80 Part 1: The World of Numbers
Rational and Irrational Numbers
When you met rational numbers, you might have wondered if there were any numbers that aren’t
rational, and there are. They’re not whole numbers, but you can’t find a way to write them as
common fractions. It’s difficult to write an irrational number, because it can’t be written as a
fraction, and when you try to write it as a decimal, it goes on forever. There are rational numbers
whose decimal form goes on forever, but those are repeating decimals, like
2
3 0.66666...(^5) , which
you can write as 0.6. The irrational numbers don’t have repeating patterns.
Taken together, the rational numbers and the irrational numbers form the set of real numbers.
DEFINITION
Irrational numbers are numbers that cannot be written as the quotient of two
integers. Real numbers is the name given to the set of all rational numbers and all
irrational numbers.
Some of the irrational numbers come up when you try to take the square root of a number that
isn’t a perfect square number. Taking the square root of a number means finding some number
you can multiply by itself to give you this answer. The square root of 9 is 3, because 3^2 = 9.
We w r ite 93 .
If one number can be squared to produce another, the first number is the square root of the
second. The square root of 16 is 4 because 4^2 = 16. The symbol for square root, , is called a
radical. 16 4^5 , 6.25 2.5^5 , and 2 is an irrational number approximately equal to 1.414.
When you try that with a number that isn’t a square number, like 8, the answer isn’t an integer,
and many times, it’s an irrational number. 8 2.828427125. When that happens, you either have
to round that number and say, for example, that 8 is approximately 2.83 or just leave the number
in square root notation: 8.
Many times your first encounter with an irrational number comes when you work with circles
and meet the number called pi, or S. The reason it has a name is that it is difficult to write. It’s
a number a little larger than 3, but there’s no fraction that’s exactly right, and its decimal form
goes on forever without a pattern. People have used approximate values like^227 or 3.14, and those
come close, but they’re not exactly the number we call pi. Mathematicians have worked at finding
many of the decimal digits of pi. We can say that pi is approximately 3.14159265359... but it just
keeps going, and as you can see, there’s no repeating pattern.
Those are the characteristics of an irrational number. It can’t be written as a fraction, because it’s
not rational. When you try to find a decimal expression for it, it goes on and on and doesn’t show
you any pattern.