational numbers are harder to understand than rational numbers, but I consider it worth the time and effort because they have some fascinating properties.  I just have to wonder in awe when I see the facts laid in front of me because it all sounds unbelievable, yet proven true.

To study irrational numbers one has to first understand what are rational numbers.  In short they are whole numbers, fractions, and decimals - the numbers we use in our daily lives.

In mathematical terms a number is rational if you can write it as a ratio of two integers, or in other words in a form a/b where a and b are integers, b not zero.  Clearly all fractions are of that form.  Terminating decimal numbers can easily be written in that form, and also all non-terminating decimal numbers that repeat a sequence of digits are rational.  Click here to review about rational numbers, and how to convert repeating decimal numbers to fractions.

Irrational numbers: non-repeating non-terminating decimal numbers

After discussing how terminating decimal numbers and repeating decimal numbers are rational, we can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERS.

Can you imagine a line through origin that does NOT touch ANY of the points with whole number coordinates?????  It's hard, but those kind of lines do exist.  They just avoid touching any of the points with whole number coordinates, and their slope is an irrational number!!!  Difficult to fathom.  Of course when you are drawing lines on paper or on computer, you are limited in your accuracy and even a line y = Pi*x probably to go through a point with whole number coordinates, namely the point (7,22).  It really wouldn't go throuhg it if we could draw extremely accurately, it would just go close.  But since it goes close, 22/7 is a nice approximation to Pi.

Line y = Pi*x goes through the point (7,22)
The line y = Pi * x indeed looks like it goes through the point (7,22) since the graphics program cannot draw accurately enough.

What does that mean in practical terms?  If you can make up a sequence of digits that never ends and that never runs into a pattern, then you have an irrational number.  Sounds easy?  Well, you try.  You can't take several small patterns and just shift them around, because that would just creata a longer pattern.  And if you just do it randomly, how can you be sure that it is not creating a very long pattern of maybe million digits long?

Now, take Pi.  They tell children in school that Pi is approximately 3.14.  In reality Pi is an unending, never repeating decimal, or an irrational number. Here are the first digits: 3.1415926535897932384626433832795

Another example is square root of 2, whose first decimals are 1.4142135623730950488016887242097.  Many other square roots are irrational, in fact most are. Yet another example is the sine of an angle. For some special angles the sine of the angle is a rational number, but in most cases it's an irrational number.  You can read more about understanding sine and why does a calculator give me a sine wave.

Practice a little: Explore which one of these square roots are rational or irrational:
  √7 ,   √8 ,   √49 ,   √65 ,   √121 ,   √100 ,   √101 .
Make a guess about which square roots are irrational and which are not!

Now, you may wonder that how do you know that √2 wouldn't have a pattern in the decimal sequence?  Maybe the pattern is very well hidden and is really long, billions of digits?  Even if you check it till million first digits, maybe the pattern is just longer than you were able to print the digits out with your computer?

Here is where mathematical proof comes in.  The proof that √2  is indeed irrational is usually found in college level texts, but it isn't that difficult to follow.  It does not rely on computers at all, but instead it is a 'proof by contradiction' - if √2  WAS a rational number, then we'd get a contradiction.  I encourage you to let your high school students study this proof since it is very illustrative of a typical proof in mathematics and is not very hard to follow: Proof that square root of 2 is an irrational number.

Where can you find special irrational numbers?

The answer to this depends on what you consider 'special'.  Mathematicians have proved that certain special numbers are irrational, for example Pi and e.  The number e is the base of natural logarithms.   It is irrational, just like Pi, and has the value 2.718281828459045235306....  

It's not easy to just "come up" with such special numbers.  But you can easily find more irrational numbers after you've found that most square roots are irrational.  For example, what do you think of √2 + 1?  Is the result of that addition a rational or an irrational number?  How can you know?  What about other sums where you add one irrational number and one rational number, for example √5 + 1/4?

You can also add two irrational numbers, and the sum will be many times irrational.  Not always though; for example, e + (-e) = 0, and 0 is rational even though both e and -e are irrational.  Or, take 1 + √3 and 1 - √3 and add these two irrational numbers - what do you get?

Or, multiply/divide an irrational number by a rational number, and you get an irrational number.  For example, √7/10000 is an irrational number that is quite close to zero.  Yet another possibility to find irrational numbers is to multiply square roots or other irrational numbers.  Sometimes that results in a rational number though (when?).  Mathematicians have also studied what happens if you raise an irrational number to a rational or irrational power.

Yet more irrational numbers arise when you take logarithms, or calculate sines, cosines, and tangents.  They don't have any special names, but are just called "sine of 70 degrees" or "base 10 logarithm of 5" etc.  Your calculator will give you decimal approximations to these.

Which are more: rational or irrational numbers?

Now comes another one of the amazing things: there are multitudes of terminating decimal numbers and multitudes of non-terminating repeating decimals, but there are EVEN MORE non-terminating non-repeating decimals, or in other words irrational numbers. That is usually shown in college level mathematics courses.  In mathematical terms we say that the rational numbers are a countable set whereas the irrational numbers are an uncountable set.

In other words, there are more irrational numbers than there are rational.  Can you understand that?  It seems like something I accept as a proven fact but that is not tangible or easily illustrated in concrete terms - often the case with these irrational numbers, which seem to elude us and yet are kind of fascinating to think about, right?


More information

The Evolution of Real Numbers
An excellent tutorial on the difference of rational and irrational numbers and how the thinking on those has "evolved" in history; covers topics such as ratio of natural numbers, continuous versus discrete, unit fractions, measurement, common measure, squares and their sides etc.

Doesn't Pi Ever Repeat?
How do people know that pi goes on forever without repeating?

Are All Digit Strings in Pi?
Do all possible digit combinations appear in pi? For example, is it possible that somewhere in the decimal expansion there are a million 2's in a row?

Pi = 3.14159...
What is Pi? Who first used Pi? How do you find it? How many digits is it?

Meaning of Irrational Exponents
y^x is the product of x factors of y. Where do irrational exponents fit in this? How can you have an irrational number of factors?

Non-integer Powers and Exponents
How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?

BRILLIANT. I especially like the sensitivity to representing the idea visually. Truly a gem in the world of explaining the inexplicable.


What are other names for the square root line and the slash in a fraction? Not the surd, or slash or divisor.
Pat Wilmore

Surd can be referred to as either a "radical sign" or a "surd symbol" or "radical symbol". The fraction slash is also called a solidus.

what is the longest repeating decimal ?

lou simon

I assume you mean what is the longest cycle. Well, there is no limit to the length of the cycle. Let's say I make a repeating decimal with a huge amount of 4's in a row, followed by one 7. Well, you can always make the cycle to be just one longer by adding one more 4 into it, and you could keep doing that indefinitely. In other words, if there was such a thing as the longest cycle, one could still always make it longer by adding one 4 into it, and so it wouldn't be the longest after all.

How do you draw the square root of a number eg 10 on a number line using a compass and a ruler?

One method is explained at Dr. Math: Construct a line segment the length square root of X.

All this talk about how fantastic pi is, as irrational and nonrepeating as it is in its pattern, yet never referring to the fact that it is the constant by which 2 pi R = circumference of a circle. It never repeats just like a circle is never-ending and perfectly round. Perhaps there is something to be understood about such a constant that results in perfection. You can never attain it... or it's always fresh -- like nothing repeating the same way twice in time. Does a circle have anything to do with time? Interesting how we have time measured by things that are round like pivoting clock hands in a circle. Something to think about. Also the geometric shape itself.

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