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. 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
More to read
1.1 Real Numbers: Algebra Essentials
Numbers come in different families. Counting numbers like 1, 2, 3 are called natural numbers. Add zero and you get whole numbers. Add negatives like -1 and -2 and you get integers.
A rational number is any number you can write as a fraction of two integers, like 3/4 or -5/2. When you turn a rational number into a decimal, it either stops (like 0.75) or repeats forever in a pattern (like 0.333...).
An irrational number can't be written as a fraction of two integers. Its decimal goes on forever without ever settling into a repeating pattern. Famous examples are pi and the square root of 2.
Put rational and irrational numbers together and you get the real numbers, the numbers you can place anywhere on the number line.
Key Concepts
Rational numbers can always be written as a fraction, or as a decimal that either terminates (stops) or repeats forever in a pattern. To check if a number is rational, try writing it as a decimal. If the decimal stops or repeats, it's rational. If it goes on forever without repeating, it's irrational. Rational numbers and irrational numbers together make up the set of real numbers.
1.8 The Real Numbers
Every real number has a home on the number line. Integers like -2, 0, and 5 sit at whole-number spots. Fractions like 1/2 sit exactly halfway between two integers, and you can find the spot for any fraction by dividing the space between integers into equal parts.
Decimals work the same way. A decimal like 0.6 sits six tenths of the way from 0 to 1. Some decimals, like 0.25, stop after a few digits, these come from rational numbers. Others, like the decimal form of the square root of 2, never stop and never settle into a repeating pattern, these come from irrational numbers.
Knowing whether a number is an integer, a rational number, an irrational number, or a real number helps you predict how its decimal will behave and exactly where to place it on the line.
Introduction to the Properties of Real Numbers
Think about rearranging the pieces of a quilt. The same pieces can make very different patterns depending on how you arrange them. Numbers have their own kind of building blocks too.
Real numbers split into two big groups: rational numbers, which can be written as a fraction of two integers and have decimals that stop or repeat, and irrational numbers, which can't be written as a fraction and have decimals that never stop or repeat. Together these two groups make up every real number you'll ever use, from the integers you count with to numbers like pi that show up in circles.
Projects: Cardinality of Infinite Sets
Not all infinities are the same size. Mathematicians have shown that the set of irrational numbers is actually a bigger kind of infinite than the set of natural numbers (1, 2, 3, and so on).
The natural numbers are called countably infinite, you could list them one at a time forever. The irrational numbers are uncountably infinite, there are so many of them packed onto the number line that you could never list them one at a time, even given forever to try.
This idea took mathematicians a long time to prove and figure out.
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2 · Watch
En anglaisTake 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.