The World of Numbers

1Integers: Expanding the Horizon

Integers expand the whole-number line to the left of 0: …, −2, −1, 0, 1, 2, … Opposites sit equally far from 0. Adding is a move; subtracting is add-the-opposite — the horizon is the whole line, not only the right half.

A minus that only means “owe later” without a place on the line is a horizon you skipped.

How it works

1. Place a few negativesLeft of 0.
2. Name the opposite of 7 as −7Same distance.
3. Add as a move on that lineThe expand.

2Filling the Spaces: Fractions and Rational Numbers

Rationals fill many (not all) gaps: a/b with b ≠ 0. Between 0 and 1 sit 1/2, 1/3, 2/3, … Decimals that stop or repeat are rational. Filling the spaces means you can always find another rational between two rationals — not that every point is rational.

0.1010010001… with longer zero-runs (as a later-or-aside look) is not this repeating-or-stopping family.

How it works

1. Write a/bA rational.
2. Place it on the lineA gap-filler among wholes.
3. Name that another rational sits between any twoThe fill — still not every point.

3Irrational Numbers

An irrational is a real that is not a/b: √2, π as taught. Their decimals do not stop and do not cycle. You can bound them (1.4 < √2 < 1.5) without finishing the expansion.

A long decimal you have not tested for a cycle is not yet proved irrational.

Figure. √2 sits between 1 and 2 and is not a ratio of two integers. The mark is a place, not a fraction tick.

How it works

1. Name a taught irrational√2, or π.
2. Say: not a/b; decimal neither stops nor cyclesThe sort.
3. Bound it between two rationals if you need a sizeA working box.

4Real Numbers: Decimals and Cyclic Patterns

Reals are the completed line for this class: rationals plus irrationals. A decimal expansion is a name of a real; a cycle (0.1818…) is a rational in disguise. The journey does not end — later classes refine proofs; here you sort and place.

Calling every long decimal irrational is a rush.

Figure. A repeating decimal is still a fraction. 0.333… names one of three equal parts, not a new kind of number.

How it works

1. See a decimalThe name.
2. Stop or cycle → rational; taught non-cycle → irrationalThe sort.
3. Place it on the one lineReals.

5Conclusion: The Never-Ending Journey

The never-ending journey is a claim-size note: you can always name a closer rational to √2; you do not finish √2 as a fraction today. The journey is the line, not a new number-system each year.

Pretending √2 = 22/7 is the π-steal and a fake close.

Figure. Each box sits inside the next: counting numbers, then integers, then rationals, then reals. A new box is a larger world, not a replacement.

How it works

1. Bound, do not fake-close1.41 < √2 < 1.42 if that is your box.
2. Keep 22/7 as a π-approximate if the lesson used it — labelled approximateA box, not equality.
3. Stop — the line still has roomThis class.

6A definition is a test you can run

Rational is a test: a/b with b ≠ 0, or a stop/cycle decimal. If you cannot write the fraction or the cycle, you have a digit-string.

A pretty √ badge is not the test for rational.

Figure. The rational test is “is it p/q with q not zero?”. 3/4 passes. √2 fails. The test is the definition.

How it works

1. Write a/b or show a cycleThe object.
2. If you cannot, do not call it rationalThe test.
3. Irrational needs the lesson’s examples or a given non-cycleHumble.

7Name the given before the unknown

The given is the writing (√2, 0.18 repeating, −7). The unknown is the drawer. Copy the writing before you say irrational because it “looks long”.

0.333… is rational — a silent given you skipped if you said all long decimals are irrational.

Figure. Given: the whole is 1, cut into 4 equal parts. The unknown count is 3. Name those two facts before you write 3/4.

How it works

1. Copy the symbol or the decimalThe given.
2. Name integer / rational / irrational / realThe unknown.
3. Then sortGiven first.

8One worked case is enough at this class

One sort is enough. A page of √2 clones does not add a new fill-the-gap.

Ten twins of 22/7 labelled as π are still one approximate.

Figure. One worked nest: 1.4²=1.96 and 1.5²=2.25, so √2 sits between 1.4 and 1.5 on this zoomed line.

How it works

1. Sort one numberThe case.
2. A second only as a checkOptional.
3. StopThis class.