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What Is A Real Number Calculator

Reviewed by Calculator Editorial Team

Real numbers are fundamental in mathematics and science. This guide explains what real numbers are, how to identify them, and provides a calculator to check if a number is real.

What Are Real Numbers?

Real numbers are all the numbers that can be found on the number line. They include both rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions.

Real Number Definition

A real number is any number that can be represented on the number line. This includes all positive and negative numbers, zero, fractions, decimals, and irrational numbers like √2 or π.

Types of Real Numbers

  • Natural numbers: Positive integers (1, 2, 3, ...)
  • Whole numbers: Natural numbers including zero (0, 1, 2, ...)
  • Integers: Whole numbers and their negatives (...-2, -1, 0, 1, 2, ...)
  • Rational numbers: Numbers that can be expressed as a fraction of two integers (a/b where b ≠ 0)
  • Irrational numbers: Numbers that cannot be expressed as fractions (e.g., √2, π, e)

Real Numbers vs. Imaginary Numbers

Real numbers are distinct from imaginary numbers, which include the square root of negative numbers (e.g., √-1 = i). Imaginary numbers are not part of the real number system.

How to Identify Real Numbers

Identifying real numbers involves checking if a number can be represented on the number line. Here are some criteria to determine if a number is real:

  1. Check if the number can be expressed as a fraction of two integers.
  2. Verify if the number has a terminating or repeating decimal expansion.
  3. Ensure the number is not the square root of a negative number.

Identification Tips

If a number can be written as a fraction or has a decimal expansion that either terminates or repeats, it is a real number. Numbers like √-1 or i are not real numbers.

Examples

Real Number Examples

  • 5 (integer)
  • -3.7 (decimal)
  • 2/3 (fraction)
  • √9 (square root of a perfect square)
  • π (irrational number)

Non-Real Number Examples

  • √-1 (imaginary unit)
  • i (imaginary number)
  • Complex numbers like 3 + 2i

FAQ

What is the difference between real and imaginary numbers?

Real numbers can be found on the number line and include all positive and negative numbers, zero, fractions, and irrational numbers. Imaginary numbers include the square root of negative numbers and are not part of the real number system.

Can all fractions be considered real numbers?

Yes, all fractions that can be expressed as a ratio of two integers are real numbers. They are part of the rational number subset of real numbers.

Is zero a real number?

Yes, zero is a real number. It is an integer and can be found on the number line between negative and positive numbers.

Are all irrational numbers real numbers?

Yes, all irrational numbers are real numbers. They cannot be expressed as fractions but can be represented on the number line.