Your kid’s homework says “find the product of 6 and 7”, and you pause. The product definition in math is straightforward: a product is the result you get when you multiply two or more numbers or expressions together, and the numbers being multiplied are called factors. So 6 × 7 = 42, and 42 is the product.
What is the product definition in math?
The outcome of the operation where two or more numbers (or variables or expressions) are multiplied is termed the product. The factors are the things which are multiplied.
The formal version is slightly wider than the classroom one. Wikipedia’s definition of a product covers both the result of a multiplication and the expression that identifies the objects to be multiplied. So “3 × 5” and “15” are both correctly called the product.
The parts of a multiplication sentence: factors, multiplicand, multiplier, product
Take 4 × 3 = 12:
- Multiplicand: 4 — the number which is multiplied.
- Multiplier: 3 — the number which does the multiplication.
- Product: 12 — the answer.
- Factors: 4 and 3 together.
Let us read it concretely: if you have four boxes of three apples, then you have 12 apples. One thing which is worth saying out loud, because it saves a lot of worry, is that the labels multiplicand and multiplier are positional only. Multiplication is commutative, so almost never does which number gets which name change the final answer you write down.
Worked examples with single-digit factors
- 2 × 5 = 10 — the product of 2 and 5 is 10.
- 4 × 7 = 28 — the product of 4 and 7 is 28.
- 9 × 3 = 27 — the product of 9 and 3 is 27.
This is why “product” is the word to hunt for in a word problem. It is always the case that “the product of” means to multiply, every time, with no exceptions.
Where does the word “product” come from, and why is a supermarket word in your times tables?
What explainer pages skip is, “product” first appeared in English in the context of mathematics.
From Latin prōdūcō to the times table
It comes from Middle English product, from Latin prōductus, the perfect participle of prōdūcō meaning to bring forth. Wiktionary records the word as first attested in English in the mathematics sense. The Latin sense of the word suits completely to what multiplication does, that is to bring forth a new quantity from the factors which are given.
How the everyday meaning drifted in afterward
When the word ‘product’ meets a shop, it is there that a ‘thing made or sold’ is defined. That sense is the later, derived one. It is the exact opposite of what most readers assume, and there is a bit of friction because most students read ‘find the product’ to mean going to name an object as opposed to performing a calculation. Explaining the history of the word resolves the friction. Bringing forth is the reason the ‘product’ definition in math was able to stretch to operations that are quite removed from arithmetic, like ring multiplication, as well as a categorical product.
How do you find the product of two or more numbers?
- Two factors: multiply them directly. 6 × 7 = 42.
- Three or more factors: multiply two at a time, left to right, carrying the running result. 9 × 3 = 27, then 27 × 4 = 108.
- Regroup when it’s easier: multiplication is associative as well as commutative, so 9 × 3 × 4 can also be done as 9 × 12 = 108.
Products of fractions and decimals
- Fractions: multiply the numerators, then the denominators. 2/3 × 3/4 = 6/12 = 1/2.
- Decimals: multiply as whole numbers, then count the decimal places in the factors and place the point. 0.4 × 0.3 = 0.12.
Products of variables and algebraic expressions
- The product of 3 and x is 3x.
- The product of x and x is x².
- The product of (x + 1) and 2 is 2x + 2.
So, products are not a whole number idea. Whatever is multiplied has a product.
Product vs. sum, factor, multiple, and quotient: which word means what?
Most of the mix-ups in the product definition in math is due to vocabulary as opposed to the math itself.
| Keyword | Operation | Example |
|---|---|---|
| Product | Multiply | 3 × 5 = 15 |
| Sum | Add | 3 + 5 = 8 |
| Difference | Subtract | 5 − 3 = 2 |
| Quotient | Divide | 15 ÷ 3 = 5 |
Product vs. factor: inputs and output
Factors are the inputs; the product is the output. In 3 × 4 = 12, the factors are 3 and 4, and the product is 12.
Product vs. multiple: the integer condition nobody states
A multiple is the product of a quantity and an integer. The integer condition is the part that the school pages leave out. It’s what links the two words. If at least one factor is a whole number, then the product is always going to be a multiple of the other factor. So, 12 is the product of 3 and 4, so it’s also going to be a multiple of both of those factors.
But 12 is not a multiple of 3.5 — even though 3.5 × (12 ÷ 3.5) = 12. The multiplier has to be an integer.
A quick contrast to keep in mind: factors of a number are finite and never larger than the number itself; multiples are infinite and never smaller.
What properties does every product follow?
Commutative, identity and zero properties
- Commutative: reversing the factors leaves the product unchanged. 2 × 3 = 3 × 2 = 6.
- Identity: the product of any number and 1 is that number. 35 × 1 = 35.
- Zero: the product of any number and 0 is 0. 5 × 0 = 0 — and a single 0 anywhere in a chain zeroes the lot, so 9 × 3 × 0 × 4 = 0.
Zero property vs. the zero-product property
These two get conflated constantly, and only one of them matters in algebra. The zero property says a × 0 = 0. The zero-product property is the converse, and it is far more powerful: if a product equals zero, then at least one factor must be zero.
That is the rule behind solving a factored equation. For (x − 2)(x + 3) = 0, set each factor to zero: x = 2 or x = −3. Keep them straight — the first is the elementary rule, the second is the one you will actually use.
The product of no numbers: the empty product is 1
What is the product of no numbers at all? It is 1 — the identity element of multiplication. That is the same fact sitting underneath 35 × 1 = 35, and it closes the loop the school pages leave open once they have taught you that multiplying by 0 gives 0.
Where do you actually use products outside the classroom?
- Total cost: 7 tickets at £12 each is the product 7 × 12 = 84.
- Area of a rectangle: length × width, so a 4 m by 3 m room is 12 m².
- Equal groups and arrays: 4 rows of 3 chairs is 4 × 3 = 12 chairs — the picture multiplication is first taught with.
- Three-factor jobs: scaling a recipe, converting units, and volume (length × width × height).
This is the practical reason the vocabulary is worth knowing. The word “product” tells you which operation to reach for before you touch any numbers.
How far does “product” go beyond arithmetic?
The same word extends all the way up, and what changes is the type of thing it gives you back.
| Kind of product | Example | What it returns |
|---|---|---|
| Product of two numbers | 3 and 5 → 15 | A number |
| Product of a sequence (capital pi notation) | ∏ from i = 1 to 6 of i² = 1 · 4 · 9 · 16 · 25 · 36 | A number |
| Matrix product | Entry c_ij = Σ a_ik b_kj | A matrix |
| Dot product | Two vectors | A number |
| Cross product | Two 3-D vectors | A vector |
| Cartesian product | A = {1, 2}, B = {x, y} → {(1, x), (1, y), (2, x), (2, y)} | A set of ordered pairs |
| Categorical product | Objects in a category | An object with projection maps. |
The two places the school definition stops working
School’s definition of “the number you get when you multiply” breaks in exactly two places, and both happen unexpectedly:
- The cross product. Multiply two 3-dimensional vectors and the answer is itself a vector, and it’s going to be perpendicular to both the factors, with length equal to the area of the parallelogram they span. Not a number.
- The Cartesian product. A × B is a set of ordered pairs (a, b) with a ∈ A and b ∈ B. Not a number either.
The definition that survives both is the generalised one Wiktionary records: a product is the result of any operation that generalises multiplication of numbers — the multiplicative operation in a ring, a product of types, a categorical product. The version of the product definition in math learned in school is, of course, a specific case of this; it is not wrong. Recognizing this stops you from being initially confused by the term “dot product” that you will encounter later, and assuming you have run into a different word.
The three things to take away
- In multiplication you take two or more factors and you combine them to obtain a product.
- The word was a mathematics term before it was a shop term — it originally comes from Latin prōdūcō, meaning “to bring forth”.
- In mathematics, beyond basic operations, a product is the output of any operation that generalizes multiplication (e.g. a cross product returns a vector, a Cartesian product returns a set).
Hold the plain definition and its boundary together, and the word stays useful from the times table all the way to linear algebra. There’s a whole lot to explore on Read Hive HQ.
People Also Ask
Is “product” only used for multiplication in math?
Yes — in math, “product” always refers to multiplication or to an operation that generalises it. In arithmetic, it is the result of multiplying factors, so the product of 3 and 5 is 15. At higher levels it names the output of multiplication-like operations: dot products, cross products, matrix products, Cartesian products and categorical products. It never means add, subtract or divide.
What is a multiplicand and a multiplier, and does the order matter?
A multiplicand is the number being multiplied and a multiplier is the number you multiply by. For example, in the equation 4 × 3 = 12, the multiplicand is 4 and the multiplier is 3. The product is 12. Order almost never matters because multiplication is commutative, so multiplying two numbers is the same as multiplying the numbers in the other order: 4 × 3 and 3 × 4 both give 12. The two names are positional labels, not different kinds of number.
Why does the product of a number and 0 equal 0?
Because multiplication counts equal groups, and zero groups of anything makes nothing, so for example 5 × 0 = 0. One zero anywhere in a chain of factors zeroes out the whole product, so 9 × 3 × 0 × 4 = 0. Don’t get this confused with the zero-product property, which runs the other way: if a product equals zero, at least one factor must equal zero. That is how you solve (x − 2)(x + 3) = 0 to get x = 2 or x = −3.
