Problem 24 - A Graphical Approach to College Algebra

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

The decimal is an approximation, use \\(\sqrt{2} \approx 1.414213562\\).

Step 1: Understand the Nature of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 2, \(\sqrt{2}\), should multiply by itself to equal 2.

Step 2: Determine the Exactness

The number \(1.414213562\) is a decimal representation. Numbers like \(\sqrt{2}\) are irrational, meaning they cannot be precisely written as a finite or repeating decimal. Therefore, the number found is a result of rounding used by the calculator and not an exact number.

Step 3: Use Proper Notation

Since the decimal 1.414213562 is not infinite, it cannot be exact. This means \(\sqrt{2}\) is not equal to 1.414213562, but rather \(\sqrt{2} \approx 1.414213562\). The symbol \(\approx\) signifies 'approximately equal to' which should be used in the student's answer.

Key Concepts

Irrational Numbers

Irrational numbers are fascinating because they cannot be represented as simple fractions. These numbers, like \( \sqrt{2} \), do not have terminating or repeating decimal forms. This unpredictability is what makes them 'irrational.' When multiplied by themselves, they give us a whole number, but their decimal expressions go on forever without repeating. This means you can never write out the whole decimal; there's always more to discover!

• They cannot be exactly represented as fractions.
• Examples include \( \pi \), \( e \), and \( \sqrt{2} \).

Consider \( \sqrt{2} \): It's impossible to write it out completely in decimal form because its digits continue infinitely without a pattern. Understanding this characteristic is essential because it impacts how we express these numbers, especially when using calculators, which can only provide approximations.

Decimal Approximation

Decimal approximations help us handle irrational numbers in practical situations. Since these numbers have infinite, non-repeating decimals, a calculator rounds them to a manageable number of decimal places. For \( \sqrt{2} \), your calculator might show \( 1.414213562 \), which is just an approximation.

• This rounded number is useful in calculations, even though it is not exact.
• It allows us to work with irrational numbers in daily scenarios.

Practically, we write \( \sqrt{2} \approx 1.414 \) or something similar, using fewer decimal places for ease and to fit the context of the problem. Remember that approximation is an important skill, enabling us to use irrational numbers effectively in equations and measurements.

Mathematical Notation

Mathematical notation provides a clear and concise way of writing mathematical ideas, which is especially useful when dealing with approximations. When a number is not exact, as with irrational numbers, we use the symbol \( \approx \) to indicate that it is approximate. In contrast, the symbol \( = \) is reserved for exact equality, which we cannot attribute to the approximation of \( \sqrt{2} \).

• \( = \) shows that two values are exactly the same.
• \( \approx \) tells us that a value is very close to, but not exactly, another value.

Using \( \sqrt{2} \approx 1.414213562 \) communicates that although we have calculated a decimal value for \( \sqrt{2} \), it represents just an approximation. Properly using these symbols helps keep our mathematical statements honest and clear, ensuring everyone understands the level of precision involved.