what is a "real" number

7th grade



Hi Sirena,

The numbers that we are accustomed to using are all real numbers.

  • Posivive integers, like 1, 2, 3... are real numbers
  • Negative integers, -3, -5... are real numbers
  • Fractions are real numbers, for example  3/5 and  -5/8
  • Decimal numbers are real numbers, for example 3.56 and -45.45

So what numbers are not real numbers? One problem with the real numbers is that, if you take any real number and multiply it by itself you never get a negative number. In particular there is no real number n that satisfies n  n = -1.

Mathematicians have defined a "number" usually designated i, or sometimes j, that has the property that i i = -1. This then generates a new type of numbers called the complex numbers.

Complex numbers are useful in many areas of mathematics, Science and Engineering. Some aspects of electricity and magnetism are best understood using complex numbers. The place where students in high school see complex numbers is in the study of quadratic equations. If you consider real numbers only, then some quadratic equations have solutions and some do not. If you include complex numbers then every quadratic equation has a solution, and this fact helps unify the study of such equations.



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