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
1 · Read
Numbers come in families, and each family was invented because someone needed it. Counting numbers like 1, 2, 3 came first, for tallying sheep and grain. Zero arrived later, and negative numbers later still, giving the integers: ..., -3, -2, -1, 0, 1, 2, 3, ... Fractions came along for sharing and measuring. All of these families together, plus one more you will meet below, make up the real numbers.
A rational number is any number you can write as a fraction of two integers, one integer over another, with the bottom one never zero. Every integer counts: 7 is just 7/1. Here is the part worth remembering. Divide the top by the bottom and the decimal always does one of two things. It either stops completely, or it repeats a block of digits forever. It works the other way too: any decimal that stops or repeats can be turned into a fraction of integers.
Try 2/5. Dividing 2 by 5 gives 0.4, which stops right there. Now try 4/11. Dividing gives 0.363636..., and the block 36 repeats forever. Even when the repeating block is several digits long, it still counts. Both decimals follow the stop-or-repeat rule, so 2/5 and 4/11 are both rational.
Irrational numbers refuse to play by that rule. No fraction of two integers ever lands exactly on one, so its decimal runs forever with no repeating block at all. The square root of 2 is famous: it is the length of the diagonal of a square with sides of length 1, and no fraction measures it exactly. Pi is another one. People often write pi as 3.14, but that is only an estimate, since a decimal that stops is rational and pi is not.
Between any two numbers, however close, there are infinitely many others: 5.1, 5.01, 5.001, and on and on. That holds for a huge gap like 0 to 1000 and for a tiny one like 0.5 to 0.6. Every gap on the number line holds the same kind of endless crowd.
Integers, fractions, and more all belong to the real numbers. Rational numbers are fractions of two integers, with decimals that stop or repeat a block forever. Irrational numbers like the square root of 2 and pi never stop and never repeat, and between any two numbers there are infinitely many others.
Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org · license
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 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.