Irrational Numbers

Overview

Master the logic of real numbers. Learn to convert fractions and decimals, identify repeating patterns, and define irrational constants like π and e. Gain practical skills in bounding square roots and ordering number sets to solve real-world estimation problems with precision.

Syllabus

Course 1: Rational Numbers and Terminating Decimals
Course 2: Mastery of Repeating Decimals
Course 3: The Real Number System and Irrationality
Course 4: Estimating and Comparing Real Numbers

Courses

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You will establish the foundational definition of rational numbers and master the relationship between fractions and terminating decimals. You will use long division to observe how decimals behave and learn to move fluently from terminating decimals back to their simplest fraction forms.
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You will dive deep into the world of repeating decimals. You will learn to use professional notation to represent infinite loops, distinguish between pure and mixed repeating blocks, and use algebraic methods to convert these infinite strings back into exact rational fractions.
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You will meet the numbers that refuse to be written as fractions. You will learn why their decimal expansions never terminate or repeat, recognize famous examples like π and e, and identify irrationals generated by square roots. Finally, you will learn to bound these numbers between integers to understand their place on the number line.
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You will turn irrational numbers into practical tools by producing rational approximations. Using trial squaring, calculator rounding, and number line plotting, you will learn to compare irrationals with rationals and estimate the results of real-world arithmetic expressions.