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Letters for numbers: writing expressions from words

In 1591 François Viète printed a book in which letters stood for numbers, vowels for the unknown ones and consonants for the known ones.

BackgroundFrançois Viète was the son of a lawyer and studied law himself. The MacTutor biography at the University of St Andrews quotes him describing himself as someone who did not profess to be a mathematician but delighted in mathematics whenever there was leisure. He was good enough that Henry IV of France turned to him to read the coded letters being sent to his enemy, Philip II of Spain. The code took Viète some time, and he broke it. In 1591, at Tours, he published In artem analyticam isagoge, which MacTutor calls the first systematic algebraic notation. Letters stood for quantities, both known and unknown: vowels for the unknowns, consonants for the known ones. The habit we use now, with early letters such as a, b, c for known values and late letters such as x for unknowns, came later from René Descartes in La Géométrie, printed at Leiden in 1637 as an appendix to his Discourse on Method.

One letter can stand for every age at once

OpenStax starts its algebra chapter with two people who share a birthday. Greg is 20 and Alex is 23 this year. When Greg was 12, Alex was 15; when Greg is 35, Alex will be 38. Whatever Greg’s age, Alex is 3 years older. The ages change, so in algebra they are variables. The gap of 3 never changes, so it is a constant. Call Greg’s age g and you can describe every row of that table with one expression, g + 3. A variable is a letter that represents a number or quantity whose value may change. Letters often used for variables are x, y, a, b and c, but any letter works if you say what it means.

An expression is a number, a variable, or a combination of numbers, variables and operation symbols: 3 + 5, n − 1, 6 · 7 and g + 3 are all expressions. OpenStax compares an expression to a phrase in English, such as “the sum of three and five”, which has no verb. An equation joins two expressions with an equal sign, and the equal sign works like the verb that turns a phrase into a sentence: 3 + 5 = 8 says something that is either true or false. When you translate words into algebra, check first which of the two you are being asked for. “Five more than n” is a phrase, so the answer is the expression n + 5. “Five more than n is 12” is a sentence, so it becomes the equation n + 5 = 12.

Short ways to write multiplication and division

Algebra avoids the cross sign for multiplication, because × looks like the letter x. OpenStax asks: does 3xy mean 3 × y or 3 · x · y? Write a raised dot or brackets instead, or simply put the factors side by side. A number comes in front of the letters, so x · 3 is written 3x and b · 5 · a becomes 5ab, with the letters usually in alphabetical order. A factor of 1 is not written: 1 · a is just a, and −1 · a is −a. Repeated factors become a power, so a · a · a is a³, read “a cubed”. Division is usually written as a fraction, so a ÷ 4 becomes a over 4, which can also be typed as a/4. None of this changes the value. It only makes expressions shorter and easier to compare.

Long form	Short form
x · 3	3x
b · 5 · a	5ab
1 · a	a
(−1) · a	−a
a · a · a	a³
a ÷ 4	a/4 (a over 4)
(x + y) · 2	2(x + y)

Turning words into an expression

Most word phrases name an operation and then two numbers. “The sum of a and b” is a + b, “the difference of 20 and 4” is 20 − 4, “the product of 6 and n” is 6n and “the quotient of x and 3” is x/3. Look for the words of and and to find the two numbers. Two phrases need care because they reverse the order you read them in. “Eight more than y” means 8 added to y, so it is y + 8. “Seven less than 9z” means 7 subtracted from 9z, so it is 9z − 7, not 7 − 9z. A quick test with numbers settles any doubt: seven less than 10 is 3, and only 10 − 7 gives 3.

Longer descriptions work the same way once you name the unknown. OpenStax describes a window whose height is 6 inches less than its width. Let w be the width; then the height is w − 6. Another example: the number of dimes in a purse is 2 less than 5 times the number of quarters. Let q be the number of quarters. Five times the quarters is 5q, and 2 less than that is 5q − 2. Shopping questions are the same. If notebooks cost $3 each and a pen costs $2, then n notebooks and one pen cost 3n + 2 dollars. Write what the letter means in a short sentence before you start. A letter whose meaning you never wrote down is the most common reason a correct-looking expression turns out wrong.

An expression has no equal sign; an equation has one. y + 8 is an expression, y + 8 = 40 is an equation. If a question says “write an expression”, an answer with “=” in it is answering a different question.

the sum of a and b	a + b
the difference of a and b	a − b
the product of 6 and n	6n
the quotient of x and 3	x/3
eight more than y	y + 8
seven less than 9z	9z − 7
twice n, plus 1	2n + 1
variable	a letter for a value that may change
constant	a number whose value stays the same

Eight more than y:
- y + 8
- 8y
- y − 8
- 8 − y
Seven less than 9z:
- 9z − 7
- 7 − 9z
- 9z + 7
- 63z
Write the product of 6 and n the short way.
Write x · y · 3 the short way.
Write a · a · a as a power.
How is a ÷ 4 usually written in algebra?
- a/4
- 4/a
- 4a
- a − 4
Which one is an equation?
- y + 8 = 40
- y + 8
- 4 · 2 + 1
- x ÷ 25
Alex is 3 years older than Greg, who is g years old. Alex’s age is:
- g + 3
- g − 3
- 3g
- g/3
Notebooks cost $3 each and a pen costs $2. Write the cost in dollars of n notebooks and one pen.
The number of dimes is 2 less than 5 times the number of quarters, q. The number of dimes is:
- 5q − 2
- 2 − 5q
- 5(q − 2)
- 5q + 2
Which is the usual way to write b · 5 · a?
- 5ab
- b5a
- ab5
- 5 + a + b

Common mistakes

“Seven less than x is 7 − x.” The phrase starts from x and takes 7 away: x − 7. Put in x = 10. Seven less than 10 is 3, and 7 − 10 is −3, so 7 − x gives the wrong sign.
“3x means 3 + x.” Side by side means multiplied. 3x is 3 · x. With x = 4, 3x is 12 and 3 + x is 7.
“1x and x3 are fine ways to write x and 3x.” They give the right value, but by the usual conventions the 1 is dropped and the number goes first: x and 3x. Tests and other people read expressions faster in the standard form.

Teacher's noteIn the 2022 Korean national curriculum, letters enter in the middle school band (grades 1 to 3 of middle school): the standard asks learners to write various situations as expressions with letters, to recognise why letters are useful, and to find the value of an expression. Its notes also ask teachers to compare letters with numbers and with everyday language, looking at what they share and where they differ. Before that, primary grades 5 to 6 already write relations with boxes and triangles, so the box-to-letter step is a good bridge: write □ + 3 first, then a + 3. The phrase “less than” is where most translations go wrong. Ask learners to test every translation with one number before moving on.

Key points

A variable is a letter for a number that can change; a constant stays the same.
An expression has no equal sign; an equation joins two expressions with =.
Short forms: 3x, 5ab, a for 1a, a³ for a · a · a, a/4 for a ÷ 4.
“Seven less than 9z” is 9z − 7. Test translations with a number.

Sources for this course (10)

Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수02-01] 다양한 변화 규칙을 찾아 설명하고 수나 식으로 나타내기(규칙을 식으로 나타낼 때 혼합 계산식, 일반항을 나타낸 식 등을 이용해야 하는 복잡한 문제는 다루지 않음); 초등학교 5~6학년군 [6수01-01] 혼합 계산에서 계산하는 순서(계산 순서에 중점, 지나치게 복잡한 혼합 계산은 다루지 않음), [6수01-04] 약수·공약수·최대공약수, [6수02-01] 대응 관계를 □, △ 등을 사용하여 식으로(덧셈식, 뺄셈식, 곱셈식, 나눗셈식 중 하나로 표현되는 간단한 경우만); 중학교 1~3학년군 [9수01-02] 소인수분해로 최대공약수, 수와 연산 용어 거듭제곱·지수·밑·분배법칙, [9수02-01] 다양한 상황을 문자를 사용한 식으로 나타내어 유용성을 인식하고 식의 값 구하기, [9수02-02] 일차식의 덧셈과 뺄셈, [9수02-09] 다항식의 덧셈과 뺄셈, [9수02-10] (단항식)×(다항식), [9수02-14] 함숫값, [9수02-19] 다항식의 곱셈과 인수분해(역관계 이해); 변화와 관계 용어 대입, 다항식, 항, 단항식, 상수항, 계수, 차수, 일차식, 동류항, 등식, 전개, 변수, 인수, 인수분해; '식의 값', '전개식'은 교수·학습 상황에서 사용 가능; 문자와 수, 문자와 일상 언어의 공통점과 차이점 탐색)
Textbooks: OpenStax Prealgebra 2e, 2.1 Use the Language of Algebra (Greg 20 and Alex 23, g + 3; variable and constant; × not used in algebra; expression like a phrase, equation like a sentence; 4 + 3·7 gives 25, not 49; PEMDAS and "Please Excuse My Dear Aunt Sally"; multiplication and division left to right, 18 ÷ 9·2; 18 ÷ 6 + 4(5 − 2); 5 + 2³ + 3[6 − 3(4 − 2)]; 2³ + 3⁴ ÷ 3 − 5²; work down the page: openstax.org/books/prealgebra-2e/pages/2-1-use-the-language-of-algebra)
2.2 Evaluate, Simplify, and Translate Expressions (evaluate by substituting; 9x − 2 at x = 5 and x = 1; term, coefficient, coefficient of x is 1; 9b + 15x² + a + 6; like terms 5x, 7, n², 4, 3x, 9n²; 3x + 7 + 4x + 5; 7x² + 8x − x² − 4x; eight more than y, seven less than 9z; window height 6 inches less than width; dimes 2 less than 5 times the quarters: .../pages/2-2-evaluate-simplify-and-translate-expressions)
7.3 Distributive Property (three friends at the movies, 3(9.25) = 3(9 + 0.25) = 27.75; a(b + c) = ab + ac, a(b − c), (b + c)a; 3(x + 4); 2(x − 3); 6(5y + 1); 100(0.3 + 0.25q): .../pages/7-3-distributive-property)
9.4 (perimeter of a rectangle P = 2L + 2W: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
OpenStax Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping (GCF of 54 and 36 is 18; GCF of 27x³ and 18x⁴ is 9x³; four steps, distributive property in reverse, check by multiplying; 4x + 12, 5a + 5, 12x − 60, 4y² + 24y + 28, 5x³ − 25x², 21x³ − 9x² + 15x = 3x(7x² − 3x + 5), −8y − 24 = −8(y + 3), 5q(q + 7) − 6(q + 7): openstax.org/books/elementary-algebra-2e/pages/7-1-greatest-common-factor-and-factor-by-grouping)
2.6 (distance, rate and time formula d = rt: .../pages/2-6-solve-a-formula-for-a-specific-variable)
Microsoft Support, "Calculation operators and precedence in Excel" (negation before %, ^, then * and /, then + and −; =5+2*3: support.microsoft.com/en-us/office/calculation-operators-and-precedence-in-excel-48be406d-4975-4d31-b2b8-7af9e0e2878a)
History for the story block only: MacTutor History of Mathematics, University of St Andrews, "François Viète" (1540 to 1603; studied law, son of a lawyer; decoded letters to Philip II of Spain for Henry IV; In artem analyticam isagoge, Tours 1591, first systematic algebraic notation, vowels for unknowns and consonants for knowns; convention of early letters for knowns and late letters for unknowns introduced later by Descartes in La Géométrie: mathshistory.st-andrews.ac.uk/Biographies/Viete/)
"René Descartes" (1596 to 1650; La Géométrie an appendix to the Discours de la méthode, published at Leiden in 1637: mathshistory.st-andrews.ac.uk/Biographies/Descartes/).

Checked: 2026-10-19