Problem 13 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\sqrt{-4}\) is not a real number, because no real number _____________equals \(-4\).

Step-by-Step Solution

Answer

The blank is filled with 'squared'.

Step 1: Identify the square root nature

The expression \( \sqrt{-4} \) involves finding a number that, when squared, equals \(-4\).

Step 2: Understanding squares of real numbers

A square of a real number, whether positive or negative, results in a non-negative value. Therefore, no real number squared can result in a negative value.

Step 3: Conclusion from observations

Given that squaring a real number cannot produce a negative, it follows that \( \sqrt{-4} \) is not a real number. Therefore, no real number squared results in \(-4\). Thus, the blank to be filled is 'squared'.

Key Concepts

Real Numbers

Real numbers are the numbers you typically use in everyday life. They include integers like -3, 0, and 5, as well as fractions and decimals like 0.75 or -2.5.
Real numbers also encompass all positive and negative numbers, except fractions and decimals that repeat not ending, which are known as irrational numbers. Examples include numbers like \(a\), \(-3/4\) and so much more.
Real numbers play a critical role in our understanding of mathematics because they form what we think of as the number line, a visual representation of all these numbers in a continuous sequence.

• They are denoted by the symbol \(R\).
• A real number can be positive, negative, or zero.

They are important because they help us understand various mathematical concepts like square roots and negative numbers, both of which often lead students to new branches of mathematics.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \( 3 0\) equals 9. This means that finding the square root is essentially the reverse operation of squaring a number.
When considering real numbers, square roots of positive numbers are straightforward. For example, the square root of 16 is 4. But finding square roots of negative numbers in the real number system is not possible because a negative number squared always results in a positive number.
In the context of real numbers:

• Positive numbers have two square roots: positive and negative.
• Zero has a single square root which is itself.
• Negative numbers would require us to explore the realm of imaginary numbers.

This is because no real number exists that can revert a negative outcome when squared.

Negative Numbers

Negative numbers are values less than zero and appear on the left side of the number line. These numbers have unique properties and rules, especially when dealing with operations like addition, subtraction, multiplication, and division.
One key characteristic of negative numbers is their behavior within multiplication. When a negative number is squared (multiplied by itself), the result is a positive number. For instance, \( (-3)^2 = +9 \). Similarly, when two negatives are multiplied, the outcome is positive.

• Negative times Negative = Positive
• Negative times Positive = Negative

This helps explain why taking the square root of negative numbers isn't possible in the real number system. Since a negative multiplied by a negative always results in a positive, you can't find a real square root for a negative number. This concept leads to the introduction of imaginary numbers, bridging gaps where negative square roots cannot be understood through real numbers alone.