The fascinating irrational numbers
Most children learn about Pi and square roots somewhere during the middle school. They may hear the term "irrational number" and some even remember it, 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, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.
In mathematics, a number is rational if you can write it as a ratio of two integers, in other words in a form a/b where a and b are integers, and b is not zero. Clearly all fractions are of that form. Terminating decimal numbers can easily be written in that form, and also all non-terminating repeating decimals (decimals that repeat a sequence of digits) are rational.
Irrational numbers: non-repeating non-terminating decimal numbers
After discussing with students how terminating decimals and repeating decimals are rational, you can then announce that NON-repeating NON-terminating decimal numbers are 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, the drawing accuracy is limited, and even a line y = πx probably goes through a point with whole number coordinates, namely the point (7,22), because 22/7 is a fairly close approximation to Pi. If you could draw extremely accurately, the line wouldn't actually go through (7, 22) — it would just go close.
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, go ahead and give it a 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?
Do you remember Pi? We are told in school that Pi is approximately 3.14. In reality, Pi is an unending, never repeating decimal, which means it is an irrational number. Here are the first digits: 3.1415926535897932384626433832795
Another example of an irrational number is square root of 2, whose first decimals are 1.4142135623730950488016887242097. Most square roots are irrational. Yet another example is the sine of most angles. You can read more about sine here.
Practice a little: Make a guess about which square roots below are irrational and which are not. Then find out using a calculator. If you see a long decimal number, that square root IS irrational.
√7 , √8 , √49 , √65 , √121 , √100 , √101
Now, you may wonder that how do we know that √2 doesn'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 the first million digits, maybe the pattern is longer than that?
Here is where mathematical proof comes in. The proof that √2 is indeed irrational does not rely on computers at all but instead 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 approximate value 2.718281828459045235306....
It's not easy to just "come up" with such special numbers. But you can easily find more irrational numbers using most square roots. 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?
If you multiply or divide an irrational number by a rational number, you get an irrational number. For example, √7/10000 is an irrational number. Yet another possibility to find irrational numbers is to multiply square roots and 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 amazing thing: 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 grasp that? It seems like something I accept as a proven fact but that is not tangible or easily illustrated in concrete terms — which is often the case with irrational numbers. They seem to elude us, yet are fascinating to think about.
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.
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?
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?