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Chapter 0: Problem 2

In Exercises determine whether the real number is rational or irrational. $$-3678$$

Short Answer

Expert verified
-3678 is a rational number.

Step by step solution

01

Understand the definition of Rational and Irrational numbers

Rational numbers are numbers that can be expressed as a ratio of two integers (where the denominator is not zero), while irrational numbers cannot be expressed as such a ratio.
02

Apply the definition to the given number

The number provided, -3678, can be expressed as a ratio of two integers. For example, it can be written as -3678/1. The numerator (-3678) and the denominator (1) are both integers, and the denominator is not zero.

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Key Concepts

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

Real Numbers
Did you know that real numbers are a huge family of numbers that include almost every type of number you can think of? They encompass several different kinds of numbers, such as whole numbers, fractions, irrational numbers, and many more. Real numbers are the vast universe in which all other number types exist.

Here are some characteristics of real numbers:
  • Real numbers include all rational numbers. These are numbers that can be written as a fraction, like \( \frac{1}{2} \) or \(-3678\).
  • They also include irrational numbers, like \( \pi \) and \( \sqrt{2} \), which cannot perfectly be written as a fraction.
  • Owning a property called 'being continuous,' real numbers can be represented as points on a number line, filling every possible gap without a break.
This explains why real numbers are so fundamental and versatile in mathematics. When we determine whether a number is rational or irrational, we are examining its place within the larger set of real numbers.
Integers
Integers are like building blocks in the world of mathematics. They make up the set of whole numbers that include all positives, negatives, and zero. In math, integers serve as a basic foundation for operations and equations.

Here鈥檚 what makes integers special:
  • Integers are all around us, from \(-10\) to \(0\) to \(245\), and keep on going infinitely in both directions.
  • They are straightforward to calculate with, making addition, subtraction, and multiplication quite predictable.
  • When forming a ratio, such as \(-\frac{3678}{1}\), integers are used in the numerator and the denominator to express rational numbers.
Understanding integers helps ease the way into grasping more complex concepts, like rational numbers and how they fit into the real number system.
Ratio of Integers
The ratio of integers is a fundamental concept that is central to defining rational numbers. It involves expressing a number as a division or fraction involving two integers. This is the concept that tells us whether a number is rational or not.

Consider these points about the ratio of integers:
  • A rational number, such as \( \frac{a}{b} \), is made when two integers, \( a \) and \( b \), are used, with \( b eq 0 \).
  • This relationship allows for almost every fraction you can think of, turning complex decimal expansions into something simpler to handle.
  • Even whole numbers can be seen as rational numbers. For example, the number \(-3678\) can be written as \(-\frac{3678}{1}\), showing it as a ratio of integers.
By learning how numbers can be created from a simple ratio of integers, we gain a deeper understanding of the structure and classification of numbers in math.

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Most popular questions from this chapter

Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. $$ x^{3}-6 x^{2}+11 x-6 $$

evaluate the expression for the given value of x. $$ \left(x^{2 / 3}\right)^{3} \quad x=10 $$

simplify by removing all possible factors from the radical. $$ \sqrt[4]{\left(3 x^{2} y^{3}\right)^{4}} $$

Rationalize the numerator or denominator and simplify. $$ \frac{2}{\sqrt{x}+\sqrt{x-2}} $$

simplify by removing all possible factors from the radical. $$ \sqrt{8} $$
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