The counting numbers, the integers, the rationals and the reals, each built from the one before. A number is a place on the line, not the notation you happened to write it in, and comparing two of them means comparing places.
Core ideas
Algebra puts a letter where a number goes. Before that is useful you have to know what the letter is allowed to stand for.
Every natural number is an integer, every integer is a rational (5 is 5/1), every rational is a real. The containments are strict in one direction only: −3 is an integer and not a natural, 2/3 is rational and not an integer, √2 is real and not rational.
1/2, 2/4, 0.5 and 3/6 are four ways of writing one number, and it occupies one point on the line. Almost every question in this course is a question about the point; the spelling is chosen for convenience.
a < b means a lies to the left of b. Deciding it from a rounded decimal is guesswork — 1.414 and √2 agree to three places and are different numbers. Compare by a common denominator, or by squaring.
The material
Four systems, how you tell a rational from an irrational, and the two facts about the line the rest of the course leans on.
The systems are built outward, and each one is built because the previous one could not answer a question. The naturals cannot answer 3 − 5. The integers cannot answer 3 / 5. The rationals cannot answer "what is the side of a square of area 2". Each answer forces a larger system.
The natural numbers are 1, 2, 3, …. The integers are the naturals, their negatives, and 0. A rational number is one that can be written p/q where p and q are integers and q ≠ 0. The real numbers are all the numbers on the number line; a real number that is not rational is called irrational.
Whether 0 counts as a natural number is a convention and it differs between books. This course starts the naturals at 1 and nothing later depends on the choice.
"Can be written" is the operative phrase, exactly as it will be for polynomials in Polynomials and Factoring. 4 is rational because 4 = 4/1. 0.25 is rational because 0.25 = 1/4. You do not have to be shown the fraction; it has to exist.
Every rational number has a decimal expansion that either stops or falls into a repeating block, and every such decimal is a rational. An irrational number is exactly one whose expansion does neither.
The dots are not an approximation. 1/3 is 0.333… exactly; 0.333 is a different number, and a smaller one, because it is 333/1000. Throughout this course the fraction is the number and the decimal is a report on it.
The square root of a positive whole number is a whole number when the radicand is a perfect square, and is irrational otherwise. Thus √9 = 3, while √2 and √10 are irrational.
The proof of that theorem uses prime factorisation and a contradiction; neither is a prerequisite for this course. The act required here is the usable test: look for a whole number whose square is the radicand. “Roots and Radicals” supplies the factor method used to simplify a root after the classification is made.
The rationals therefore have gaps, and the line does not. That gap is the whole reason the real numbers are needed.
Fix a point for 0 and a length for 1. Every real number then corresponds to exactly one point, and every point to exactly one real number. The second half of that sentence is an assumption about the reals, not something proved here; constructing the reals so that it holds belongs to analysis.
a < b means a lies to the left of b on the line.
For any two reals exactly one of a < b, a = b, a > b holds. And between any two distinct reals there is another one — their average will do — so no real number has a next one. The integers do: after 4 comes 5.
To compare two numbers exactly, put them over a common denominator, or square them when both are positive. 1.41 against √2: 1.41² = 1.9881, which is less than 2, so 1.41 < √2. Squaring is safe here only because both numbers are positive — on negatives it reverses the order, since −3 < −2 but 9 > 4.
−5/4, −1.2, 0.6, 2/3, 1.41, √2. Over a common denominator −5/4 = −25/20 and −1.2 = −24/20, so −5/4 is the leftmost. Likewise 0.6 = 9/15 and 2/3 = 10/15. And 1.41 < √2 by the squaring test. The order is −5/4 < −1.2 < 0.6 < 2/3 < 1.41 < √2.
Interactive
Ordered by exact comparison, not by decimal
Method
Identify it, then place it. The two jobs use different information.
If you can, it is rational and its decimal will stop or repeat. The square root of a whole number is irrational unless that number is a perfect square, which is what settles √2 and √10. A radical sign on its own proves nothing: √9 is 3, and √(4/9) is 2/3.
A decimal you can see is always a rounding of an irrational and often a rounding of a rational. It tells you roughly where to look and never settles a close comparison.
For fractions, a common denominator. For positive square roots, square both sides — but only when both are positive, because squaring reverses the order of two negatives.
Left of 0 for negatives, right for positives. Among negatives the number with the larger digits is further left: −5/4 sits left of −1.2.
No decimal is trusted beyond the digits it actually has.
Only one of these six is irrational. The other five are fractions, and every comparison between them was settled by a common denominator rather than by a decimal.
The squaring step is the one with a condition attached. It compared 1.41 and √2 correctly because both are positive. Applied to −5/4 and −1.2 it would give 25/16 > 36/25 and suggest the wrong order, so the negatives were handled by the denominator instead.
Now fade the support: order −7/6, −1.1, 3/4 and √3. Start only with −1.1 = −11/10, then finish the two comparisons yourself before checking: −7/6 < −1.1 < 3/4 < √3. If the first pair reverses, the minus signs were ordered by their digits; if the last pair is uncertain, square the two positive numbers.
Classify and order −9/8, −1.12, 5/6, 0.84 and √3, writing the exact comparison used for each close pair. Reach for a common denominator or for squaring, not for a decimal expansion. That habit is what makes “Roots and Radicals” — where √72 becomes 6√2 — an exact calculation rather than an approximate one.
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