Numbers and algebra, from the ground up

This is the first of two guides that rebuild school mathematics from the beginning, in the order the ideas actually depend on each other rather than the order a classroom happens to teach them. This one covers number and algebra: what the numbers are, where each new kind comes from, and the moves that turn arithmetic into algebra. The second, Shape and change, picks up with geometry and carries on to the edge of calculus. Every rule here is motivated before it is stated and worked by hand after, and where an idea has a shape, there is a picture of it. Read it in sittings, with paper next to you — maths is not a thing you can read the way you read prose, and the understanding lives in your hand as much as on the page. There is not a line of code anywhere; this is mathematics, taught as mathematics.

/ part one · number

Every number is a place on a line

Mathematics starts with counting, so this guide does too. The counting numbers are 1, 2, 3, and on forever — the numbers you count things with. Add zero, the count of nothing, and you have the whole numbers: 0, 1, 2, 3, ….

Everything in this first part grows out of that small beginning, and it grows for one reason, over and over. An operation asks a question the numbers you have cannot answer, and a new kind of number is invented to answer it. That is the whole plot of Part I, and it is worth watching for.

There is one picture that will carry not just this chapter but the entire guide: the number line. Every number is a point on it. Bigger numbers sit further right, smaller ones further left, and the gaps between the whole numbers are evenly spaced.

The number line: 7 is a place, not a process. Every number this guide meets — negative, fraction, or stranger — will turn out to be a point on this line.

Hold on to that habit: whenever a new kind of number shows up, the first question to ask is where does it live on the line? Negatives, fractions, and numbers no fraction can name are all coming, and all of them are points on this same line.

Writing a number down is already an idea

The way we write whole numbers is a piece of machinery so familiar it has gone invisible, and it is worth making visible again. In the number 4,507, each digit's value depends on where it sits. Every place is worth ten times the place to its right.

place value · what 4,507 actually says

4,507  =  4 × 1000  +  5 × 100  +  0 × 10  +  7 × 1

The zero is not nothing here — it is doing a job, holding the tens place open so the 5 lands in the hundreds and the 4 in the thousands. This is why arithmetic on paper works at all: when you line up columns to add, you are lining up matching units, ones under ones and tens under tens. The same idea comes back in the decimals chapter, where the places simply keep going to the right of a dot.

Two operations, each with an undo

You know four operations: add, subtract, multiply, divide. It is more honest, and more useful, to say there are really two, and each one comes with an undo.

Subtraction undoes addition. The problem 12 − 5 = 7 is asking one precise question: what do I add to 5 to get 12? Division undoes multiplication in the same way. 12 ÷ 4 = 3 is asking what do I multiply 4 by to get 12?

This is not word-play. Solving equations, later in the guide, is nothing more than undoing operations in a systematic way, and the undo questions are also what force new numbers into being. What do I add to 6 to get 2? has no whole-number answer, and that gap is exactly where the next chapter's negative numbers come from.

The three laws the rest of maths refuses to give up

Addition and multiplication obey three laws. They are worth naming now, at the start, because almost every rule later in the guide is these three laws wearing a disguise.

the three laws · shown on whole numbers, where they are obvious

commutative    3 + 5 = 5 + 3              3 × 5 = 5 × 3
associative    (2 + 3) + 4 = 2 + (3 + 4)  (2 × 3) × 4 = 2 × (3 × 4)
distributive   7 × 104 = 7 × 100 + 7 × 4 = 700 + 28 = 728

Commutative means order does not matter. Associative means grouping does not matter — when everything is the same operation, where you put the brackets changes nothing. Distributive is the one law that connects multiplication to addition: to multiply a sum, you may multiply each part separately and add the results.

The distributive law is the busiest of the three, so it is worth seeing rather than just reading. Draw a rectangle whose height is a and whose width is split into b and c. Its total area is a × (b + c). But it is also just two rectangles side by side, a × b and a × c.

a × (b + c) is one rectangle; a·b + a·c is the same rectangle split down the middle. The distributive law just says these are equal.

You already use this law without naming it, every time you do arithmetic in your head. 25 × 12 is easier as 25 × 10 + 25 × 2 = 250 + 50 = 300. And 7 × 11 is easier as 7 × 10 + 7 = 77. That is the distributive law, quietly at work.

Subtraction and division obey none of this. 5 − 3 is not 3 − 5, and (20 − 8) − 3 = 9 while 20 − (8 − 3) = 15. Order matters for them, and so does grouping — one more reason to treat them as the undo operations rather than as free-standing ones of their own.

When negatives arrive next chapter, the rule that a negative times a negative is a positive will be derived from the distributive law, not simply announced. That is the pattern for the whole guide: the laws are the commitments we refuse to give up, and the rest is forced to fit around them.

Order of operations is a reading rule, not a maths rule

Write 2 + 3 × 4 and there are two ways to read it: do the addition first and get 20, or the multiplication first and get 14. Something has to decide, and that something is the order of operations.

Notice what kind of rule this is. It is not a fact about numbers; it is a convention for reading notation, agreed so that everyone gets the same answer from the same marks on paper. There are four levels, not the six that most people half-remember:

Grouping symbols — brackets first, always.
Exponents — repeated multiplication, like 3² = 3 × 3, taken up properly in the powers chapter.
Multiplication and division — one single level, and ties are read left to right.
Addition and subtraction — one single level, and ties are read left to right.

by hand · the convention applied

2 + 3 × 4 = 2 + 12 = 14        (2 + 3) × 4 = 5 × 4 = 20
8 ÷ 4 × 2 = 2 × 2 = 4          6 − 2 + 3 = 4 + 3 = 7

The six tiers that aren't there

The mnemonics — PEMDAS in the US, BODMAS in the UK — read as if multiplication beats division, and addition beats subtraction. They don't. Each pair shares one level, and ties break left to right. Reading six strict tiers turns 8 ÷ 4 × 2 into 1 and 6 − 2 + 3 into 1. Both are wrong, as the line above shows.

There is a viral expression, 8÷2(2+2), that people argue is 16 or 1. The honest answer is that it is badly written. Textbooks and calculators genuinely disagree about whether the 2(2+2) binds tighter than the division sign. Mathematics does not stage this fight; it writes a proper fraction bar or one more pair of brackets, and the ambiguity vanishes.

Below zero, where numbers gain a direction

The whole numbers cannot answer 2 − 6. There is no count you can add to 6 to arrive at 2. And yet the question is a perfectly ordinary one: spend 6 when you have 2 and you are 4 short; cool something 6 degrees starting from 2 degrees and it has a real temperature. The question is sensible, so mathematics invents numbers to answer it.

The fix is to extend the number line the other way, to the left of zero. Every number gains a mirror image on the far side. The whole numbers together with these new negatives are called the integers.

Zero gets a mirror. −2 and 2 are the same distance from it, one on each side — reflections of each other through zero.

The most useful way to read a negative number is as a direction, not just a smaller quantity. −3 is "3, but the other way". Money owed rather than money held; degrees below freezing rather than above.

On the line, bigger still means further right, and that produces the one ordering fact that feels wrong the first time: −5 < −2, where "<" means "is less than". Five degrees below zero is colder than two below, even though, stripped of its sign, 5 looks like the bigger number.

One symbol, three jobs

The − sign does three different jobs, and most sign errors come from blurring them.

It is an operation: 5 − 3, meaning "subtract". It is part of a number's name: −4, meaning "negative four". And it means "the opposite of": −x is the mirror image of x through zero, whatever x is — so if x is already negative, −x is positive. When a minus sign confuses you, stop and ask which of the three jobs this particular one is doing.

Adding and subtracting integers is just movement along the line. Adding a positive moves you right; adding a negative moves you left. And subtracting a number is the same as adding its opposite — which is exactly what the undo pairing from the last chapter demands, and it quietly turns every subtraction into an addition.

by hand · movement on the line

−4 + 9 = 5           start at −4, move 9 right
−3 + (−2) = −5       start at −3, move 2 further left
 5 − (−3) = 5 + 3 = 8    subtracting −3 is adding its opposite

"Two negatives make a positive"

That slogan is about multiplication, and only multiplication. In addition, two negatives pile up: −3 + (−2) = −5, not +5. Debt added to debt is more debt.

Why a minus times a minus is a plus

Multiplying by a positive whole number is repeated addition, so 3 × (−2) = −6 comes easily: three lots of −2. The genuinely puzzling case is (−2) × (−2). Why should that be positive?

Start by watching a pattern. Hold the second factor at −2 and let the first factor step down through the integers, one at a time.

the pattern · each step down adds 2 to the product

 3 × (−2) = −6
 2 × (−2) = −4
 1 × (−2) = −2
 0 × (−2) =  0
−1 × (−2) =  2
−2 × (−2) =  4

Each time the first factor drops by 1, the product rises by 2. The pattern is perfectly steady, and there is no reason for it to suddenly break when it crosses zero. So (−1) × (−2) = 2 and (−2) × (−2) = 4.

But a pattern is a suggestion, not a proof. Here is the proof, and it rests entirely on the distributive law from the last chapter. Start from something undeniably true — anything times zero is zero — and take (−1) × (1 + (−1)), since 1 + (−1) is zero.

the proof · −1 × −1, forced by the distributive law

(−1) × (1 + (−1)) = (−1) × 0 = 0

distribute the left side:
(−1) × 1 + (−1) × (−1) = 0
    −1 + (−1) × (−1) = 0        so  (−1) × (−1) = 1

The only thing assumed was that the distributive law keeps working when negatives are involved. Give that up and you would lose all of arithmetic. Keep it, and a minus times a minus must be a plus. The rule is the price of keeping the laws, not a decree handed down from nowhere.

Division inherits everything, because it undoes multiplication: same signs give a positive, different signs give a negative. So (−12) ÷ (−3) = 4, and 12 ÷ (−4) = −3.

−3² is not 9

By convention the square binds before a bare minus sign: −3² means −(3²) = −9. If you mean "negative three, squared", the brackets have to say so: (−3)² = 9. This looks like fussy punctuation until algebra, where −x² and (−x)² are genuinely different expressions. When in doubt, bracket.

The integers close subtraction: any two of them can now be subtracted and the answer is always an integer. But division is still open — 2 ÷ 3 has no integer answer — and that gap is the next chapter.

One number wears many names

The integers closed subtraction, but they left division open: 2 ÷ 3 has no integer answer. Fractions are the numbers invented to close that gap, and they deserve more patience than anything else in this guide.

There is a reason for the patience. When a US national panel reviewed why learners stall in mathematics, it named difficulty with fractions as a major obstacle on the road to algebra, and adults returning to the subject report hitting the same wall. If fractions have always felt like a bundle of unrelated rules, the aim of this chapter and the next is to show that there is really just one idea underneath them — renaming — and that every rule grows out of it.

A fraction is a single number, one point on the line. Read 3/4 as an instruction: take the step from 0 to 1, cut it into 4 equal pieces, and keep 3 of them. The bottom number, the denominator, names the size of the piece — quarters. The top number, the numerator, counts how many you took.

Here is the one idea the whole subject rests on: the same point has many names. Cut each quarter in half, and the very same length is now 6 pieces out of 8. Nothing about the quantity changed — only the size of the pieces you counted it in.

Cut the same length into four and take three, or into eight and take six — the shaded part is identical. 3/4 and 6/8 are two names for one point.

The rule for renaming is exactly what the picture shows: multiply the top and bottom by the same number, or divide them both by the same number, and the value is untouched.

by hand · renaming, in both directions

2/3 = 4/6 = 6/9         top and bottom both scaled — by 2, then by 3
12/18 = 2/3            top and bottom both divided by 6

Dividing top and bottom by the largest number that goes into both gives the simplest name — "lowest terms". That is all "simplifying a fraction" ever means. (The largest number dividing both is the greatest common factor; for 12 and 18 it is 6. It is worth a moment's thought only because renaming downward needs it.)

Renaming also settles comparison at a stroke. Which is bigger, 5/8 or 2/3? Eighths and thirds are different-sized pieces, so you cannot compare the counts directly. Rename both into a shared piece — a size both can express. The smallest that works is 24ths.

by hand · comparing 5/8 and 2/3

5/8 = 15/24        2/3 = 16/24        so 2/3 is bigger, by just 1/24

That shared unit — the smallest number both denominators divide into — is the least common multiple, 8 and 3 giving 24. It has one more job in the next section, and then you have met the whole toolkit fractions need.

Add, times, and "flip to divide"

Every operation on fractions comes back to renaming. Watch it happen three times.

Adding: the units have to match

You cannot add 2 metres and 3 minutes — the question doesn't mean anything until the units agree. A fraction is a count of a unit too, so the same rule holds: fractions only add when their denominators match.

When they already match, addition is just counting. 2/6 + 1/6 is two sixths plus one sixth, which is three sixths. When they don't, rename first. For 1/4 + 1/6, the shared unit is twelfths — a quarter is 3 twelfths, a sixth is 2 twelfths — and now they add like anything else.

A quarter is 3 twelfths and a sixth is 2 twelfths. Re-cut to the same piece, and they simply add: 3 + 2 = 5 twelfths.

1/4 + 1/6 = 3/12 + 2/12 = 5/12

Subtraction works identically: 3/4 − 1/6 = 9/12 − 2/12 = 7/12. The common denominator is not a ritual to memorise; it is the step that makes the two counts count the same thing.

Adding tops and bottoms

The tempting shortcut 1/3 + 1/6 = (1+1)/(3+6) = 2/9 fails a quick sanity check: 2/9 is smaller than 1/3, yet you added something positive to 1/3. Nothing that shrinks a number when you add to it can be addition. Add the counts, never the units.

Multiplying: "of" is what it means

Multiplying by a fraction means taking a portion of something. 3/4 × 20 is three quarters of 20, which is 15. This is the honest picture of fraction multiplication — scaling a quantity up or down, not repeated addition.

The mechanical rule (tops multiply, bottoms multiply) falls straight out of that picture. To see 2/3 × 3/4, draw a unit square. Shade 3/4 of it across, then 2/3 of it down. The part caught by both shadings is the product.

Three quarters across, two thirds down. The doubly-shaded corner is 6 of the 12 cells — so 2/3 × 3/4 = 6/12 = 1/2.

2/3 × 3/4 = 6/12 = 1/2      two thirds of three quarters is one half

"Multiplying always makes things bigger"

True for whole numbers above 1, and quietly false the moment you leave them: 8 × ½ = 4, and 6 ÷ ½ = 12. Multiplying by a number below 1 shrinks; dividing by one grows. If a fraction calculation ever feels backwards, this buried instinct from primary school is usually the reason.

Dividing: flip, because it undoes

Ask division's own question. 2 ÷ ¼ means "how many quarters fit into 2?" There are 4 in every whole, so 8. Notice that dividing by ¼ multiplied the answer by 4.

That is the whole rule, and here is why it holds. Dividing by 2/5 has to undo multiplying by 2/5. What undoes multiplying by 2/5 is multiplying by 5/2, because 2/5 × 5/2 = 10/10 = 1 — the two cancel out completely. Every fraction has such a partner, found by flipping it top-for-bottom, and it is called the reciprocal. So dividing by a fraction is multiplying by its reciprocal.

by hand · 3/4 ÷ 2/5

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8

The rule you may have memorised in school as "keep, change, flip" was real all along. It just never arrived with the question it answers.

One last piece of naming. 15/8 is heavier than a whole, and can be written as the mixed number 1⅞, meaning 1 + 7/8. Both names are fine, but for calculation the single-fraction form is almost always the one to work with.

Fractions close division: any number can now be divided by any other. (Zero is the one exception — 2 ÷ 0 has no answer, because no number times 0 gives 2.) The numbers built so far — the integers together with all these fractions — are called the rational numbers: everything you can write as one integer over another. For the next two chapters they only change costume, first as decimals, then as percentages.

Decimals are fractions in disguise

Decimals are not a new kind of number. They are place value — the idea from the very first chapter — continued past the ones column. Each place is still worth ten times the place to its right, and the decimal point is just the marker for where the whole-number places stop.

One step right of the point is tenths, then hundredths, then thousandths. So a decimal is really a fraction whose denominator is a power of ten, written in a way that hides the fraction.

place value · 0.347, unpacked

0.347 = 3/10 + 4/100 + 7/1000 = 347/1000

Reading a decimal as "keep zooming in" makes this concrete. The digit after the point picks which tenth of the line you are in; the next digit picks which hundredth of that tenth; and so on. Every extra digit is one more tenfold magnification, pinning the point down more precisely.

Each digit zooms in by a factor of ten. 0.3 says "in this tenth"; magnify it and 0.34 says "in this hundredth"; 0.347 sits seven-tenths of the way across that.

Because a decimal is a fraction, its arithmetic is inherited, not new. Adding decimals means lining up the point — which is exactly the fractions rule that units must match, wearing columns: tenths add to tenths, hundredths to hundredths. So 1.2 + 0.35 lines up as 1.20 + 0.35 = 1.55.

Multiplication is inherited too, and it explains the rule about counting decimal places that you may once have memorised without reason:

by hand · why 0.3 × 0.4 = 0.12

0.3 × 0.4 = 3/10 × 4/10 = 12/100 = 0.12

Tenths times tenths are hundredths, so one decimal place plus one decimal place makes two. The digits say 3 × 4 = 12; the places say hundredths.

0.3 × 0.4 is not 1.2

Multiplying the digits and keeping a single decimal place gives 1.2 — a number bigger than both things you multiplied, when multiplying by 0.4 has to shrink (the scaling picture from the fractions chapter). Counting the places is what keeps the size honest.

Which fractions give tidy decimals

Turn a fraction into a decimal by long division and exactly one of two things happens: the digits stop, or they fall into a repeating cycle. Which one happens can be read straight off the denominator.

A decimal that stops is really a fraction over a power of ten, and 10 = 2 × 5. So a fraction in lowest terms terminates exactly when the only prime factors of its denominator — the primes it is built from, numbers like 2, 3, 5 and 7 that are divisible only by 1 and themselves — are 2s and 5s, nothing else.

read the bottom · which fractions stop, which repeat

1/8  = 0.125          8 = 2 × 2 × 2      terminates
7/40 = 0.175          40 = 2×2×2 × 5     terminates
1/3  = 0.3333…        3 in the bottom    repeats
1/6  = 0.1666…        6 = 2 × 3          repeats
1/7  = 0.142857 142857…                  repeats, cycle of six

A repeating decimal is not a failure or an approximation; it is an exact name that happens to need infinitely many digits. And any repeating decimal can be turned back into a fraction with one trick: shift it left by a full cycle, then subtract, so the infinite tails cancel each other out.

by hand · the shift trick on 0.3636…

let x = 0.363636…
100x  = 36.363636…       one cycle is two digits, so shift two places
100x − x = 36            the endless tails line up and cancel
99x = 36,  so  x = 36/99 = 4/11

Long-divide 4 by 11 and out come the digits 36 36 36 …, closing the loop.

0.999… = 1

Run the same trick on x = 0.999…: 10x − x = 9, so 9x = 9, so x = 1. Not "close to 1" — equal to 1. They are two names for one point, exactly as 1/2 and 0.5 are.

Most people flinch here, picturing 0.999… as a process forever creeping toward 1 rather than a finished number sitting on the line. It is a finished number. The fully honest version of this argument — what an endless string of digits even means — is a limit, and that is a story the second guide tells at the edge of calculus.

Percent is just hundredths

A percentage is a fraction whose denominator has been fixed at 100. Per cent means "per hundred", and that is the entire definition. 45% is 45/100, which is 9/20, which is 0.45 — three names for one number.

A hundred-square makes "per cent" literal: 45% is 45 of the 100 cells. A percentage is a fraction that has already agreed on its denominator.

Because percentages share one denominator, they are easy to compare and combine — which is exactly why the adult world runs on them. But the shift worth carrying out of this chapter is how you picture a percentage change.

Do not think of a 20% rise as "find 20%, then add it on", in two steps. Think of it as one multiplication: growing by 20% is multiplying by 1.20, and shrinking by 15% is multiplying by 0.85. The multiplier folds the whole change into a single number.

by hand · the multiplier view

a 20% rise on 40:   40 × 1.20 = 48
a 15% fall on 40:   40 × 0.85 = 34

Once you see changes as multipliers, a whole family of everyday traps becomes one-line arithmetic.

A rise then an equal fall does not return you home

Go up 50% and then down 50% and you do not end where you started: ×1.5 then ×0.5 is ×0.75, so you land at three-quarters of the original. Two changes multiply, they do not cancel by adding to zero.

The same maths debunks "20% off, then 30% off is 50% off": ×0.8 then ×0.7 is ×0.56, a 44% discount, not 50%. And "125 is 25% more than 100" does not flip to "100 is 25% less than 125" — it is 20% less, because the 25 is now measured against 125, not 100.

Ratio: comparing by dividing

A ratio compares two quantities by division rather than subtraction. If a paint mix is 2 parts blue to 3 parts white, written 2 : 3, the claim is about how many times as much, not how much more. Double the batch and it is 4 : 6 — the same ratio, because the multiplicative relationship is unchanged even though the difference grew from 1 to 2.

That word "multiplicative" is the whole game, and getting it wrong is one of the most common errors in all of school mathematics. Enlarge a 3-by-5 photo so the short side becomes 6, and the short side has been doubled — so the long side must double too, to 10. The tempting move is to add: the short side gained 3, so add 3 to the long side as well, giving 8. That keeps the difference the same and wrecks the shape.

The short side doubles from 3 to 6. Doubling the long side too (6 × 10) keeps the shape; adding 3 to it instead (6 × 8) makes a visibly fatter, wrong rectangle. Ratios scale by multiplying.

A unit rate is a ratio with the second quantity reduced to 1 — 90 miles in 3 hours is 30 miles per 1 hour — and it is the form that makes ratios comparable and usable. Prices per litre, speeds per hour, and exchange rates are all unit rates.

When two ratios are claimed equal, a/b = c/d — a statement called a proportion — there is a shortcut for solving one that has an unknown in it: cross-multiply to ad = bc. This is not a spell — it is what you get by multiplying both sides of a/b = c/d by b and by d, so the fractions clear. If 3/4 = 9/d, then 3d = 36, so d = 12.

Percent, ratio and proportion are all the same underlying idea — comparison by multiplication — which is why the multiplier from the last chapter and the scaling here keep reappearing. The next chapter leaves comparison behind and asks a different question: what happens when you multiply a number by itself, over and over.

Powers, and how fast they climb

An exponent is shorthand for repeated multiplication, exactly as multiplication is shorthand for repeated addition. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The 2 being multiplied is the base; the small raised 5, counting how many copies, is the exponent or power.

Three rules govern powers, and none of them needs memorising, because each is just careful counting of how many copies of the base are in play.

the laws · each one is counting the base's copies

2³ × 2⁴ = 2⁷        three 2s times four 2s is seven 2s      (add the exponents)
(2³)⁴  = 2¹²        four groups of three 2s is twelve 2s    (multiply them)
2⁷ ÷ 2³ = 2⁴        seven 2s, three cancelled, four left     (subtract them)

Now push those rules somewhere they were not obviously meant to go, and watch them define new kinds of exponent. What could 2⁰ mean — zero copies of 2? On its own the question sounds empty. But the subtraction rule answers it: 2³ ÷ 2³ is plainly 1, and the rule says it equals 2³⁻³ = 2⁰. To keep the rule working, 2⁰ has to be 1. And it is not just a 2 thing: any non-zero number to the power 0 is 1.

A negative exponent is forced the same way. 2³ ÷ 2⁵ is 8/32 = 1/4, and the rule says it is 2³⁻⁵ = 2⁻². So 2⁻² must mean 1/2² = 1/4. A negative exponent is a reciprocal — a division — not a negative number.

2⁻³ is not −8

The minus in the exponent flips, it does not negate. 2⁻³ = 1/2³ = 1/8, a positive number smaller than

Think of the exponent as a dial: each step down divides by the base, running 8, 4, 2, 1, ½, ¼, … — it never crosses into negative values.

Even a fractional exponent is pinned down by the same rule. Whatever 8^(1/3) means — the caret ^ is just "to the power of", written this way because the power is itself a fraction — cubing it gives (8^(1/3))³ = 8^(3/3) = 8, so 8^(1/3) is the number that cubed gives 8 — namely 2. A power of 1/3 is a cube root, a power of 1/2 is a square root, and that is the hinge into the next chapter.

Why this matters: growth that leaves everything behind

Multiplying repeatedly builds numbers at a speed that repeated adding cannot touch, and it is worth seeing rather than being told. Compare 2ⁿ, which doubles at each step, with n², a respectable growing quantity from ordinary algebra.

Both start together, but by n = 10 the doubling has reached 1024 while n² has only crawled to 100. Exponential growth doesn't just win the race — it makes the other runner look stationary.

And the gap only widens. At n = 10 the two stand at 1024 against 100; by n = 20 it is 1,048,576 against 400. Doubling does to numbers what adding never can, and that gulf is why exponential growth deserves its reputation — and why a whole chapter of the second guide is devoted to it.

Roots, and the gaps between the fractions

A root undoes a power, the way subtraction undoes addition. The square root of a number asks: what, multiplied by itself, gives this? √9 = 3 because 3 × 3 = 9. The cube root asks the same with three copies: the cube root of 8 is 2. As the last chapter showed, these are just fractional powers in other clothing — √9 is 9^(1/2), the cube root of 8 is 8^(1/3).

√9 is 3, not ±3

The radical sign √ names one number, the positive one: √9 = 3. It is a different question from "which numbers square to 9?" — that one has two answers, 3 and −3. The distinction looks like hair-splitting now, but it is exactly what the ± in the quadratic formula is there to repair, several chapters from here. Keep it filed.

Roots simplify by pulling out square factors, because a root passes through multiplication: √50 is √(25 × 2) = √25 × √2 = 5√2. But a root does not pass through addition — √(9 + 16) is √25 = 5, which is nowhere near √9 + √16 = 7. Multiplication yes, addition no; that difference will catch people again in the algebra chapters, so it is worth banking now.

A number no fraction can name

Here is a genuinely unsettling fact: √2 is not a fraction. There is no pair of whole numbers whose ratio, squared, gives exactly 2 — the decimal 1.41421356… runs on forever with no repeating cycle, so none of the decimals-into-fractions machinery can catch it. Numbers like this are called irrational: not "unreasonable", just not a ratio.

That is a strong claim — no fraction, not one, ever — and a claim that strong deserves a proof rather than your trust. The proof is short and clever, but it leans on a piece of algebra you meet a few chapters from now: multiplying out a bracket like (2k + 1)². Once you can do that, √2's irrationality falls in three lines, so it is set aside here as a promise and collected there.

For now, meet √2 where it lives. It is a perfectly definite point on the number line — you can even construct it exactly. Draw a square one unit on each side; its diagonal has length exactly √2 — that is Pythagoras, which the second guide proves — and swinging that diagonal down onto the line lands precisely on the number no fraction could name.

A unit square has a diagonal of length √2. Swing it down onto the line and it lands just past 1.4 — a real point, between 1 and 2, that no fraction will ever name.

Once you allow numbers like √2, the number line finally has no holes in it. Line up the families built so far. The counting numbers sit inside the whole numbers, which sit inside the integers, which sit inside the rationals — and the rationals, dense as they are, still leave gaps, which the irrationals fill. Everything together is the real numbers (written ℝ), the full, unbroken line.

Each family sits inside the next. The rationals fill the line densely but still miss points like √2 and π; the reals are the whole line, holes included.

One compact piece of notation before algebra, since it rests on powers. Scientific notation writes a big or small number as a digit between 1 and 10 times a power of ten: 3.2 × 10⁴ is 32,000, and 6.1 × 10⁻³ is 0.0061. It is nothing more than place value written multiplicatively — the exponent counts how many places to shift the point — and it keeps very large and very small numbers readable.

Part I is done. Every kind of number school ever shows you now sits on one line, each invented to answer a question the previous ones could not. Part II takes the finished number system and learns to make general statements about all of it at once — which is what algebra is.

/ part two · algebra

A letter is a number not yet pinned down

Part I built the numbers. Part II learns to talk about them in general. Two ideas open the door, and both look too obvious to bother stating. Neither is, and getting either one slightly wrong quietly breaks everything that follows.

First: the equals sign is a claim, not a command. When researchers gave sixth-graders the problem 8 + 4 = □ + 5, every one of the 145 children tested answered 12 or 17. They read = as "now write the answer" — the calculator's = key, the worksheet's answer box — and so either computed 8 + 4, or piled all the numbers together.

But = does not mean "compute". It asserts that the two sides name the same number. Here it claims that □ + 5 is the same number as 8 + 4, which is 12 — so □ has to be 7. A balance that has to stay level, not a button that produces an answer.

The equals sign is a balance. 8 + 4 and □ + 5 name the same number, so the pans hold the same weight — which forces □ to be 7, not 12 or 17.

Every equation you will ever solve is a claim of this kind, and the whole method of the next chapter — "do the same thing to both sides" — is meaningless under the answer-box reading and obvious under the balance reading. Whatever you do to equal things, keeping them equal, leaves them equal.

Second: a letter stands for a number. Not an object, not an abbreviation — a number that simply has not been pinned down yet.

Asked what 4c + 3b might mean, students famously answer "4 cakes and 3 buns". It is not that. c and b each hold a number, and 4c + 3b is the single number you get once they do. If c = 10 and b = 2, then 4c + 3b = 40 + 6 = 46. The letter is a held breath, not a label on a thing.

So before doing anything clever with letters, practise merely evaluating — substitute a number and compute, using nothing but Part I:

by hand · substitute, then compute

3x + 5    at x = 4:      3 × 4 + 5 = 17
x²        at x = −3:     (−3)² = 9        brackets — the whole number is −3
2l + 2w   at l = 5, w = 3:   10 + 6 = 16

The notation, spelled out

Writing a number and a letter side by side means multiply: 2x is 2 × x. The × is dropped mainly because it looks too much like the letter x. So 2x at x = 3 is 6, not the two-digit string 23 — digits jammed together are place value, but a number jammed against a letter is a product.

One genuine inconsistency, worth knowing about so it can't trip you: the mixed number 2½ means 2 plus ½, yet 2x means 2 times x. The same "side by side" notation means add in one place and multiply in the other. It is a historical accident; just be aware of it.

One distinction of vocabulary sets up the next chapter. An expression is a recipe for a number, with no claim attached: 3x + 5. An equation is a claim that two expressions name the same number: 3x + 5 = 20. You evaluate an expression; you solve an equation. Asking "what does 3x + 5 equal?" has no answer until x is fixed — only tidier or messier ways of writing the same recipe.

Arithmetic speaks in specific numbers; algebra speaks in general laws. a + b = b + a states the commutative law for every pair of numbers at once, in five symbols. That is the whole upgrade, and it is why the three laws from Part I were worth naming so early: they are about to become the only moves you are allowed to make.

Solving is undoing, in reverse

An equation claims that two expressions name one number. Solving it means finding the value of the letter that makes the claim true, and the entire method rests on the balance reading of = from the last chapter.

If the two sides really are the same number, then adding 5 to both, or halving both, keeps them the same number. Picture the old balance scale again: whatever you do to one pan, do to the other, and it stays level. The art is choosing moves that leave the letter standing alone on one side.

That art is just Part I's inverse pairs, run in reverse. Read 3x + 5 = 20 as a short story about x: it was multiplied by 3, then 5 was added. To free x, undo the story backwards — last step first — doing each undo to both sides.

by hand · 3x + 5 = 20, peeled

3x + 5 = 20
3x = 15          subtract 5 from both sides       (undo the "+ 5")
x = 5            divide both sides by 3            (undo the "× 3")

check:  3 × 5 + 5 = 20  ✓

The check is not optional politeness. Substituting the answer back asks "is the claim actually true at x = 5?" — it catches slips in seconds, and it is the single habit that separates people who trust their algebra from people who hope.

Brackets change only the length of the story. In 2(x − 3) + 4 = 18, x had 3 subtracted, was doubled, then had 4 added; peel in reverse.

by hand · 2(x − 3) + 4 = 18, peeled

2(x − 3) + 4 = 18
2(x − 3) = 14        subtract 4
x − 3 = 7            divide by 2
x = 10               add 3          check: 2 × 7 + 4 = 18  ✓

Fractions in an equation are cleared, not endured: multiply every term by a common denominator and they vanish. And when the letter shows up on both sides, gather it onto one side first — subtracting a term works just like subtracting a number. In 4x − 3 = 2x + 7, subtract 2x from both sides to get 2x − 3 = 7, then peel as usual to x = 5.

It is the same act as rearranging a formula

Solving is not only for finding mystery numbers. The temperature conversion F = 9C/5 + 32 is an equation in two letters, and "solving it for C" — rearranging it to get C by itself — is the very same peeling, last operation first, applied to both sides.

by hand · rearranging F = 9C/5 + 32 for C

F = 9C/5 + 32
F − 32 = 9C/5           subtract 32
5(F − 32) = 9C          multiply by 5
C = 5(F − 32)/9         divide by 9        check: F = 212 gives C = 100  ✓

Rearranging a formula and solving 3x + 5 = 20 are the same skill; only the number of letters differs. If you can do one, you can do the other.

The dangerous step is writing the equation, not solving it

Most real problems arrive as words, and the standard failure happens before any algebra. Told "there are six times as many students as professors", about a third of engineering freshmen in a famous study wrote 6S = P — copying the word order, six-then-students. But 6S = P says the professors are the bigger group. The students are the many, so you multiply the small count to reach the big one: S = 6P.

The cure is mechanical: test your equation with easy numbers before trusting it. Two professors should mean twelve students; S = 6P gives 6 × 2 = 12 ✓, while 6S = P demands 6 × 12 = 2 ✗. Name your letters precisely — "P is the number of professors", never "P is professors" — and let a concrete number vet the sentence before you solve.

Sometimes the answer is a stretch

Not every claim is an equality. x + 3 ≤ 5 claims that one number is no bigger than another, and its answer is not a single value but a whole stretch of the line: every x up to and including 2.

The symbols point the way they mean: < is "less than", > is "greater than", and ≤, ≥ add "or equal to". An inequality is solved almost exactly like an equation — same peeling, both sides — with one single exception that has to be understood, not memorised.

Adding or subtracting the same thing on both sides slides both numbers along the line together, so their order is preserved. Multiplying or dividing by a positive stretches the line away from zero, and order survives that too. But multiplying or dividing by a negative reflects the whole line through zero, and reflection turns left into right — it reverses the order.

Multiplying by −1 reflects every point through 0. 3 sits left of 5, but their reflections land the other way round: −3 sits right of −5. That flip is why the inequality sign reverses.

So the one extra rule: when you multiply or divide both sides by a negative, flip the inequality sign. The solution set is drawn straight onto the line — a filled dot for an endpoint that is included, an open dot for one that is not:

by hand · −2x < 6, and its picture

−2x < 6
x > −3           divide both sides by −2, so < flips to >

     ──○━━━━━━━━━▶      open dot at −3 (not included), everything to its right
      −3   0

The way to be sure of the flip is to test one number from inside your answer and one from outside. Take x = −2 (inside): −2 × (−2) = 4, and 4 < 6 ✓. Take x = −4 (outside): −2 × (−4) = 8, and 8 < 6 ✗. Inside works, outside fails — the direction of the sign is confirmed.

Flipping at the wrong moment

The flip belongs to exactly one move: multiplying or dividing both sides by a negative number. Subtracting 7, adding a negative, or merely having negative numbers sitting in the problem flips nothing. x + 3 ≤ 5 becomes x ≤ 2 with no drama at all. Flip on the reflection, never on the scenery.

Two claims that must hold at once

Step up to two letters and two claims that have to be true together — a system of equations, like 2x + 3y = 12 together with x − y = 1. A solution is now a pair of values, one for x and one for y, that satisfies both claims at the same time.

Each equation alone allows endlessly many pairs; the system is the extra demand that one pair serve both. There are two standard ways to find it, and both work by boiling two letters down to one.

Substitution. Solve one equation for a letter, then let that stand in for the letter in the other. From x − y = 1 you get x = y + 1, so the first equation becomes 2(y + 1) + 3y = 12, which is 5y + 2 = 12, so y = 2, and then x = 3.

Elimination. Scale the equations until a letter's counts match, then add or subtract the whole equations to cancel it.

by hand · eliminating y

2x + 3y = 12
3x − 3y =  3          the second equation, multiplied through by 3
5x      = 15          add the two — the y-terms cancel
x = 3,  then y = 2     from x − y = 1

check in both:  2×3 + 3×2 = 12 ✓    3 − 2 = 1 ✓

Check in both equations, always — a pair can satisfy one beautifully and fail the other.

Not every system lands on a single pair, and the two ways it can fail are worth seeing. Run elimination on x + y = 1 and x + y = 3 and everything cancels to 0 = 2, a flat falsehood — so no pair works; the claims contradict each other. Run it on x + y = 1 and 2x + 2y = 2 and you get 0 = 0, always true, because the second equation is just the first in disguise — so every solution of one solves the other, endlessly many.

For two straight-line equations, those are the only three possibilities. The picture makes them obvious, and the second guide draws each equation as a line: one solution is two lines crossing, no solution is two parallel lines that never meet, and infinitely many is one line drawn twice.

Two lines can cross once (one solution), run parallel forever (none), or be the very same line (every point solves both). There is no fourth option.

Multiplying out, and putting it back together

Algebra's legal moves are the three laws from Part I, now applied to letters. The busiest by far is the distributive law, and the reason it is busy is that it runs in both directions.

Read left to right, 3(x + 4) = 3x + 12 is expanding: multiplying the bracket out. Read right to left, 3x + 12 = 3(x + 4) is factoring: spotting that 3x + 12 is 3 of something and writing it back as a product. One law, two directions, and most of this chapter is learning to walk it fluently each way.

Expanding is also what makes "collecting like terms" legal. 5x + 2x = (5 + 2)x = 7x is just the distributive law read backwards, so 5x + 3 + 2x − 7 tidies to 7x − 4 — x-terms with x-terms, plain numbers with plain numbers. Signs travel with their term: in 5 − 3x + 2 the minus belongs to the 3x, and a bare minus in front of a bracket multiplies everything inside by −1, so −(x − 4) = 4 − x.

Multiplying two brackets is the distributive law used twice — every term in the first bracket multiplies every term in the second.

by hand · (x + 2)(x + 3), all four products

(x + 2)(x + 3) = x·x + x·3 + 2·x + 2·3
               = x² + 3x + 2x + 6
               = x² + 5x + 6

All four products, not only the ends. Multiplying just the firsts and lasts gives x² + 6, and a one-second test kills it: at x = 1 the real product is 3 × 4 = 12, while x² + 6 gives 7.

Three expansions turn up so often they are worth knowing on sight — as patterns you recognise, not formulas you recite. The first is the one to really see, because a square built on a sum is a picture.

The square on (a + b) breaks into four pieces: a², b², and two equal ab rectangles. Those two are why the middle term is 2ab — the mistake of writing just a² + b² simply forgets them.

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²        the middle terms cancel

The third, the difference of squares, is worth its own picture too, because it is really a shape being rearranged. Take a square of side a and cut a smaller square of side b out of one corner. The L-shaped piece left behind has area a² − b² — and if you slice it and slide the pieces together, they form a clean rectangle, (a + b) tall and (a − b) wide.

Cut a b-square from an a-square and the leftover L rearranges into an (a + b) by (a − b) rectangle. So a² − b² = (a + b)(a − b) — which even does mental arithmetic: 51 × 49 = (50 + 1)(50 − 1) = 2500 − 1 = 2499.

Factoring: the same road, driven backwards

Factoring is expanding in reverse — given a sum, recover the product it came from. Three moves cover all the ground this guide needs.

Pull out a common factor first, always: 6x² + 9x = 3x(2x + 3), the greatest-common-factor idea from fractions, now applied to terms. Recognise a special product: x² − 9 is a² − b² with b = 3, so it is (x + 3)(x − 3). Unscramble a trinomial: to factor x² + 5x + 6, look for (x + p)(x + q); expanding that gives x² + (p + q)x + pq, so you need two numbers that add to 5 and multiply to 6 — namely 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). Expanding checks it instantly; factoring is self-correcting.

The false neighbours

Three moves that feel right are wrong, and one numeric test exposes each. (a + b)² is not a² + b² — try a = 2, b = 3: 25 versus 13. You cannot cancel across a plus sign: (4 + 2)/2 is 3, not 4. And √(a + b) is not √a + √b: √(9 + 16) = 5, but √9 + √16 = 7.

Each fake is a real rule with × quietly swapped for +: (ab)² = a²b², ab/b = a, √(ab) = √a·√b work fine. Multiplication passes through; squaring, dividing and rooting do not. Whenever a tidy-looking shortcut tempts you, feed it small numbers first.

The payoff: no fraction squares to 2

Back in the roots chapter, √2's irrationality was promised as a proof owed. You now own every tool it needs — the square-of-a-sum pattern most of all — so here is the debt, paid.

Suppose some fraction did square to 2, and write it p/q in lowest terms (any fraction has a lowest-terms form — just divide out the common factor). Square both sides of √2 = p/q, then clear the denominator:

by hand · the three lines

2 = p²/q²   →   p² = 2q²          so p² is even (it is 2 × something)

an odd number's square is odd:  (2k+1)² = 4k² + 4k + 1
so p must be even too:  write p = 2k
then (2k)² = 2q²  →  4k² = 2q²  →  q² = 2k²          so q² is even, so q is even

That middle step is where the special product earns its keep: (2k + 1)² = 4k² + 4k + 1 is exactly the (a + b)² pattern, and it proves an odd number always squares to an odd number — so anything whose square is even must itself have been even.

Now the trap springs. p and q are both even — but the fraction was in lowest terms, with nothing left to cancel. That is a flat contradiction, so the original supposition was impossible: no fraction squares to 2. The number line really does hold points no ratio will ever name, and it took nothing more than this chapter's expanding to prove it.

Quadratics: when x shows up twice

A quadratic equation has the letter squared: ax² + bx + c = 0 with a not zero, like x² − 5x + 6 = 0. And the last few chapters' method — peel the operations off in reverse — simply dies on it.

Peeling worked because the unknown appeared once, so the story of how it was built could be run backwards step by step. Here x appears twice, as x² and as x, and no sequence of undo-steps can gather those two into one place. Quadratics need genuinely new ideas, and this chapter is a chain of four, each one repairing the limit of the one before, ending at a formula that cracks every case.

Factoring, and the zero-product property

Here is the one fact that turns factoring from tidying into a way of solving equations: if two numbers multiply to zero, at least one of them must be zero.

Nothing else behaves like this. A product of 6 tells you nothing certain about its factors (2 × 3, or 1 × 6, or ½ × 12). But a product of zero is different — the only way to multiply to nothing is for one of the parts to already be nothing.

So if you can write a quadratic as a product equal to zero, the problem splits cleanly in two.

by hand · x² − 5x + 6 = 0

x² − 5x + 6 = 0
(x − 2)(x − 3) = 0        two numbers with sum 5, product 6 → 2 and 3
x − 2 = 0   or   x − 3 = 0
x = 2   or   x = 3        check: 4 − 10 + 6 = 0 ✓   9 − 15 + 6 = 0 ✓

Two solutions — and quadratics generally have two. This method is just the previous chapter's unscrambling with a zero on the right, and it inherits that chapter's limit: it only works when you can actually spot the factors.

The square-root method

One shape of quadratic can still be solved by peeling, because in it x appears only once: something squared equals a number, like (x + 3)² = 4. The square was the last thing done, so un-square first — carefully.

Two numbers square to 4, namely 2 and −2, so x + 3 = 2 or x + 3 = −2, written x + 3 = ±2. That gives x = −1 or x = −5. The ± is not decoration: √4 names only the positive root, 2 (the convention from the roots chapter), so the negative possibility has to be put back by hand — and this is the ± that the √9 warning, chapters ago, told you to keep filed.

Completing the square

The trick that turns every quadratic into that solvable shape. It has a literal picture behind it, and the picture is the whole idea.

Expand (x + 3)² and you get x² + 6x + 9. So x² + 6x is almost a perfect square — it is (x + 3)² with the 9 missing. Draw it: an x-by-x square, with the 6x split into two 3-by-x strips laid along two edges. The result is a bigger square with one corner missing, a 3-by-3 notch.

x² + 6x is a square two strips short of completion. Fill the missing 3-by-3 corner — add 9 — and it becomes the perfect square (x + 3)². That added 9 is exactly what "completing the square" adds.

So to solve x² + 6x + 5 = 0, force that missing corner into existence: add and subtract the 9 it wants.

by hand · x² + 6x + 5 = 0, completed

x² + 6x + 5 = (x² + 6x + 9) − 9 + 5      half of 6 is 3; 3² = 9
            = (x + 3)² − 4

(x + 3)² − 4 = 0  →  (x + 3)² = 4  →  x = −1 or x = −5

Half the x-coefficient, square it, adjust the constant to keep the equation honest: any quadratic submits to this (divide through by a first if the x² has a coefficient). And the completed form (x + 3)² − 4 hands you a bonus the second guide collects — since a square is never negative, the expression can never dip below −4.

The formula is completing the square, done once and for all

Run those exact steps on the general equation ax² + bx + c = 0 — divide by a, halve, square, adjust — and the algebra, done in letters, compresses into a single result.

by hand · the derivation, and the formula it produces

ax² + bx + c = 0

complete the square:   (x + b/2a)² = (b² − 4ac) / 4a²
un-square, keep the ±:  x + b/2a  = ±√(b² − 4ac) / 2a
subtract b/2a:          x = ( −b ± √(b² − 4ac) ) / 2a

That last line is the quadratic formula — completing the square performed once, symbolically, so that you never have to perform it again. Feed it 2x² + 5x − 3 = 0, where a = 2, b = 5, c = −3.

by hand · the formula applied

b² − 4ac = 25 − 4·2·(−3) = 49        √49 = 7
x = (−5 ± 7) / 4  =  1/2  or  −3      check: 2·¼ + 5·½ − 3 = 0 ✓

The quantity under the root, b² − 4ac, is called the discriminant, and it tells you what you will find before you do the work. Positive, and there are two different real solutions (the 49 above). Zero, and the ± collapses to a single one: x² − 6x + 9 = 0 is (x − 3)² = 0, giving only x = 3. Negative, and there is no real answer at all — x² + 1 = 0 would need a number whose square is −1, and no point on the real line squares to a negative.

There is one honest sentence to add about that last case. Mathematicians did to it exactly what they did to 2 − 6 and to 2 ÷ 3: they built a larger system of numbers where such equations do have solutions. That system lies past this guide's border, but the pattern is the same one you have watched all along — an impossibility is where the next kind of number gets born.

Where the solutions are, in one picture

Everything in this chapter has been algebra, but a quadratic also has a shape, and seeing it ties the chapter together. Plot y = x² − 5x + 6 — height y for each x — and you get a parabola, the U-shaped curve every quadratic makes. Solving x² − 5x + 6 = 0 is asking where that height is zero: where the curve crosses the axis.

The parabola y = x² − 5x + 6 dips below the axis and back. It touches zero at exactly x = 2 and x = 3 — the same two solutions the factoring found. Solving a quadratic is finding where its curve meets the axis.

So the practical order is: factor if the factors show themselves; un-square if x appears only once; otherwise complete the square, or just let the formula finish the job, since the formula is completing the square, frozen. Whichever route, check by substituting back.

Part II's promise is kept. Everything linear, and now everything quadratic, can be solved with the laws of Part I and a single well-placed zero. That is a real milestone: it is the whole of school algebra's core, resting on nothing but the number line and three laws you could see were true on the counting numbers.

The second guide, Shape and change, picks the story up here. It gives these curves a home in the coordinate plane, turns the input-to-output machines you have been solving into functions you can graph, and follows them all the way to the doorstep of calculus. You cross that border already speaking the language.