What is the Real Number System?
The Real Number System
Numbers can be classified in many different ways. Let’s look at some of those, starting with natural numbers. Natural numbers is this set of counting numbers, like 1, 2, 3, 4, etc.
Whole numbers include the set of natural numbers and also zero. Whole numbers would be numbers like 0, 1, 2, 3, etc. Integers include the set of whole numbers and all their negatives.
Integers would be numbers like -3, -2, -1, 0, 1, 2, 3, etc. Notice it goes on in both directions. It would continue going in both directions.
Rational numbers include this set of integers, so -3, -2, -1, 0, 1, 2, etc., but it also includes the whole set of numbers that can be expressed as a ratio of 2 numbers.
Notice that the root of rational is ratio. A ratio is a fraction, or a comparison of two numbers. Rational numbers not only include the whole set of integers, but also the set of numbers that can be written as a ratio of integers.
For instance, 1/2 is a rational number. It’s a fraction, or a ratio. Also, -4/3. That is a rational number, because it’s a ratio. It’s a fraction.
Rational numbers also include repeating and terminating decimals, like 3/10 repeating and 25/100. 3/10 repeating is the repeating decimal, and it can be expressed as a fraction (1/3).
Again, it’s a rational number, since it can be written as a ratio. 25 hundredths is a terminating decimal and it can also be written as a fraction (1/4).
Again, it’s a rational number, since it can be written as a ratio. The real number system also includes another set of numbers: irrational numbers.
Irrational numbers, if we look at the word, ir- (the prefix) means “not”. Then we see that word again. Rational. Irrational numbers is the set of numbers that are not rational, or numbers that cannot be written as a ratio, which means numbers that can not be written as a fraction.
This set of numbers includes things like pi. Pi is an irrational number. It’s 3.14159, etc., etc., etc. That number goes on forever, meaning it doesn’t repeat and it doesn’t terminate.
Therefore, it cannot be written as a fraction. It’s not rational. Another example would be like the square root of 2. The square root of 2 is 1.414, etc., etc., etc.
Again, it goes on forever, so it’s not rational. It doesn’t repeat like .3333333 and it doesn’t terminate. It never ends. It goes on forever. The real number system is made up of two different groups of numbers: rational numbers and irrational numbers.