Unit 16: Irrational Numbers
Not every number can be written as a fraction -- irrational numbers fill in the gaps the rationals leave behind on the number line.
4 topics · 12 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Classifying the Real Number System
- How it worksHow do the rational and irrational numbers together fill in every point on the real number line?
- Worked exampleShow me how to classify the numbers -3, 0.75, the square root of 2, and 7 within the real number system.
- Common mistakeWhy isn't every square root automatically an irrational number?
Your turn1 of 2
Is the square root of 72 rational or irrational?Why Some Square Roots Never Terminate or Repeat
- How it worksHow does assuming the square root of 2 is a fraction lead to a contradiction that proves it isn't?
- Common mistakeWhy doesn't a decimal that goes on forever automatically make a number irrational?
- Going furtherWhat happens to a perfect square's square root that keeps it from ever being irrational?
Your turn1 of 2
A decimal goes on forever, repeating the same two digits 76 over and over. Is this number rational?Approximating and Comparing Irrational Numbers
- How it worksHow do the nearest perfect squares above and below a number bound its square root's approximate value?
- Worked exampleFind an approximation for the square root of 40 to the nearest tenth using the nearest perfect squares.
- Worked exampleFind the correct order from least to greatest for 3, the square root of 8, and the square root of 15.
Your turn1 of 2
Find the value of the square root of 51, accurate to the nearest tenth.Operations on Rational and Irrational Numbers
- Worked exampleShow me why 4 plus the square root of 3 must be irrational.
- Common mistakeWhy isn't the sum of two irrational numbers always irrational?
- Going furtherWhat happens when you multiply the square root of 2 by itself?
Your turn1 of 2
Find the value of 7 plus the square root of 10.