Algebra 101 with Euler

Lesson 1

Euler dictates algebra to a tailor's apprentice

Leonhard Euler of Switzerland was, by common consent, the hardest working mathematician who ever lived, and by the late 1760s he was blind. He had lost the sight of one eye years earlier and then the other, yet his output barely slowed. He did his calculating in his head, held whole arguments in memory, and dictated his books to assistants. The most famous of those books for beginners, the Elements of Algebra, was dictated to his servant, a young man who had originally been a tailor's apprentice and who, the memoir printed with the book tells us, was at the start completely ignorant of mathematics. Euler quite literally had to teach his own secretary the subject while the secretary wrote it down.

That accident of history is why the book is so good. Most great mathematicians explain badly to beginners, because they have forgotten what it is like not to know. Euler could not skip a step, wave at a diagram, or say you can see it from the picture, because there was no picture and his first reader knew nothing. Every idea had to arrive in words, in order, with nothing assumed. If a step confused the apprentice, it got explained again, more slowly. When you read this course, you are sitting where that young man sat, hearing the subject built from the ground up by a patient master who refused to leave anyone behind.

One honesty note before we begin. Euler's algebra is real algebra, and most of what follows works exactly as your school textbook says. But his book was written more than 250 years ago, so in places his order and notation differ from a modern course. He spends many chapters on arithmetic and fractions before he allows himself an equation, he works his examples in crowns and louis and other old coins, and some notation that is standard now was still settling down then. Think of it as learning from a grandfather: the wisdom is sound, the accent is from another century.

Lesson 2

Numbers, signs, and negatives

Euler begins where arithmetic begins, with what a number is. A number, he says, is a comparison: how many times a chosen unit, one crown, one foot, one pound, fits into the thing you are measuring. Then come the two signs that run all of algebra. The sign + , read plus, means add: 5 + 3 is 8, and 12 + 7 is 19. The sign - , read minus, means take away: 8 - 5 is 3, and 20 - 14 is 6. So far nothing is new. The new idea, the one this whole lesson exists for, arrives when you try to take away more than you have. What is 2 - 5? Arithmetic with pockets and purses says it cannot be done. Algebra says it can, and the answer is -3.

Euler explains negatives with money, and his example is still the clearest one ever given. What a person really possesses, he counts as positive. What a person owes, he counts as negative. A man who has 100 crowns but owes 50 really has 100 - 50, which is 50. A man who has 2 crowns and owes 5 has not merely nothing: he still owes 3, so his position is -3, and in Euler's own words, negative numbers are less than nothing. Give that man 50 crowns to pay a debt of 50 and he rises only to zero, richer than before but still owning nothing. Zero is not the bottom of the number world. Below it stretches a whole mirror world: 0, -1, -2, -3, on and on.

Working with negatives follows a few plain rules, and each rule is just the debt story told again. Adding a negative is owing more: 7 + (-4) is 3, as if you earned 7 and paid out 4. Subtracting a negative is having a debt forgiven: 5 - (-3) is 8, because removing an IOU of 3 leaves you 3 better off. Multiplying takes one more idea. A debt times a number of times grows the debt: 3 x (-4) is -12. And a negative times a negative turns forgiveness on forgiveness, so (-3) x (-4) is +12. Check it with the pattern 3 x 4 is 12, 3 x 3 is 9, 3 x 2 is 6: as the second number drops by 1 the answer drops by 3, and continuing past zero gives 3 x (-1) is -3, then flipping the first sign flips everything back to positive.

Lesson 3

Letters for numbers

Now Euler makes the move that turns arithmetic into algebra. In order to generalise, he writes, we make use of letters instead of real numbers. The letter a does not stand for one fixed number. It stands for any number you choose to put there, great or small. So a + b means the sum of two numbers, whichever they are, and a - b means the first with the second taken away. If a is 10 and b is 4, then a + b is 14 and a - b is 6. Change the numbers and the letters still tell the truth. One line of letters does the work of a thousand lines of arithmetic, because it states the pattern once for every case at once.

Letters obey the same rules as the numbers they replace, which is the whole secret. You can add like letters together: 3x + 4x is 7x, just as 3 apples and 4 apples make 7 apples, whatever x turns out to be. You can multiply letters by numbers, and by each other: 3 times x is written 3x, and x times x is written with a small raised 2, as x squared, meaning x multiplied by itself. Watch out for the beginner trap Euler warns about in his own patient way: 3x and x + x + x are the same thing, but 3x and x + 3 are not. One is three copies of the unknown. The other is the unknown with 3 added, a completely different amount unless x happens to be small.

Why bother? Because letters let you talk about a number you do not know yet, and that is where algebra gets its power. Suppose a purse holds some crowns, and a second purse holds twice as many. With letters you can write the whole situation before anyone counts a coin: the first purse holds x, the second holds 2x, and together they hold x + 2x, which is 3x. The letters are not the answer. They are a handle on the problem, a way to carry an unknown around, combine it, and reason about it, until the moment it gives up its value. That moment is called solving an equation, and it is only two lessons away.

Lesson 4

Fractions and powers

Fractions are numbers too, and Euler insists you master them before equations, because equations keep producing them. A fraction like 3/4 means three parts of a whole cut into four equal parts. To add fractions, the parts must be the same size, so you recut them to a common denominator: 1/2 + 1/3 is 3/6 + 2/6, which is 5/6. To multiply, simply multiply top by top and bottom by bottom: 2/3 x 3/4 is 6/12, which simplifies to 1/2. To divide by a fraction, flip it and multiply: 1/2 divided by 1/4 is 1/2 x 4/1, which is 2, and that makes sense, because there are exactly two quarters inside a half.

Powers are shorthand for repeated multiplication, the way letters were shorthand for numbers. 2 x 2 x 2 is written 2 to the power 3 and equals 8. 2 to the power 4 is 16, doubling again, and 10 to the power 3 is 1000, which is why a thousand is such a tidy number in our counting system. The small raised number is called the exponent, and it counts the copies. Two rules cover most of what a beginner needs. First, multiplying powers of the same base adds the exponents: x squared times x cubed is x to the power 5, because that is two copies of x followed by three more copies. Second, a power of a power multiplies the exponents: (x squared) cubed is x to the power 6, three groups of two copies each.

Every power has an undoing, called a root. The square root of 9 is 3, because 3 x 3 is 9, and the cube root of 8 is 2, because 2 x 2 x 2 is 8. Euler also shows you a beautiful tie between this lesson and the last one: fractions can live in the exponent. A number to the power 1/2 means its square root, so 9 to the power 1/2 is 3. Nothing here is decoration. In two lessons, when an equation leaves you with x squared equals 49, the root is what hands you the answer, x equals 7. Fractions and powers are the tools. Equations are the job they were sharpened for.

Lesson 5

Simple equations

An equation is a balance. It says the thing on the left of the equals sign weighs exactly the same as the thing on the right. Solving it means finding what value of the unknown makes that true, and the golden rule is simple: whatever you do to one side, do the same to the other, so the balance stays level. Start with Euler's gentle kind of first problem. x + 5 = 12. The unknown has 5 added to it. Undo that by subtracting 5 from both sides: x + 5 - 5 = 12 - 5, so x = 7. Check it, always check it: 7 + 5 is 12. The balance holds.

Each kind of knot has its own undoing, and they are all the reverse operations you already know. If the unknown is multiplied, divide: 3x = 21, divide both sides by 3, and x = 7, since 3 x 7 is 21. If it is divided, multiply: x/4 = 5, multiply both sides by 4, and x = 20. If two knots are tied together, undo them in reverse order, like unwrapping a parcel. Take 2x + 4 = 18. First undo the + 4: subtract 4 from both sides to get 2x = 14. Then undo the times 2: divide by 2 to get x = 7. Check: 2 x 7 + 4 is 18. Correct. That two step rhythm, undo the addition first, then the multiplication, solves a huge family of real problems.

Here is the same idea wearing a disguise, because the unknown sometimes appears on both sides. Take 5x = 3x + 8. Gather the unknowns together by subtracting 3x from both sides, which keeps the balance honest: 2x = 8, so x = 4. Check: 5 x 4 is 20, and 3 x 4 + 8 is 20. You have just done the thing this course promised. You can solve a simple equation: state the balance, undo operations one at a time on both sides equally, and verify your answer by putting it back in. Euler builds from here to problems with dozens of steps, but the method never changes. It is balance, patience, and checking.

Lesson 6

Word problems

Word problems are equations that arrive dressed as stories, and the skill is translation. Euler's book is full of them: fathers dividing money among sons, diners splitting a tavern bill, partners sharing a profit. His method, and ours, has three steps. First, choose a letter for the thing you do not know. Second, translate each sentence of the story into algebra, one fact at a time. Third, solve the equation you built, then translate the answer back into the story and check that the story makes sense. The translation step is the whole art, and it rewards slow, literal reading.

Try one in Euler's own style. Two numbers add to 20, and one is 4 more than the other. Find them. Let the smaller number be x. Then the larger is x + 4, because it is 4 more. Together they make 20, so x + (x + 4) = 20, which is 2x + 4 = 20. Subtract 4: 2x = 16. Divide by 2: x = 8. So the smaller number is 8 and the larger is 12. Check against the story, not just the equation: 8 + 12 is 20, and 12 is indeed 4 more than 8. Notice there is even a shortcut hidden here, which Euler himself points out in a similar problem: the larger number is half of (sum + difference), (20 + 4)/2 = 12, and the smaller is half of (sum - difference), (20 - 4)/2 = 8.

One more, an inheritance, the kind Euler returns to again and again. A father leaves 90 crowns to two sons, the elder to receive twice as much as the younger. Let the younger son's share be x. The elder then receives 2x. Together: x + 2x = 90, so 3x = 90, and x = 30. The younger son gets 30 crowns, the elder gets 60, and 30 + 60 is 90, as it must be. Nearly every word problem in business, building, cooking, and travel is one of these in costume: something unknown, some relationships stated in words, a total that ties them together. Name the unknown, write the relationships honestly, and the equation does the rest.

Lesson 7

Quadratic equations

Sometimes the unknown is squared, and the equation becomes a quadratic. The simplest kind hides nothing: x squared = 49. Take the square root of both sides and x = 7, since 7 x 7 is 49. A careful note: -7 also squares to 49, so the full answer is x = 7 or x = -7, though when x is the side of a field or the price of a coat, only the positive answer fits the story. Euler treats these double answers seriously, and so should you. A quadratic can have two answers, and both can be mathematically right even when only one makes sense in the world.

The standard shape is x squared + bx + c = 0, and the friendliest way in is factoring, which is multiplication run backwards. Take x squared - 5x + 6 = 0. Hunt for two numbers that multiply to 6 and add to -5: those are -2 and -3. So the equation splits into (x - 2)(x - 3) = 0. A product is zero only when one of its factors is zero, so either x - 2 = 0, giving x = 2, or x - 3 = 0, giving x = 3. Check both: 4 - 10 + 6 is 0, and 9 - 15 + 6 is 0. Both work. Two answers, both earned, neither a mistake.

When factoring refuses to cooperate, Euler reaches for the method that always works: completing the square. Take x squared + 6x = 16. The trick is to add the square of half the x coefficient. Half of 6 is 3, and 3 squared is 9, so add 9 to both sides: x squared + 6x + 9 = 25. The left side is now a perfect square, (x + 3) squared = 25. Take roots: x + 3 = 5 or x + 3 = -5, giving x = 2 or x = -8. Check x = 2: 4 + 12 is 16. Check x = -8: 64 - 48 is 16. The famous quadratic formula your textbook prints is nothing but this method done once, in letters, for every equation at once. You now own the idea behind the formula, which is worth more than the formula.

Lesson 8

Why algebra opens every door in math

Step back and look at what you can now do. You can read signs and handle numbers less than nothing. You can let a letter carry an unknown and reason about it before you know its value. You can split wholes into fractions, climb powers, and climb back down with roots. You can balance an equation, untie it step by step, translate a story into symbols, and solve even when the unknown is squared. That is not a party trick. It is the operating system of nearly every other kind of mathematics, and of a good share of the modern world built on mathematics.

Euler knew exactly where his little book was pointing, because he had personally built so much of what lies beyond it. The same letters and balances grow up into the algebra of polynomials, then into calculus, where letters stand for changing quantities; into geometry done with equations, where a curve becomes a formula you can interrogate; into physics, where falling stones, planets, and electric currents all submit to being written as equations and solved. Euler himself used these tools on everything from the orbit of the Moon to the tuning of musical instruments to the famous bridges of Königsberg puzzle. Every one of those journeys starts on the road you just walked: numbers, signs, letters, balance.

There is a last lesson in how this book came to be. The greatest mathematician of the age, blind, near the end of his life, spent his strength not on a monument for experts but on a beginner's book, dictated slowly to a young man who had sewn clothes for a living. Knowledge, in Euler's practice, was not complete until an ordinary person could use it. You are now that person for algebra. Keep the habits he taught you: go step by step, keep the balance honest, and always, always check your answer. Every door in mathematics opens with those three keys.

Sources

The Book Behind This Course

This course is taught from one great book, free for anyone to read in full at the Internet Archive, in the English translation by John Hewlett.

• Elements of Algebra, by Leonhard Euler, translated by John Hewlett (1822 edition), Internet Archive

Graduation

You finished Euler algebra

You followed a blind master and a tailor apprentice from counting and debts all the way to quadratic equations, and you solved real problems with your own hand along the way. Euler wrote this book so an ordinary beginner could learn algebra, and you just proved him right. Congratulations. Keep the balance honest, and keep checking your answers.

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