Real numbers are all numbers that can be found on the number line, including positive numbers, negative numbers, and zero. Imaginary numbers involve the imaginary unit i, where i equals the square root of negative one. In the complex plane, real numbers lie on the horizontal axis, while pure imaginary numbers lie on the vertical axis.
Real numbers include all numbers that can be placed on the number line: positive numbers, negative numbers, zero, fractions, and irrational numbers like pi and square root of two. Imaginary numbers are defined using the imaginary unit i, where i equals the square root of negative one. Unlike real numbers, imaginary numbers cannot be placed on the traditional number line.
The squaring test is a simple way to distinguish real from imaginary numbers. When you square any real number, the result is always non-negative. For example, five squared equals twenty-five, and negative four squared equals sixteen. However, when you square a pure imaginary number, the result is always negative. For instance, three i squared equals negative nine, and negative i squared equals negative one.
The complex plane provides a visual way to understand the difference between real and imaginary numbers. Real numbers lie entirely on the horizontal axis, while pure imaginary numbers lie on the vertical axis. For example, the number 3 is plotted at coordinates (3, 0), and 2i is plotted at (0, 2). Complex numbers like 1 plus 3i combine both real and imaginary parts.
To summarize the key differences: if a number contains the letter i, it's imaginary; if it doesn't, it's real. The squaring test is also useful - real numbers squared give non-negative results, while imaginary numbers squared give negative results. In the complex plane, real numbers lie on the horizontal axis and imaginary numbers on the vertical axis. These simple rules help distinguish between real and imaginary numbers in any mathematical context.