/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Is the number rational or irrati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Is the number rational or irrational? $$\sqrt{5}+1$$

Short Answer

Expert verified
The number \(\sqrt{5} + 1\) is an irrational number.

Step by step solution

01

Identify the individual terms

The expression \(\sqrt{5}+1\) consists of two terms, namely \(\sqrt{5}\), which is an irrational number, and \(1\), which is a rational number.
02

Determine the irrationality

The square root of a non-square number such as 5 is always irrational. Therefore, \(\sqrt{5}\) is an irrational number.
03

Add the irrational and rational numbers

Add \(\sqrt{5}\) and \(1\). No matter the operation, the sum of a rational and an irrational number is always an irrational number.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Square Roots
Square roots are fascinating because they allow us to find a number which, when multiplied by itself, gives us the original number. For instance, the square root of 9 is 3, because \(3 \times 3 = 9\). However, not all numbers have simple square roots.
  • A perfect square like 9 has an integer square root.
  • When we have a non-perfect square, like 5, its square root \( \sqrt{5} \) is not a simple integer.
In such cases, the square root is often an irrational number. This is because it cannot be precisely expressed as a fraction or a neat decimal.
Irrational Number
An irrational number is one that cannot be expressed as a fraction \( \frac{a}{b} \), where both \(a\) and \(b\) are integers and \(b eq 0\). Instead, irrational numbers have non-repeating, non-terminating decimal expansions. Common examples include pi (蟺) and the square root of non-perfect squares like \( \sqrt{5} \).
  • Irrational numbers cannot be precisely written as a fraction.
  • They are often found as roots of non-perfect squares or as transcendental numbers like 蟺.
These numbers are crucial for many calculations in geometry, engineering, and the sciences. They help us understand real-world quantities that aren't neatly compartmentalized.
Rational Number
A rational number is essentially the opposite of an irrational number. It can be written as a simple fraction \( \frac{a}{b} \), where both \(a\) and \(b\) are integers and \(b eq 0\). This includes whole numbers, integers, fractions, and repeating or terminating decimals. For example, the number 1 is rational because it can be expressed as \( \frac{1}{1} \).
  • Includes whole numbers and simple fractions.
  • Examples include fractions like \( \frac{3}{4} \), integers like -2, and decimal numbers that stop or repeat like 0.333...
  • These numbers are widely used in everyday calculations.
Rational numbers form a straightforward and concrete part of mathematics, easily expressed and understood.
Addition of Rational and Irrational Numbers
When you add a rational number to an irrational number, the result will always be irrational. In the exercise, the addition of \(\sqrt{5}\) (an irrational number) to 1 (a rational number) results in \(\sqrt{5} + 1\). Since \(\sqrt{5}\) cannot be neatly expressed as a fraction or a terminating or repeating decimal, the sum combines both characteristics:
  • The irrationality of \(\sqrt{5}\) overshadows the rationality of 1.
  • The sum is non-repeating and non-terminating as well, ensuring the result is irrational.
This principle is a helpful guideline in mathematics: mixing different types of numbers carefully reflects their properties in the result.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the product of two rational numbers is a rational number.

Give the domain and range of the function. $$f(x)=|2 x+1|$$

Form the composition \(f \circ g\) and give the domain. $$f(x)=\sqrt{1-x}, \quad g(x)=2 \cos x \text { for } x \in[0,2 \pi]$$

Express the volume of a cube as a function of the total surface area.

(a) Express the perimeter of a semicircle as a function of the diameter. (b) Express the area of a semicircle as a function of the diameter.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.