Real & Complex Number Systems

Numbers can be organized into sets according to their properties. Some number sets are contained within larger sets, so one number may belong to several sets at the same time.nnNatural numbers are the counting numbers: 1, 2, 3, 4, and so on. Whole numbers include all natural numbers and zero. Integers include the whole numbers and their opposites, such as -3, -2, -1, 0, 1, 2, and 3.nnRational numbers can be written as a fraction a/b, where a and b are integers and b is not zero. Rational numbers include fractions, integers, terminating decimals, and repeating decimals. For example, 3/4, -5, 0.25, and 0.333 repeating are rational numbers.nnIrrational numbers cannot be written as a ratio of two integers. Their decimal representations neither terminate nor repeat. Examples include pi, the square root of 2, and the square root of 7.nnRational and irrational numbers together form the real number system. Every real number can be represented as a point on a number line. For example, -4, 2/3, and the square root of 5 are all real numbers.nnSome equations do not have real-number solutions. For example, no real number multiplied by itself equals -1. Mathematicians define the imaginary unit i so that i squared equals -1. Therefore, the square root of -1 equals i.nnA complex number is written in the form a + bi, where a and b are real numbers. The value a is the real part, and bi is the imaginary part. Examples include 4 + 3i, -2 - 5i, and 7i. Every real number is also a complex number because it can be written with an imaginary part of zero. For example, 6 can be written as 6 + 0i.nnTo classify a number, begin with the smallest set to which it belongs and then identify the larger sets containing it. For example, 8 is natural, whole, integer, rational, real, and complex. The number -3 is an integer, rational, real, and complex number, but it is not whole or natural.nnCommon misconceptions and corrections: Students may believe that all square roots are irrational, but perfect-square roots such as the square root of 25 are rational. Students may also assume that imaginary numbers are not useful or that real numbers cannot be complex. Remember that complex numbers include both real and imaginary components, and every real number can be written in complex form.