Number Sets & Infinity
Appreciate the infinite nature of the sets of integers, real numbers, and rational numbers; position integers on a number line and distinguish between rational and irrational numbers
What a learner can do afterwards
- Explain that rational numbers can be written as a fraction of two integers and have terminating or repeating decimals
- Give examples of irrational numbers and explain why their decimal expansions neither terminate nor repeat
- Appreciate that between any two numbers there are infinitely many other numbers
The lesson
Every number you have used so far, like 3, 1/2, or 0.75, is a rational number. A rational number can always be written as a fraction of two integers, a over b. Its decimal either stops completely or repeats a pattern forever.
Look at 1/2 and 1/3. 1/2 equals 0.5, and the decimal stops there. 1/3 equals 0.333..., and the 3 repeats forever. Both are rational, because both come from a fraction of integers.
Some numbers can never be written as a fraction of integers. These are irrational numbers. Their decimals go on forever without ever settling into a repeating pattern. Square root of 2 and pi are two famous examples.
You can never finish counting all the numbers between two points, no matter how close together they are. That is what makes the rationals and irrationals infinite sets.
Rational numbers are fractions with decimals that stop or repeat, irrational numbers like square root of 2 and pi never do, and infinitely many numbers sit between any two points on the line.
Watch it
Where it sits
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.