Overview: Types of Numbers
There are many different ways to describe numbers as they are used in operations and in algebra. When students look at the number line, most numbers on that line are rational. They can be expressed as fractions or as decimals that divide exactly and are terminating. Other numbers (just as real), never do divide exactly into a neat ratio. Still other numbers can be imagined, but they do not have a real solution at all.
Rational Numbers Can Be Written as Fractions
Rational numbers include the integers (counting numbers) and all fractions. They are called “rational” because they can be also written as an exact ratio, which is another way of saying that they can be written as a fraction. The fraction will always mean exactly the same thing as the rational number, no matter how many decimal points that are used. Repeating decimals, such as those that were discussed in Fractions to Decimals and Decimals to Fractions are one type of rational number.
Pi Is an Irrational Number
Irrational numbers are a special type of number that can never be expressed exactly by a fraction. For example, decimals that do not repeat, such as pi, and any square roots that do not come out even, such as the square root of 2 are irrational numbers. No matter how many decimal places that pi is calculated to, there’s never a repeating pattern, but it is useful for determining the circumference of a circle. The ancient Greeks used geometric proofs, such as the Pythagorean theorem, to describe the lengths of line segments that were irrational.
Density of the Number Line and Real Numbers
If all rational numbers are plotted on a number line that stretches out infinitely, the line will be densely populated. One can always find a point that will fall between them, and there is still room between those rational numbers to plot the irrational numbers. Rational and Irrational numbers together form the set of real numbers. All the operations and properties apply to real numbers, so they can be added, subtracted, multiplied, and divided, according to number theory.
Imaginary Numbers Have Applications
If the number line is expanded to become a number plane, some numbers that are neither rational nor irrational can be plotted. These are “imaginary numbers” which are defined as multiples of the square root of -1. It has no real solution, because the square root of a number is always positive. They have many applications as complex numbers in quantum mechanics and fluid dynamics.
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