They have many applications as complex numbers in quantum mechanics and fluid dynamics. It has no real solution, because the square root of a number is always positive. These are “imaginary numbers” which are defined as multiples of the square root of -1. If the number line is expanded to become a number plane, some numbers that are neither rational nor irrational can be plotted. All the operations and properties apply to real numbers, so they can be added, subtracted, multiplied, and divided, according to number theory. Rational and Irrational numbers together form the set of real numbers. 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. If all rational numbers are plotted on a number line that stretches out infinitely, the line will be densely populated. The ancient Greeks used geometric proofs, such as the Pythagorean theorem, to describe the lengths of line segments that were irrational.ĭensity of the Number Line and Real 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. ![]() 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. Irrational numbers are a special type of number that can never be expressed exactly by a fraction. Repeating decimals, such as those that were discussed in Fractions to Decimals and Decimals to Fractions are one type of rational number. The fraction will always mean exactly the same thing as the rational number, no matter how many decimal points that are used. 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. Rational numbers include the integers (counting numbers) and all fractions. ![]() Rational Numbers Can Be Written as Fractions Still other numbers can be imagined, but they do not have a real solution at all. Other numbers (just as real), never do divide exactly into a neat ratio. They can be expressed as fractions or as decimals that divide exactly and are terminating. When students look at the number line, most numbers on that line are rational. There are many different ways to describe numbers as they are used in operations and in algebra.
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