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Real Numbers

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Real Numbers

Study Guide

Key Definition

Real numbers include all rational and irrational numbers. Every real number can be written as a decimal expansion (terminating, repeating, or non-repeating), and they are used to measure continuous quantities like temperature and distance.

Important Notes

Real numbers include both <a href="rational-numbers">$rational$</a> and <a href="irrationals">$irrational$</a> numbers
Rational numbers can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers
Irrational numbers cannot be expressed as a fraction
Every real number can be written as a decimal expansion (terminating, repeating, or non-repeating).
Real numbers are used in $continuous measurements$

Mathematical Notation

$\mathbb{R}$ represents the set of real numbers

$0.\overline{3}$ denotes repeating decimals

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents a square root

$x^2$ represents a squared number

$+$ represents addition

$-$ represents subtraction

Remember to use proper notation when solving problems

Why It Works

Real numbers allow us to perform all arithmetic operations and represent all possible distances on a number line with $arbitrary precision$.

Remember

Real numbers are the $foundation$ of arithmetic and are essential in measuring real-world quantities.

Quick Reference

Rational Numbers:$\frac{a}{b}$ where $a$ and $b$ are integers and $b ≠ 0$

Irrational Numbers:Numbers like $\sqrt{2}$, $\pi$, and $e$

Understanding Real Numbers

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Real numbers are all the numbers on the number line, including <a href="integers">$integers$</a>, <a href="rational-numbers">$fractions$</a>, and <a href="irrationals">$irrationals$</a>.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is a rational number? $\frac{1}{2}$, $\sqrt{2}$, $\pi$, $e$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Write 50.75 in fractional form.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Is $\sqrt{3}$ a rational or irrational number? Explain your reasoning.

Challenge Quiz

Single Choice Quiz

Advanced

Which statement about real numbers is true? $\text{Real numbers include complex numbers}$, $\text{Real numbers are only rational numbers}$, $\text{Real numbers are only irrational numbers}$, $\text{Real numbers include both rational and irrational numbers}$

Recap

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Review key concepts and takeaways