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Lesson 89: Real Numbers (Integers, Rational, Irrational)
In this lesson you'll learn about real (as opposed to imaginary) numbers.
Important: This lesson ties in closely with Lesson 91. It is suggested that you study both lessons during the same session.
We've been working quite a bit with whole numbers: ones that are positive, negative, and zero. These whole numbers are called integers. That's just a fancy name for them that you need to memorize, since it comes up again and again.
We've also worked a lot with fractions. A fraction is considered to be a rational number, since it is the ratio of two integers. Rational numbers can always be represented as repeating decimals, or terminating decimals. For example, 1/3 = 0.33333, with the threes continuing forever, and 7/10 = 0.7, stopping right there.
Some decimals never repeat at all. A famous example is the decimal equivalent of π. We've been using 3.14 as an approximate value of π, and sometimes people use 3.14159, but the truth is, the decimal portion goes on and on without any type of pattern.
This means that π is not the exact equivalent of any fraction, although we sometimes say that it is approximately 22/7. Still, it is not exact, and that means we call it an irrational number. It is not based on any ratio.
Another example of an irrational number is √5, as well as square roots of other numbers that are not perfect squares. If you compute them on a calculator, you will see that the decimals go on and on with no repeating pattern. Even if your calculator runs out of room, the decimals still go on forever without repeating or terminating.
In math, we say that the world of real numbers is made up of rational and irrational numbers. Any integer can easily be converted to a rational number or fraction by placing it over a denominator of 1. Students usually want to know what other possible numbers there can be. If you go on to study advanced math in college, you'll learn about the world of what are called imaginary numbers. These are special numbers that are neither rational nor irrational.
For now, just memorize the definitions in this lesson, and understand the concepts. These terms and ideas come up very frequently in math.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.