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# Multiply|Definition & Meaning

## Definition

In mathematics, adding the same number **again and again**, is termed **multiplication**. It is one of the four common** arithmetic operations** in math, alongside addition, subtraction and division. In other words, if you add **equal groups repeatedly**, the number of elements in that group increases by the same amount **every time**, and we say that the original group has **multiplied** by however many times we repeatedly added it.

The process of **multiplication** has a minimum of **three** **elements**, which include two factors and a product. The product comes when the first factor is added into itself a certain **number **of times, in our case, the second factor. The **number **of factors goes on increasing as per the problem you come across.

**What Is Multiplication?**

To better understand, let’s take an example where we are to **multiply **the **number **of **oranges **contained in the two crates.

As per the figure, there are two crates each having **4** **oranges **in them, the crates are known as groups and thus there are two such groups.

If we make an **addition** here, the total **number **of **oranges **becomes:

**4 + 4 = 8**

However, since we have added the two groups of 4 **oranges **each, therefore we have **multiplied **four **oranges **by the value of two. So, we can also write is as:

**2 x 4 = 6**

This gives us a clear idea that addition and **multiplication** are very closely related to each other, and also **4 + 4 **comes to be the same as **2 x 4.** When we **multiply **any two numbers, the resulting **number **is called a product. The total **number **of **elements **in an individual group is known as a **multiplicand**, whereas the total **number **of the same groups is known as a **multiplier**. In our event, **4 **stands to be the **multiplicand**, **2** comes to be the **multiplier** and **8** is the result of that **multiplication** known as **the product**.

There are several possible ways to deliver an equation that includes **multiplication**.

For instance,** 2 x 4 = 8**. It can be delivered as,

- Two
**multiplied**by four is eight. - Two
**times**four is eight. - Two fours are eight.

**Symbols of Multiplication**

Every piece of writing has its own way of **representing multiple **symbols. Some common symbols for multiplication are **a cross** (×), **an asterisk** (*), or **a dot** (·).

Usually, when we are writing in our notebooks, we most commonly use the **cross** sign as it is easier to draw and thus read. The other two symbols (**asterisk** and **dot**) are more commonly used in **computer** **languages** and **higher** **mathematics,** such as **algebra** and **calculus**.

A better visual of these symbols can be **illustrated** in the below figure.

**How To Multiply Integers?**

**Integers **can be **positive** as well as **negative**, so when **multiplying **two integers, we need to be careful about the signs they are carrying. There can be three possible scenarios,

### Multiplication of Two Positive Integers

When two positive **Integers **are **multiplied**, the **product** will always be a positive number.

For instance, **2 x 4 = 8.**

### Multiplication of One Positive Integer and One Negative Integer

When one **positive** and one **negative** **integer **are **multiplied**, the **product** will always be a negative number. Here the position of the negative sign has no effect on the overall result.

For instance, **(**–**2) x 4 = (-8).**

### Multiplication of Two Negative Integers

When two negative **Integers **are multiplied, the product will be a **positive** **number**, as minus signs will cancel out each other.

For instance, **(**–**2) x (-4) = 8.**

**How To Multiply Fractions and Decimals?**

**Fractions** are a combination of two numbers sitting on top of each other. The upper one is known as the **numerator**, and below the **numerator** lies the **denominator**. To **multiply **a pair of fractions, we just simply **multiply **the **numerators** with **numerators** and **denominators** with **denominators**, such that

Let’s say we have two fractions $\dfrac{1}{2}$ and $\dfrac{3}{4}$, then,

\[ \dfrac{1}{2} * \dfrac{3}{4} = \dfrac{1*3}{2*4} = \dfrac{3}{8} \]

Decimals can be **multiplied **in the same way as **multiplying **integers.

**104 x 56 = 5824**

Let’s say we have two decimals, 10.4 and 5.6. Multiplying** **these two **decimals** as **whole** **numbers** to see the result,

**10.4 x 5.6 = 58.24**

Now since the product is of two decimal point numbers, the **product** will have a decimal point add up to two places starting from the right, such that

## Multiplying Numbers With Powers

In mathematics, **powers** represent the relation of a **number **to itself, defined by a base raised to an exponent. The exponent tells us how many times the **number **is being **multiplied **by itself concurrently, e.g., $2^2,\, (-5)^6$.

- If the bases are the same but the
**powers**are different.- The
**powers**add up, whereas the bases merge to become one common base. i.e. $x^m \times x^n = x^{m+n}$.

For instance, $2^2 \times 2^4 = 2^{2+4} = 2^6 = 64$

- The

- If the exponents are the same, but the bases are different.
- The two different bases are
**multiplied,**whereas the**powers**merge to become one common exponent. i.e. $x^m \times y^m = (xy)^m$.

For instance,$2^2 \times 3^2 = 6^{2} = 36$

- The two different bases are

- If the bases and
**powers**are different.- Since nothing is common here, each expression is first evaluated before being
**multiplied**by each other. i.e. $x^m \times y^n = x^m y^n$.

For instance,$2^2 \times 3^3 = (2)^2 (3)^3 = 4 \times 27 = 108$

- Since nothing is common here, each expression is first evaluated before being

**Some Common Properties When Using Multiplication**

**Multiplication** also follows the same pattern as an addition when it comes to having a set of properties, which are given as follows,

**Commutative Property of Multiplication**

The **commutative** **property** tells us that when two **Integers **are multiplied, the order of the factors will have no effect on the final product. Two **multiplied **by three is the same as **multiplying **three by two,

**2 x 3 = 3 x 2**

**Associative ****Property of Multiplication**

This **Associative** **property** says that if three or more numbers are **multiplied **one after, then their position will have no effect on the result, even if they are shuffled back and forth.

**2 x (3 x 6) = 36**

**(2 x 3) x 6 = 36**

**Distributive** **Property of Multiplication**

The **Distributive** **property** says that if there lies a sum of two numbers and we are to **multiply **another **number **with them, then the product will be the same as if we were to **multiply **those two numbers separately. For instance, if the sum of numbers is (2 + 4) and we **multiply **this sum by 6, we get:

**6 x (2 + 4) = 36**

- We can add 2 and 4 to get 6. Therefore, 6
**multiplied**by 6 gives 36. - If we
**multiply**6, individually with each number,**6 x 2 = 12 + 6 x 4 = 24**, - We get the sum equal to 36. Hence, the distributive property is proven to be true.

An amazing thing about **multiplication** is that if you **multiply **any **number **in the universe by **1**, you will get the exact **number **as the product. Also, if you **multiply **any **number **in the universe by **0**, the end result will always be a **0**, no matter how large the **number **might be.

**Multiplication** is a fun tool to learn despite being an **arithmetic** headache sometimes.

**Solved Example**

Find the solution of the following **equation**, also figure out which property is being used.

**y = (3 x 6)(2+4)**

**Solution**

First, compute the sum inside the braces **(2+4) = 6.**

Next, multiply the next closest digit to the sum, which is **6:**

**y = 3 x 3 x 6**

**y = 3 x 36**

Now finally, **multiply **the remaining two digits to get the final product:

**y = 108**

The property used here is **the distributive** **property** of **multiplication**.

*All images/mathematical drawings were created with GeoGebra.*