Short answer first: yes. The three expressions 2 \cdot x, x \cdot 2, and 2x all denote the same product. Plug any number in for x and all three give the same answer. They are equal to each other in the strongest possible sense — same value, every time, no exceptions.

So here is the puzzle. If all three are equal, why does your textbook always write 2x and never x \cdot 2? Why do teachers circle x2 in red even though x \cdot 2 is a legitimate expression? The answer is that 2x is a convention, not a mathematical fact. The three forms are identical in arithmetic; only one of them is standard on the page. That gap — between what is mathematically legal and what is conventionally written — is what this article is about.

The three notations, side by side

There are three ways you might see a coefficient written next to a variable:

  1. 2 \cdot x — the explicit dot. A multiplication sign (\cdot or \times) sits between the two factors. Unambiguous; every student can parse it; nobody gets confused.
  2. x \cdot 2 — the commuted version. Same two factors, swapped order. By the commutative property of multiplication, a \cdot b = b \cdot a for any numbers, so this equals 2 \cdot x.
  3. 2x — juxtaposition. No sign at all. Writing the number next to the variable is itself a shorthand for "multiply them." By definition, 2x means 2 \cdot x.

All three produce the same value. Pick x = 5:

The mathematical object is one single thing — "two times x." What differs is only the typographic dress it is wearing.

Why juxtaposition is even allowed: in algebra, when a number and a variable are written next to each other with nothing between them, the default operation is multiplication. There is no other operation that makes sense there — you cannot add 2 and x without writing +, you cannot subtract without writing -. Multiplication is the silent default, and the dot is dropped as visual clutter. This is the same reason xy means x \cdot y and 5\pi means 5 \cdot \pi — two factors side by side, multiplication implied.

Why the convention puts the number first

If all three forms are mathematically equal, why did mathematicians settle on 2x? Two reasons, both about reading.

Reason 1 — the coefficient is what you scan for. When you read a polynomial like

7x^3 + 3x^2 - 2x + 1,

your eye is looking for the coefficients: 7, 3, -2, 1. They tell you how much of each power is present. Putting the number first means every coefficient sits at the left edge of its term — the spot your eye lands on after each plus or minus sign. Flip the convention and you get

x^3 7 + x^2 3 - x 2 + 1,

where the coefficients are buried mid-term and you have to hunt for them. The conventional form makes them pop.

Reason 2 — single-purpose positions. Once you commit to "coefficient before the variable, exponent after," each position on the page has exactly one job. The slot in front is for how many; the slot behind (raised) is for how many times it multiplies itself. A student scanning 3x^2 parses it instantly: three copies of x-squared. No overlap between the roles, no ambiguity.

The failure modes of the wrong conventions

The three alternative notations are mathematically fine, but each one causes a specific type of error when students use it.

The x2 trap — ambiguous with x^2

The single worst thing you can write is x2. Is that "x times 2" or "x squared"? In printed text with a proper font, the exponent x^2 is visibly smaller and raised; the multiplication x \cdot 2 would print with a clear dot. But in hurried handwriting on a phone screen or a rough notebook, the two look almost identical.

The ambiguous x2 notation compared to x squared and 2xThree boxes in a row. The left box shows 2x clearly with a caption reading standard, unambiguous. The middle box shows x2 with the 2 at normal baseline, flagged with a red question mark and captioned is this x squared or x times 2. The right box shows x with a small raised 2 in superscript position, captioned this one is x squared. An arrow connects the middle box to both outer boxes to show it could be read as either. 2x standard, clear coefficient first x2 ? ambiguous — never write this x² x squared — exponent, not factor the middle form could be misread as either outer form — so do not use it
The reason $2x$ is standard is not mathematical — it is about avoiding the ambiguity of $x2$. Put the coefficient in front and there is no possible confusion with an exponent.

Rule: never write a number to the right of a variable unless you mean an exponent, and in that case write it as a proper superscript (x^2, x^3). The slot on the right of a variable belongs to exponents. Reserve it for them.

Juxtaposition fails between two numbers

A subtler confusion: juxtaposition only works when at least one of the factors is a variable. You cannot drop the dot between two numbers. Writing

2 \cdot 3

as simply 23 does not mean "two times three" — it means the number twenty-three. Decimal notation got to that slot first. Adjacent digits, with no operation symbol, form a single multi-digit number. There is no way to override this with "well, I meant multiplication."

So the shorthand rule is more precise than "adjacent means multiply." It is: adjacent means multiply, when at least one factor is a variable (or an irrational constant like \pi, or a bracketed expression). 2x, 3\pi, 5(x+1) — all fine. 2 \cdot 3 — write the dot, or write 2 \times 3, or write (2)(3) with brackets. Never 23 when you mean multiplication.

xy \cdot 2 instead of 2xy

Putting the coefficient last is legal but weird. If you write xy \cdot 2, a reader will pause, mentally rearrange it to 2xy, and then continue — and that pause is an error tax you have imposed on them. The book, the exam, the answer key all use 2xy. Match them. When your work sits next to theirs on the page, consistent notation makes comparison trivial; inconsistent notation adds friction to every single step.

This one is not wrong the way x2 is wrong — it does not risk being misread. It is just non-standard, and non-standard looks like inexperience.

Worked example — same answer, three ways

Let x = 5. Compute each of the three forms.

All three give 10. This is not a coincidence and it is not a lucky choice of x — it will work for every value of x, because the three forms denote the same mathematical object. The commutative property of multiplication says a \cdot b = b \cdot a for any numbers, and the juxtaposition rule says 2x is an abbreviation for 2 \cdot x. Those two facts together make the three forms equal for every x, forever.

Why this matters in practice: when you are simplifying an expression and you see a mix of forms — one line writes x \cdot 3, the next line writes 3x — you should feel comfortable that these are the same thing and move on. Do not rewrite every step just to match convention; do not suspect your algebra is broken because the order changed. Commutativity guarantees the swap is free.

The takeaway

The three forms 2 \cdot x, x \cdot 2, and 2x are mathematically identical — same value for every x, guaranteed by the commutative property of multiplication and the juxtaposition shorthand. The preference for 2x is a typographic convention, not a mathematical one: coefficient in front, no dot, variable next, exponent (if any) as a raised superscript behind. That convention exists because it makes coefficients easy to scan, it keeps the slot behind the variable reserved for exponents, and it prevents the specific ambiguity of x2 being read as x^2.

Write 2x, not x2 and not x \cdot 2. You will never be wrong mathematically if you stick with convention, and your work will read the way the textbook, the exam, and the rest of algebra read. The mathematics does not care; the reader does.