Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Using her calculator, a student found the decimal 1.414213562 when she evaluated \(\sqrt{2} .\) Is this decimal the exact value of \(\sqrt{2}\) or just an approximation of \(\sqrt{2} ?\) Should she write \(\sqrt{2}=1.414213562\) or \(\sqrt{2} \approx 1.414213562 ?\)

Step-by-Step Solution

Answer

\(\sqrt{2} \approx 1.414213562\) because it is an approximation.

Step 1: Understanding the Problem

We need to determine if the decimal 1.414213562 is the exact value of \(\sqrt{2}\) or just an approximation.

Step 2: Identifying a Property of \(\sqrt{2}\)

The square root of 2 (\(\sqrt{2}\)) is an irrational number, which means it cannot be expressed as a finite or repeating decimal expansion.

Step 3: Examining the Given Decimal

The given decimal 1.414213562 is finite with a specific number of digits, suggesting it is a representation in decimal form.

Step 4: Concluding the Nature of \(\sqrt{2}\)

Since \(\sqrt{2}\) is irrational and cannot be expressed as a finite or repeating decimal, the provided decimal (1.414213562) must be an approximation of \(\sqrt{2}\).

Step 5: Answering the Question

The student should write \(\sqrt{2} \approx 1.414213562\) because the given decimal is an approximation and not an exact value.

Key Concepts

Square Root

The square root is one of the fundamental operations in mathematics. It signifies a number which when multiplied by itself results in the original number. For instance, the square root of 2, noted as \(\sqrt{2}\), is a number that provides 2 when squared, or \(\sqrt{2} \times \sqrt{2} = 2\). Understanding square roots is essential because they appear frequently in algebra and geometry, such as in formulas to calculate areas and angles.

The challenge with numbers like \(\sqrt{2}\) arises because they can't be easily written as exact decimal numbers. \(\sqrt{2}\) belongs to a group of numbers known as irrational numbers. These are numbers that cannot be exactly formulated as a simple fraction or decimal. For example, unlike the square root of 4, which is exactly 2, \(\sqrt{2}\) doesn't have a neat decimal representation.

Decimal Approximation

In many mathematical contexts, we need to use an approximation of numbers like \(\sqrt{2}\). An approximation is a number that's close enough to another number for some practical purposes. The decimal 1.414213562 represents one such approximation for \(\sqrt{2}\).

Approximations help to use complex numbers for calculations without needing to be infinite or overly complex. They provide:

• A simpler way to work with irrational numbers in computations.
• A practical means to estimate and convey values.

For example, when calculating the diagonal of a square, \(\sqrt{2}\) might be annually approximated as 1.414.

It's important to remember that approximations are not exact. They carry a degree of error depending on how many decimal places are used. The more digits you keep, the closer your approximation is to the real number.

Exact Value vs Approximation

Recognizing the difference between exact values and approximations is crucial in mathematics. An exact value, like \(\pi\) or \(\sqrt{2}\), remains constant and doesn't change regardless of situation. It represents the true value of a number without deviation.

In contrast, an approximation denotes a rough or near-correct value of the actual number. This is often necessary because some numbers, like \(\sqrt{2}\), are irrational and cannot be perfectly expressed in decimal form.

So, when it comes to \(\sqrt{2}\), though the calculator might display 1.414213562, this is just the best estimate given certain decimal limits. Hence, the proper way to depict it in calculations is \(\sqrt{2} \approx 1.414213562\), signifying the number is nearly but not exactly this decimal value.

Understanding this distinction helps ensure precision in mathematical communication and problem-solving. It also underlines the importance of context when choosing between exact values and approximations in calculations.