Properties

Terms

When multiplying two terms, they take the names notated above. However, when we have more than 2 terms, they take on the more common names 'multiples' or 'factors.' The result is always known as the product.

Notation

Above are the various ways one can represent the multiplication operation.

Negative Coefficient

A negative in front of a value is equivalent to multiplying the value by a negative one.

Negative Notation

Above are the various ways one can represent a negative fraction.

Multiplicative Identity

Any value multiplied by the multiplicative identity (1) is equal the value itself.

Multiplicative Inverse

Any value multiplied by its reciprocal (1/a) is 1.

Zero Property

Anything multiplied by zero is zero.

Commutative Law of Multiplication

The ordering of terms does not matter when multiplying elements together, as long as the final product includes all terms.

Associative Law of Multiplication

The grouping of terms does not matter in multiplication as long as the final product includes all terms.

Distributive Law of Multiplication Over Addition

Multiplying a sum or difference by a quantity is equal to multiplying each term by the quantity than performing the sum or difference.

Double Negatives

Multiplying a negative value by another negative value equals a product which is positive.

Multiplying Fractions

When multiplying multiple fractions, we multiply all the numerators together; then, we multiply the denominators together. Go to Arithmetic-Division to learn more about fractions.

Example Problems

Definition of Multiplication

The operation of multiplication represents the act of adding the same value multiple times. Anytime we sum the same quantity any number of times, we may instead multiply that quantity by the number of times it was added.

Both the quantity, and the number of times it was multiplied by itself make up the multiplicand and the multiplier in multiplication, with the final sum being the product:

If a1, a2, a3 to an represent the same value, then:

Multiplying by Zero

The product between anything and zero is always zero.

Zero is also the additive identity, in which any value added to it is the value itself.

Multiplicative Identity

The multiplicative identity is simple, anything multiplied by one is equal to itself.

Anytime we multiply by a quantity equal to one, we can get rid of it. This is how we 'cancel terms' in multiplication.

Sign of Product

The product of two positive values is always positive.
The product of a positive and negative value is negative.
However, the product of two negative values is positive.

When we are multiplying by more than 2 negative values, these terms can be combined two-by-two until the final sign is determined.

Multiplicative Inverse

The multiplicative inverse states that any value divided by itself is equal to one.

Using the commutative and associative laws of multiplication, we may rearrange the final example like so:

Arranging equations so that like quantities are divided by like allows us to simplify them, making solving significantly easier.

Long Multiplication

Long multiplication is used to solve problems that prove difficult or impossible to solve in your head. Generally, these problems will simply be solved with a calculator, but understanding how to solve these problems by hand will aid in your understanding of how these numbers are put together. This ability will bedrock your understanding in factoring equations, which is a fundamental skill in algebra.

Exercises

To Keep Learning

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