Which Products Are Real Numbers Calculator

This calculator helps you determine which products of two numbers are real numbers. Real numbers are numbers that can be found on the number line, including integers, fractions, decimals, and irrational numbers.

What Are Real Numbers?

Real numbers are all the numbers that can be found on the number line. They include:

Natural numbers (1, 2, 3, ...)
Whole numbers (0, 1, 2, 3, ...)
Integers (..., -2, -1, 0, 1, 2, ...)
Rational numbers (fractions like 1/2, 3/4)
Irrational numbers (numbers that cannot be expressed as fractions, like √2, π)

Real numbers are distinct from complex numbers, which include an imaginary component (like 2 + 3i).

How to Determine Real Products

The product of two numbers is real if both numbers are real numbers. If either number is complex, the product will also be complex.

If a and b are real numbers, then a × b is a real number. If either a or b is complex, then a × b is complex.

To determine if a product is real:

Check if both numbers are real numbers
If both are real, their product is real
If either is complex, the product is complex

Examples

Example 1: Real Product

Numbers: 3 (real) and 5 (real)

Product: 3 × 5 = 15 (real)

Example 2: Complex Product

Numbers: 2 (real) and 3 + 4i (complex)

Product: 2 × (3 + 4i) = 6 + 8i (complex)

Example 3: Real Product with Irrational Numbers

Numbers: √2 (irrational real) and √3 (irrational real)

Product: √2 × √3 = √6 (real)

FAQ

What is the difference between real and complex numbers?

Real numbers can be plotted on the number line and include all rational and irrational numbers. Complex numbers include an imaginary component (like i = √-1) and cannot be plotted on the standard number line.

Can the product of two real numbers be complex?

No, the product of two real numbers will always be a real number. Complex numbers only result when multiplying a real number by a complex number.

How do I know if a number is real or complex?

A number is real if it can be expressed without the imaginary unit 'i'. If it includes 'i' (like 3 + 4i), it's complex.