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Chapter 1: Problem 17

Determine whether the following real numbers are integers, rational, or irrational. $$ -7 $$

Short Answer

Expert verified
-7 is both an integer and a rational number.

Step by step solution

01

Understand the Definitions

Before we determine the classification, let's understand the definitions: - **Integer**: A whole number that can be positive, negative, or zero, such as -3, 0, 7.- **Rational Number**: A number that can be expressed as a fraction of two integers (where the denominator is not zero), such as \( \frac{2}{3} \), 4, and -5/1.- **Irrational Number**: A number that cannot be expressed as a simple fraction; it has non-repeating, non-terminating decimal forms, such as \( \sqrt{2} \) and \pi.
02

Categorize the Given Number

The number given in the exercise is -7. We need to categorize -7 using the definitions provided in Step 1.
03

Determine if the Number is an Integer

Check if -7 is an integer. Integers include negative numbers, zero, and positive numbers without any fractional or decimal part. -7 is a whole number and it's negative, so it is an integer.
04

Determine if the Number is Rational

For -7 to be a rational number, it should be expressible as a fraction of two integers. We can express -7 as \( \frac{-7}{1} \), which is a ratio of integers. Therefore, -7 is a rational number.
05

Determine if the Number is Irrational

Based on the definitions, -7 is **not** irrational because it can be expressed as a fraction (as shown in Step 4) and it is a whole number with a terminating decimal, hence it's a rational number.

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Key Concepts

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

Integer
Numbers like -7, that are whole and can be negative, zero, or positive, are called integers. Unlike fractions or decimals, integers don鈥檛 have any parts besides themselves. For example, -3, 0, and 5 are all integers.
They are the building blocks of arithmetic, forming a crucial part of the number line. Understanding integers is a fundamental step in grasping more complex math concepts.
  • An integer can be:
  • Positive (such as 1, 2, 3)
  • Negative (such as -1, -2, -3)
  • Zero, which is neutral and not positive or negative
This simplicity makes integers easy to work with and essential for basic operations like addition and subtraction.
Rational Number
A rational number is any number that you can write as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. For instance, -7 can be expressed as \( \frac{-7}{1} \), so it is a rational number.
  • Rational numbers include integers, since any integer \( n \) can be written as \( \frac{n}{1} \).
  • They also include fractions like \( \frac{3}{4} \) and terminating decimals like 0.5.
  • Some repeating decimals are also rational, like 0.333... which is \( \frac{1}{3} \).
Rational numbers cover a broad range, making them versatile for everyday math calculations.
Irrational Number
Irrational numbers are intriguing because they can't be expressed as a simple fraction. Their decimal forms go on forever without repeating. A classic example is \( \pi \), which is about 3.14159... and never ends or repeats.
  • Another example is \( \sqrt{2} \) which is approximately 1.414 and is also non-repeating and non-terminating.
  • Irrational numbers often appear unexpectedly in geometry and algebra.
Understanding irrational numbers helps in delving deeper into theoretical mathematics, where precise calculations are necessary beyond simple fractions.

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