What Does The Word Product Mean In Math?

A product is the result of carrying out the mathematical operation of multiplication. When you multiply numbers together, you get their product. The other basic arithmetic operations are addition, subtraction and division; their results are the sum, the difference, and the quotient, respectively. Each operation also has special properties governing how the numbers can be arranged and combined. For multiplication, it's important to be aware of these properties so that you can multiply numbers and combine multiplication with other operations to get the right answer.

The product of a number and one or more other numbers is the value obtained when the numbers are multiplied together. For example, the product of 2, 5 and 7 is

\(2 \times 5 \times 7 = 70\)

While the product obtained by multiplying specific numbers together is always the same, products are not unique. The product of 6 and 4 is always 24, but so is the product of 2 and 12, or 8 and 3. No matter which numbers you multiply to obtain a product, the multiplication operation has four properties that distinguish it from other basic arithmetic operations.

Commutation means that the terms of an operation can be switched around, and that the sequence of the numbers makes no difference to the answer. When you obtain a product by multiplication, the order in which you multiply the numbers does not matter. The same is true of addition. You can multiply 8 × 2 to get 16, and you will get the same answer with 2 × 8. Similarly, 8 + 2 gives 10, the same answer as 2 + 8. Written generally, this means:

\(a \times b \times c= b \times a \times c= c \times a \times b = \cdots\)

Subtraction and division do not share the commutative property. If you change the order of the numbers, you'll get a different answer.

For division,

\(3 \div4 = 0.75 \neq 4 \div3 = 1.33333\)
\(For subtraction,\)
\(7 – 5 = 2 \neq 5 – 7 = -2\)

Division and subtraction are not commutative operations.

Distribution in math means that multiplying a sum by a multiplier gives the same answer as multiplying the individual numbers of the sum by the multiplier and then adding. For example,

\(4 \times (3 + 6) = (4 \times3 + 4 \times 6) = 36\)

Adding before multiplying gives the same answer as distributing the multiplier over the numbers to be added and then multiplying before adding. Division does not have the same distributive property:

\(6 \div (3+9) \neq 6 \div 3 + 6 \div 9\)

Subtracting before dividing can give a different answer than dividing before subtracting.

The associative property means that if you are performing an arithmetic operation on more than two numbers, you can associate or put brackets around two of the numbers without affecting the answer. Products and sums have the associative property while differences and quotients do not.

For example, if an arithmetical operation is performed on the numbers 12, 4 and 2, the sum can be calculated as

\(12 \times (4\times2) = 12 \times 8 = 96\)
\(\ \text{or}\)
\((12 \times 4) \times 2 = 48 \times 2 = 96\)

But for quotients:

\(12 \div (4 \div 2) \neq (12 \div 4) \div 2\)

and for differences:

\(12 – (4 – 2) \neq (12-4)-2\)

Multiplication and addition have the associative property while division and subtraction do not.

If you perform an arithmetic operation on a number and an operational identity, the number remains unchanged. All four basic arithmetic operations have identities, but they are not the same. For subtraction and addition, the identity is zero. For multiplication and division, the identity is one. The identity property for multiplication follows:

\(a \times 1 = a\)

Here is a more explicit multiplication problem:

\(((24 \times 3) + 2 – 6)) \times 1 = ((24 \times 3) – 4) = 68\)

The basic arithmetic operations have vocabulary to describe what operation is happening to which numbers. For division and subtraction – where order and association matter – these words are very important, but for multiplication and addition these descriptors bear less importance.

‌Multiplication‌

When preforming multiplication, you multiply the multiplicand by the multiplier. However, since order doesn't matter, either number can be distinguished as each element.

\(\begin{align*}\)
\(&12 \ \ \leftarrow \ \text{multiplicand}\)
\(\times & \ \ 8 \ \ \leftarrow \ \text{multiplier}\)
\(\hline &96 \ \ \leftarrow \ \text{product}\)
\(\end{align*}\)

‌Division‌

When preforming division you divide the ‌divisor‌ from the ‌dividend‌ to get the ‌quotient.‌

\(\begin{align*}\)
\(&12 \ \ \leftarrow \ \text{dividend}\)
\(\div & \ \ 8 \ \ \leftarrow \ \text{divisior}\)
\(\hline &1.5 \ \ \leftarrow \ \text{quotient}\)
\(\end{align*}\)

We can also represent division as a fraction that still uses the original numbers, or they might be adjusted to be whole numbers (or reduced fractions). In this case, the ‌dividend‌ becomes the ‌numerator‌ and the ‌divisor‌ becomes the ‌denominator.‌

\(\begin{align*}\)
\(12 \div 8\)
\(&= \frac{12 \ \ \leftarrow \text{numerator}}{8 \ \ \leftarrow \text{denominator}} \\)
\(&= \frac{3}{2} \ \ \leftarrow \text{ reduced fraction} \\)
\(&= 1.5\)
\(\end{align*}\)

‌Addition‌

When preforming addition you add the ‌addends‌ together to get the sum.

\(\begin{align*}\)
\(&12 \ \ \leftarrow \ \text{addend}\)
\(+ & \ \ 8 \ \ \leftarrow \ \text{addend}\)
\(\hline &20 \ \ \leftarrow \ \text{sum}\)
\(\end{align*}\)

‌Subtraction‌

When preforming subtraction, you subtract the ‌subtrahend‌ from the ‌minuend.‌

\(\begin{align*}\)
\(&12 \ \ \leftarrow \ \text{minuend}\)
\(- & \ \ 8 \ \ \leftarrow \ \text{subtrahend}\)
\(\hline &4 \ \ \leftarrow \ \text{difference}\)
\(\end{align*}\)

Products show up across many fields of math, and advanced math beyond high school will only continue to push the boundaries of mathematical operations and relationships. In set theory, there is a notion of a cartesian product between sets (think of sets of coordinates (x,y) in the xy cartesian plain. There are dot products in calculus, matrix products in linear algebra, and tensor products between vector spaces.

The list can go and on, but these concepts all rely the same concept of some multiplication between ‌factors‌ (in the language that we have used these are the multiplicands and multipliers).