Numbers have names.
In this post, I’m going to talk about these names:
- REAL NUMBER
- RATIONAL NUMBER
- IRRATIONAL NUMBER
First, you must know what an INTEGER is, and what a REAL NUMBER is, and the difference between INTEGERS and REAL NUMBERS.
An INTEGER is just a name for ANY WHOLE number. Examples of INTEGERS are:
A REAL NUMBER is just a name for ANY decimal valued number. Examples of REAL NUMBERS are:
The only difference between REAL NUMBERS and INTEGERS is that REAL NUMBERS are allowed to have something after the decimal point.
INTEGERS are NEVER allowed anything after the decimal point. INTEGERS aren’t allowed to have a decimal point in them at all.
Notice in the last example for a REAL NUMBER, I put plain old 20. 20 is BOTH an INTEGER AND a REAL NUMBER AT THE SAME TIME!!! That’s an important point to understand, and we’ll talk about it again in a second.
First, try these exercises:
Question: Is 1.24 a REAL NUMBER or an INTEGER?
Answer: 1.24 is a REAL NUMBER. 1.24 is NOT an INTEGER because it has a decimal part.
Question: Is 5.929848995681948 a REAL NUMBER or an INTEGER?
Answer: 5.929848995681948 is a REAL NUMBER. 5.929848995681948 is NOT an INTEGER because it has a (huge) decimal part.
Question: Is 5000 a REAL NUMBER or an INTEGER?
Answer: 5000 is an INTEGER. 5000 is ALSO a REAL NUMBER. 5000 is BOTH an INTEGER AND a REAL NUMBER AT THE SAME TIME!!!
To understand this idea that a number can be BOTH a REAL NUMBER and an INTEGER at the same time, just think about the different nicknames you might have for one of your friends. I have a friend named Bob. Bob is really fat. So I have 2 names for Bob:
In the same way, I have TWO NAMES for the number 5000:
- REAL NUMBER
Getting to RATIONAL NUMBERS
Now here is the deal with RATIONAL NUMBERS.
A RATIONAL number is any REAL number that can be found by dividing two INTEGERS together.
For example, try this on your calculator:
1 - 10
When you do 1 divided by 10, you should see 0.1 on your calculator’s face. Your calculator should also be smiling at you because you gave it a nice short computation.
Just like my friend Bob had 2 names, “Bob” and “Fatso”, we are going to give the number 0.1 TWO NAMES:
- REAL NUMBER (because 0.1 clearly has a decimal part)
- RATIONAL NUMBER
Why do we say 0.1 is a RATIONAL NUMBER?
0.1 is called a RATIONAL NUMBER because it can be found out by dividing two integers together!!!. That’s really all there is to it.
Using your calculator, try to find out whether each of these numbers are RATIONAL (try to find two integers to divide together that makes your calculator show the number on its face):
- a) 0.5
- b) 1.2
- c) 7.8
- d) 9.8471
- a) 0.5 is found by 1 divided by 2. Therefore 0.5 is RATIONAL.
- b) 1.2 is found by 6 divided by 5. Therefore 1.2 is RATIONAL.
- c) 7.8 is found by 78 divided by 10. Therefore 7.8 is RATIONAL.
- d) 9.847 is found by 9847 divided by 1000. Therefore 9.847 is RATIONAL.
Ok, now the hard to understand part.
Notice what I did with the last one, d). I just took 9847 and divided it by 1000. That looks really cheap. Its almost like cheating.
In fact, I can do that with any decimal number that you can write down on paper.
1.1155 is just 11155 divided by 10000. Therefore 1.1155 is a RATIONAL NUMBER, because it can be found by dividing 2 INTEGERS together.
84.287158 is just 84,287,158 divided by 1,000,000. Therefore 84.287158 is a RATIONAL NUMBER, because it can be found by dividing 2 INTEGERS together.
1.1 is just 11 divided by 10. Therefore, 1.1 is a RATIONAL NUMBER, because it can be found by dividing 2 INTEGERS together.
Ok, ok. We know what a RATIONAL number is. But I have a question. What about numbers that repeat forever, like this:
- a) 0.6666666666 (6’s repeat after the decimal forever)
- b) 0.3333333333 (3’s repeat after the decimal forever)
- c) 0.099099099 (099 repeats forever)
Are these RATIONAL NUMBERS? (Remember, a RATIONAL NUMBER is any decimal number that can be found by dividing two INTEGERS together).
Here are the answers:
- a) 0.6666666666 can be found by dividing 2 by 3: Therefore 0.66666666 is a RATIONAL NUMBER
- b) 0.3333333333 can be found by dividing 1 by 3: Therefore 0.3333333333 is a RATIONAL NUMBER
- c) 0.099099099 can be found by dividing 11 by 111: Therefore 0.099099099 is a RATIONAL NUMBER
So far, we have discovered TWO TYPES of RATIONAL NUMBERS:
- The type that has a decimal value, then terminates
- such as 0.1, 0.5, and 0.89
- The type that has a decimal value that repeats a pattern forever
- such as 0.6666666666 [found by 2/3], 0.33333333333 [found by 1/3], and 0.099099099099099 [found by 11/111]
So does that mean that ANY DECIMAL NUMBER can be found by dividing 2 INTEGERS together???
ALMOST, but not quite. There is ANOTHER type of number called an IRRATIONAL NUMBER. We’ll explain that next.
OK, so we know what a RATIONAL NUMBER is now. A RATIONAL NUMBER is just any REAL NUMBER that can be found by dividing two INTEGERS together.
But, try this in your calculator:
- SQUARE ROOT of 2
- SQUARE ROOT of 5
When I punch those into my calculator, I get:
- SQUARE ROOT of 2 is found to be 1.414213562 (then it stops showing decimals, but it WOULD keep going if it had the space. Yours might cut off earlier.)
- SQUARE ROOT of 5 is found to be 2.236067978 (then it stops showing decimals)
Both the SQUARE ROOT of 2 and the SQUARE ROOT of 5 are called IRRATIONAL NUMBERS. IRRATIONAL NUMBERS are numbers that, no matter how hard you try, you CAN’T CALCULATE THEM BY DIVIDING TWO INTEGERS TOGETHER.
Now you MIGHT think, “well, I can find out the SQUARE ROOT of 2 dividing two integers! I can do:
1,414,213,562 / 1,000,000,000 = 1.414213562
Actually that is WRONG!! Using a better calculator to find SQUARE ROOT of 2, we would get 1.4142135623730950488016887242097. The SQUARE ROOT of 2 has decimal values that GO ON FOREVER, with NO REPEATING PATTERN, so therefore, the SQUARE ROOT of 2 is an IRRATIONAL NUMBER. The SQUARE ROOT of 2 can NEVER be FULLY found out by dividing ANY two INTEGERS together. You always have to go for that SQUARE ROOT button on your calculator to find the true value of the SQUARE ROOT of 2.