B. Number · Number: Number Sets, Powers and Rational Numbers in ActionDensity, Infinity and Limits: How Number Sets Behave
B1.3 A1About 70 minutes
Learning goals- I can explain what it means for a number set to be dense, and find rational numbers between any two rational numbers.
- I can explain why the natural numbers, integers and rational numbers are infinite sets.
- I can use patterns to describe a limit, such as a sequence of numbers that gets closer and closer to a value.
- I can explain why 0.999… is equal to 1.
Words to know
dense
A set is dense if between any two different numbers in the set there is always another number of the set. ℚ and ℝ are dense; ℤ is not.
infinite set
A set that goes on forever: it has no last element and cannot be completely counted.
infinity (∞)
The idea of being without end. ∞ is not a real number; it describes something that keeps growing without stopping.
sequence
A list of numbers in order, like 1/2, 3/4, 7/8, …
limit
The value a sequence gets closer and closer to (as close as you like) as it goes on.
mean (average) of two numbers
Their sum divided by 2. It is exactly halfway between them.
A question to argue about
What number comes right after 1? With the integers, the answer is easy: 2. Nothing fits between 1 and 2.
But with fractions, is it 1.1? No, 1.01 is closer. Then 1.001 is closer still. You can never name "the next" rational number after 1. This lesson is about three big ideas that explain why: density, infinity and limits.
Density: there is always room for one more
Take any two different rational numbers a and b. Their mean, (a + b) ÷ 2, is:
- rational, because ℚ is closed under addition and division by 2, and
- exactly halfway between them.
So between any two rationals there is another rational. Now repeat: between a and the mean there is another, and so on forever. That means there are infinitely many rationals between any two rationals. We say ℚ is dense.
The integers are not dense: there is no integer between 3 and 4. The natural and whole numbers are not dense either. The real numbers are dense, and so are the irrationals: between 1.4 and 1.5 is the irrational number √2 = 1.41421…
Worked exampleExample 1: Find three rationals between two fractions
Find three rational numbers between 2/5 and 1/2.
- Write both with a common denominator: 2/5 = 4/10 and 1/2 = 5/10. There is no tenth between them, so use a bigger denominator.
- Multiply by 4/4: 2/5 = 16/40 and 1/2 = 20/40.
- Now three fractions fit between them: 17/40, 18/40 and 19/40.
- Check with decimals: 2/5 = 0.4, 17/40 = 0.425, 18/40 = 0.45, 19/40 = 0.475, 1/2 = 0.5. ✓
17/40, 9/20 (= 18/40) and 19/40 (many other answers are possible)
Worked exampleExample 2: Squeeze between close decimals
Find a number between 0.3199 and 0.32.
- Write 0.32 with the same number of decimal places: 0.3200.
- Add a digit: 0.31990 and 0.32000. Any number from 0.31991 to 0.31999 works.
- Or use the mean: (0.3199 + 0.32) ÷ 2 = 0.6399 ÷ 2 = 0.31995.
0.31995 (one of infinitely many answers)
Infinity: no last number
The natural numbers have no largest element: whatever number you name, n, the number n + 1 is bigger. So ℕ is an infinite set. So are 𝕎, ℤ, ℚ and ℝ.
The symbol ∞ means "without end". It is not a real number, so you cannot do arithmetic with it the way you do with 5. When we write "n → ∞" we mean "as n keeps getting bigger without stopping".
The rational numbers are infinite in two ways: they go on forever in both directions on the number line, and there are infinitely many of them squeezed into any small interval, like between 0 and 0.001.
Limits: getting as close as you like
Look at the sequence 1/2, 3/4, 7/8, 15/16, 31/32, … Each term is closer to 1. The gap to 1 is 1/2, 1/4, 1/8, 1/16, …, and it gets halved each time. You can make the gap smaller than 0.001 or 0.000 000 1 just by going far enough. We say the sequence approaches 1, or the limit is 1. No term ever equals 1, but the limit does.
Another example: 1, 1/2, 1/3, 1/4, …, 1/n, … gets closer to 0 as n → ∞. And 0.9, 0.99, 0.999, … gets closer to 1.
Worked exampleExample 3: A limit in real life
A snowplough driver is 2 km from the depot. Each hour (in a pretend world) she drives half of the remaining distance. How far is she from the depot after 1, 2, 3 and 10 hours? What is the limit of the distance she has driven?
- After 1 h: remaining = 2 × 1/2 = 1 km. After 2 h: 2 × (1/2)² = 0.5 km. After 3 h: 2 × (1/2)³ = 0.25 km.
- Pattern: after n hours, remaining = 2 × (1/2)ⁿ km.
- After 10 h: 2 × (1/2)¹⁰ = 2/1024 ≈ 0.00195 km, about 2 m.
- The remaining distance approaches 0, so the distance driven, 2 − 2 × (1/2)ⁿ, approaches 2 km.
1 km, 0.5 km, 0.25 km, about 0.002 km; the distance driven has a limit of 2 km.
Worked exampleExample 4: Why 0.999… = 1
Show that 0.999… (9s forever) equals 1, in three ways.
- Algebra: Let x = 0.999… Then 10x = 9.999… Subtract: 10x − x = 9, so 9x = 9 and x = 1.
- Density: If 0.999… and 1 were different numbers, the mean of the two would be a number between them. But any decimal less than 1 is smaller than 0.999… once you go out enough places (for example 0.9999 < 0.99999). There is no room for a number between them, so they are not different.
- Limit: 0.999… means the limit of 0.9, 0.99, 0.999, …, and that limit is 1.
0.999… = 1. Two different-looking decimals can name the same real number.
Archimedes and the limit of π
About 2200 years ago, the Greek mathematician Archimedes drew polygons inside and outside a circle. As the number of sides grew, their perimeters closed in on the circle's circumference. With 96-sided polygons he showed that π is between 3 10/71 and 3 1/7. That is a limit at work: more sides, closer to π.
Hands-on activityThe tearing-paper limit
You need: one sheet of scrap paper.
- Tear the sheet in half. Put one half on the table. That is 1/2 of the sheet.
- Tear the piece in your hand in half. Put one half beside the first piece. Now the table has 1/2 + 1/4 = 3/4.
- Keep going: tear what you hold in half and put one half down. Record the fraction on the table each time: 7/8, 15/16, 31/32, …
- Stop when the pieces are too small to tear (usually after 7 or 8 tears).
Questions: How much is on the table after 8 tears? (255/256.) How much is still in your hand? (1/256.) If you could tear forever, what is the limit of the amount on the table? (1 whole sheet.) Why can you never actually reach it? (There is always a piece left in your hand.)
Just for funMathematicians imagine a Hilbert's Hotel with infinitely many rooms, all full. A new guest arrives. No problem: everyone moves from room n to room n + 1, and room 1 is free! (The front desk is very tired.)
Your turn
1Find a rational number between 3/4 and 4/5.
Common denominator 20: 15/20 and 16/20. Use 40ths: 30/40 and 32/40, so 31/40 works. (Or the mean: (0.75 + 0.8) ÷ 2 = 0.775.)
2Find a number between −1.25 and −1.24.
−1.245 (or any number like −1.241, −1.249).
3Is there an integer between −3 and −2? What does this tell you?
No. The integers are not dense.
4What is the next term of 1/2, 3/4, 7/8, 15/16, …, and what is the limit?
Next term 31/32; the limit is 1.
5What is the limit of 1/10, 1/100, 1/1000, …?
0. The terms get closer and closer to 0.
6Write 0.999… as a whole number and explain using algebra.
1. x = 0.999…, 10x = 9.999…, so 9x = 9 and x = 1.
7Why is ∞ not a real number?
It does not stand for a point on the number line. It describes growing without end; you cannot reach it or calculate with it like 5 or −2.
More practice: warm-up
1Find the mean of 0.4 and 0.5.
2Name a number between 2.7 and 2.8.
Infinite: there is always a next number, n + 1.
4Which set is dense: ℤ or ℚ?
5What value does 0.9, 0.99, 0.999, … approach?
More practice: on your own
1Find three rational numbers between 1/3 and 1/2.
Common denominator 12: 4/12 and 6/12. Use 24ths: 8/24 and 12/24. Three between: 9/24 (= 3/8), 10/24 (= 5/12), 11/24.
2Find an irrational number between 2 and 3.
Example: √5 ≈ 2.236 (or √6, √7, √8, or π − 1 ≈ 2.14).
3A ball is dropped from 4 m and each bounce reaches half the height of the one before. Write the heights of the first four bounces. What is the limit of the bounce height?
2 m, 1 m, 0.5 m, 0.25 m. The heights approach 0 m.
4What is the limit of the sequence 2.9, 2.99, 2.999, …? Is 2.999… equal to 3?
The limit is 3, and yes, 2.999… = 3 (it is 2 + 0.999…, and 0.999… = 1).
5Explain why there is no smallest positive rational number.
If someone names a positive rational r, then r ÷ 2 is also positive and rational, and smaller. So no positive rational can be the smallest.
Challenge
1How many rational numbers are there between 0 and 0.001? Give three of them and explain.
Infinitely many. Examples: 0.0005, 0.0001, 0.000 01. The rationals are dense, so we can keep taking means, getting a new number between any two forever.
2The sequence 1, 1 + 1/2, 1 + 1/2 + 1/4, 1 + 1/2 + 1/4 + 1/8, … has what limit? Explain using the square picture.
The part after the first 1 is 1/2 + 1/4 + 1/8 + …, which approaches 1. So the whole sum approaches 1 + 1 = 2.
3Sheila says, "Between two irrational numbers there is always a rational number." Test it with √2 ≈ 1.41421 and √3 ≈ 1.73205. Do you think it is always true?
Between √2 and √3 is 1.5 (or 1.6, 1.7). In general, yes: real numbers that are different differ at some decimal place, so you can cut off a decimal just above the smaller one. That terminating decimal is rational and fits between them. (This is another way the rationals are dense in ℝ.)
Quiz
2. Which number is between 5/8 and 3/4?
3. What is the limit of 1/3, 1/9, 1/27, 1/81, …?
4. Which is equal to 0.999…?
5. Why is the set of natural numbers infinite?
6. The mean of two rational numbers is always:
Big idea: Expectation B1.3 asks students to use patterns and number relationships to explain density, infinity and limit. These ideas are the doorway to calculus in Grade 12 (where limits are the main tool), but at Grade 9 they are explored with number lines, fractions and patterns.
Common difficulties:
- Thinking 0.999… is "just under" 1. Ask, "What number is between them?" There is none.
- Treating ∞ as a very big number. It is an idea of endlessness, not a value.
- Struggling to find a fraction between close fractions. Multiply both by the same number (×2, ×4, ×10) to get room between the numerators.
How to help: Do the tearing-paper activity together, and ask "Will the table ever have the whole sheet?" Debating 0.999… = 1 at dinner is a great way to practise mathematical argument (part of A1).