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Chapter 0: Problem 27

Determine whether each statement is true or false. Every irrational number is an integer

Short Answer

Expert verified
False

Step by step solution

01

- Understand Irrational Numbers

An irrational number is a number that cannot be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b e 0\). Irrational numbers have non-terminating and non-repeating decimal expansions.
02

- Understand Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.
03

- Compare Irrational Numbers to Integers

Since irrational numbers have non-terminating and non-repeating decimals, they cannot be expressed as whole numbers (integers). Therefore, irrational numbers and integers do not overlap.
04

- Conclusion

Based on the definitions and comparison, it is clear that an irrational number cannot be an integer.

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Key Concepts

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

Integers
Integers are one of the most fundamental concepts in mathematics. They include all whole numbers, both positive and negative, as well as zero. Examples of integers are -3, 0, 4, and 27. Importantly, integers do not include fractions or decimals. So, 1.5 is not an integer because it contains a fractional part. Similarly, integers do not have any non-terminating decimal expansion; they are always exact whole numbers.
decimal expansions
Decimal expansions refer to the way we express numbers using digits after the decimal point. There are two main types of decimal expansions: terminating and non-terminating. Terminating decimals, like 0.75 or 2.5, have a finite number of decimal places. Non-terminating decimals can be repeating or non-repeating. A repeating decimal has a sequence of digits that repeats indefinitely, such as 0.666..., which is often written as 0.6虆. Non-repeating, non-terminating decimals are typically irrational numbers, which cannot be expressed as a simple fraction. Examples include numbers like \(\sqrt{2}\) and \(\pi\ \).
Whole Numbers
Whole numbers are a subset of integers. They include all the positive integers along with zero, but not the negative ones. Therefore, examples of whole numbers are 0, 1, 2, and 3. It is crucial to distinguish between whole numbers and other types of numbers. Whole numbers do not have any fractions or decimal parts. Like integers, the decimal expansion of any whole number is finite and non-terminating decimals do not apply to them.

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Most popular questions from this chapter

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ 58 \cdot \frac{3}{2}-8 \cdot \frac{3}{2} $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ -8>-|-6| $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ 10 \geq 10 $$

Simplify each expression. $$ -7(\sqrt{36})-(-2)(-3) $$

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ w-6 x+5 y-3 z $$
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