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Chapter 9: Problem 116

Precision Is it true that \(\sqrt{2}=1.414\) ? Explain.

Short Answer

Expert verified
No, \(\sqrt{2}\) is not exactly equal to 1.414. However, 1.414 is a commonly used approximation for \(\sqrt{2}\) in many practical use cases.

Step by step solution

01

Understanding the concept

We know that \(\sqrt{2}\) is an irrational number, which means it doesn鈥檛 have an exact decimal representation. Its decimal representation is non-terminating and non-repeating.
02

Explanation of precision

The provided statement \(\sqrt{2}=1.414\) is an approximation where the number is rounded to three decimal places. This is an implicit indication of precision to the thousandths place, which is used for simplicity in many mathematical and scientific operations. However, it's important to know that it's not absolutely precise and some level of accuracy is lost.
03

Result

Hence, the statement \(\sqrt{2}=1.414\) is not precisely true, but it is an acceptable approximation for most practical purposes where high precision isn't mandatory.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Approximation
Decimal approximation is a method used to represent non-terminating decimals in a more manageable form. When dealing with irrational numbers like \( \sqrt{2} \), whose decimal form goes on forever without repeating, we often use an approximation鈥攁 finite, more user-friendly version. For instance, although the exact value of \( \sqrt{2} \) contains an infinite number of decimals, we simplify it to 1.414 for convenience.

  • This approximation is useful in calculations where exact precision is unnecessary.
  • It provides a quick and easy way to communicate values in everyday scenarios.
Approximations help deal with numbers practically, but it's essential to remember they can never fully capture the 'true' value of an irrational number.
Mathematical Precision
Mathematical precision refers to the exactness of a number in mathematical expressions and calculations. The phrase "precision to the thousandths place" implies rounding a number to three decimal points. When we say \( \sqrt{2} = 1.414 \), we're acknowledging that while this is a close approximation, it鈥檚 not the exact number.

  • High precision is critical in industries like engineering and computer science.
  • In daily life and non-critical applications, rounding is often sufficient.
Precision affects how accurately calculations reflect real-world values, and while approximations are useful, understanding their limitations is vital.
Non-terminating Decimals
Non-terminating decimals are decimals that continue endlessly without repeating a pattern. This is typical of irrational numbers like \( \pi \) and \( \sqrt{2} \). Since they never stop or repeat, these numbers cannot be fully captured in a finite decimal form.

  • This quality differentiates irrational numbers from rational numbers.
  • Calculations with these numbers require special approaches like symbolic representation.
Handling non-terminating decimals often involves using a sequence of digits that can practically represent the number to the required level of precision.
Number Representation
Number representation is how we express numbers in different forms鈥攕uch as as fractions, decimals, or symbols鈥攄epending on our needs. For irrational numbers, this can involve:
  • Using decimal approximations to simplify their use in everyday calculations.
  • Representing them symbolically (e.g., \( \sqrt{2} \)) to maintain mathematical integrity.
Choosing the right form of representation balances the need for precision with ease of communication, helping to convey complex mathematical ideas efficiently.

By understanding how numbers can be represented, learners can appreciate both the power and limitations of mathematical tools at our disposal.

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