Problem 51 - Elementary Algebra

Question

Explain why \(\sqrt{-16}\) cannot be a real number.

Step-by-Step Solution

Answer

The square root of a negative number, like \(\sqrt{-16}\), is not a real number.

Step 1: Understand the square root function

The square root function, denoted by \(\backslash sqrt{ \_ }\), involves finding a number that, when multiplied by itself, results in the given number. For example, \(\backslash sqrt{16} = 4\), because \(4 \times 4 = 16\).

Step 2: Consider the properties of real numbers

Real numbers are all the numbers that can be found on the number line, including both positive and negative numbers, as well as zero. However, when dealing with square roots, only non-negative numbers have real number solutions.

Step 3: Analyze the given expression \(\sqrt{-16}\)

For a square root to be a real number, the number inside the square root (the radicand) must be non-negative. In this case, \(\sqrt{-16}\), the radicand is \(-16\), which is negative.

Step 4: Conclude the outcome

Since \(\sqrt{-16}\) implies taking the square root of a negative number, it does not produce a real number because there is no real number that, when squared, results in a negative number. Therefore, \(\sqrt{-16}\) cannot be a real number.

Key Concepts

Real Numbers

Real numbers include all the numbers you can find on a number line. They cover:

• Positive integers (e.g., 1, 2, 3)
• Negative integers (e.g., -1, -2, -3)
• Fractions and decimals (e.g., 0.5, 2.75)
• Zero

Real numbers are essential in daily life and mathematics. However, one thing to remember is that real numbers do not include imaginary numbers. When it comes to square roots, real numbers only result from non-negative radicands, which means the number inside the square root sign must be zero or positive to have a real number as its square root.

Properties of Square Roots

Square roots have specific properties that help us understand them better.

• Multiplication: The square root of a product is the product of the square roots, \(\backslash sqrt{a \backslash times b} = \backslash sqrt{a} \backslash times \backslash sqrt{b}\).
• Division: Similarly, the square root of a quotient is the quotient of the square roots, \(\backslash sqrt{a / b} = \backslash sqrt{a} / \backslash sqrt{b}\).
• Non-negative: The square root function often only returns non-negative results, called the principal square root.

These properties apply to real numbers. When dealing with square roots, note that we cannot, under normal circumstances, take the square root of a negative number and get a real number.

Non-Negative Radicand

For the square root to result in a real number, the radicand must be non-negative. The radicand is the number inside the square root sign. For example, in \(\sqrt{16}\), the radicand is 16.

• If the radicand is positive, like \(\sqrt{25} = 5\), we get a positive real number.
• If the radicand is zero, like \(\sqrt{0} = 0\), we get zero, which is also a real number.
• If the radicand is negative, like \(\sqrt{-16}\), we cannot find a real number solution because no real number squared will result in a negative number.

This principle is key in understanding why \(\sqrt{-16}\) cannot be a real number.

Imaginary Numbers

Imaginary numbers come into play when dealing with the square roots of negative numbers. An imaginary number is a complex number where the real part is zero and the imaginary part is non-zero.
The imaginary unit is denoted by \(i\) and is defined as \(i = \sqrt{-1}\). Using this, we can express the square root of any negative number. For example:
\(\backslash sqrt{-16} = 4i\), because \(4i \backslash times 4i = -16\)
Imaginary numbers allow us to extend the concept of square roots beyond the limitations of real numbers.
They are widely used in various fields, such as engineering, physics, and complex number theory.