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Chapter 0: Problem 16

Write an English sentence to represent the algebraic statement. $$\frac{2}{5} \in \mathbb{Q}$$

Short Answer

Expert verified
The fraction 2 over 5 is a rational number.

Step by step solution

01

Identify the Mathematical Statement

The given mathematical statement is \(\frac{2}{5} \in \mathbb{Q}\).\
02

Understand the Symbols

The symbol \(\frac{2}{5}\) represents a fraction, while \( \mathbb{Q} \) denotes the set of all rational numbers.
03

Interpret the Membership Symbol

The symbol \(\in\) means 'is an element of' or 'belongs to'.
04

Formulate the Sentence

Combine the interpretations: the fraction \(2\) over \(5\) is an element of the set of rational numbers.

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Key Concepts

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

Rational Numbers
Rational numbers are fractions or ratios that can be represented as \(\frac{a}{b}\) where both \(a\) and \(b\) are integers, and \(b\) is not zero. These numbers are part of the set known as \( \mathbb{Q} \).
Examples include:
  • \( \frac{1}{2} \)
  • \( \frac{-3}{4} \)
  • \( \frac{7}{1}\) (which is simply 7)

Rational numbers can also be represented as decimals which either terminate or repeat.
For instance, \(\frac{1}{4}\) is 0.25, and \(\frac{1}{3}\) is approximately 0.333...
Understanding rational numbers is key to solving problems related to them.
Mathematical Symbols
Mathematical symbols help us communicate complex ideas in a simplified manner. Here are a few important ones:
  • The fraction symbol (\( \frac{}{} \)) is used to show a division between two numbers.
  • The element of symbol (\( \in \)) signifies membership, meaning that an item belongs to a set.
  • The set of rational numbers (\( \mathbb{Q} \)) includes all numbers that can be expressed as fractions.

Recognizing and understanding these symbols is crucial. They allow us to write mathematical sentences that convey specific meanings.
Fraction Representation
Fractions consist of two parts:
  • The numerator (top part) represents how many parts we have.
  • The denominator (bottom part) shows the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{2}{5} \, \:\text{2 is the numerator and 5 is the denominator. }\)
Fractions can be simplified if both the numerator and the denominator share common factors.
Understanding how to break down and interpret fractions helps in working efficiently with them in arithmetic and algebra.

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Most popular questions from this chapter

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.

Simplify each expression. Assume that \(m\) and \(n\) are integers and that \(x\) and \(y\) are nonzero real numbers. \(\frac{x^{4 m-3} y^{5 n+7}}{x^{m-7} y^{3 n+2}}\)

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ \left(5 a^{2} \sqrt{b}+7 b^{2} \sqrt{a}\right)^{2} $$

Explain the similarity in simplifying the given expressions. a. \((3 x+2)(4 x-7)\) b. \((3 \sqrt{x}+\sqrt{2})(4 \sqrt{x}-\sqrt{7})\)

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 v-\sqrt{17})^{2} $$
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