## Multiplicative Inverse Of Integers

Finding the multiplicative inverse of positive integers is the same as natural numbers . Just like positive integers, the product of a negative number and its reciprocal must be equal to 1. Thus, the multiplicative inverse of any negative number is its reciprocal. For example, × = 1, therefore, the multiplicative inverse of -6 is -1/6. Note that the multiplicative inverse of a negative number is always negative. And, in the multiplicative inverse of a negative integer, the negative sign will attach to the numerator, and not with the denominator.

## Additive Inverse And Multiplicative Inverse

There are two properties of numbers: multiplicative inverse and additive inverse property related to the multiplication and addition operation respectively. For a number x, – x is the additive inverse and 1/x is the multiplicative inverse. Let us understand the difference between additive inverse and multiplicative inverse with the help of the following table:

Additive Inverse |
---|

## What Is The Inverse Of A Number

A **number** can have two inverses. One **inverse** is the additive **inverse**, which is the value that when added with the original **number** will equal zero. Another **inverse of a number** is the multiplicative **inverse**, or reciprocal. When a reciprocal is multiplied by the original **number**, the product is always 1.

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## Using Properties Of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

## What Is Additive Inverse

The additive inverse of a number is its opposite number. **If a number is added to its additive inverse, the sum of both the numbers becomes zero**. The simple rule is to change the positive number to a negative number and vice versa. We know that, 7+ =0. Thus -7 is the additive inverse of 7 and 7 is the additive inverse of -7.

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## Properties Of Inverse Operations

**Inverse Additive Property**

The value, which, when added to the original number gives 0, is known as the additive inverse.

Suppose, x is the original number, then its additive inverse will be minus of x, i.e., $-$$\text$, such that:

$\text = 0$

For example, $6+=0$. Hence, $6$ is the additive inverse of 6 and vice versa.

Suppose, $-$$\text$ is the original number, then its additive inverse will be the positive value of x, i.e., x.

**Inverse Multiplication Property**

The value, which, when multiplied to the original number gives 1, is known as the multiplicative inverse.

Suppose, x is the original number, then its multiplicative inverse will be the reciprocal of $\text$, i.e.,$\frac}$, such that:

$\text\times\frac}=1$

For example, $6\times\frac=1$. Hence, $\frac$ is the multiplicative inverse of 6 and vice versa.

Suppose, $\frac$ is the original number, then its multiplicative inverse will be reciprocal of $\frac$, i.e., $x$.

For example: multiplicative inverse of $\frac$ is $\frac$.

## The Inverse Of A Trigonometric Function

The trigonometric functions are the functions that relate the right-angled triangle angle with the ratios of the side of the triangle.

Ex: sin = Opposite side/Hypotenuse. So, we find the angle of the triangle by using this formula. What if we have to calculate the hypotenuse of a triangle if an angle is given. Then we will use the inverse function to calculate the hypotenuse. So, the inverse is written as

= sin-1 is the inverse of the trigonometric sine function.

Similarly, cos = Base/Hypotenuse. So, inverse is = cos-1.

Tan = Opposite side/ Base. So, inverse is = tan-1 .

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## How Do You Explain Addition Meaning And Properties

Addition is commutative, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if a and b are any two numbers, then a + b = b + a. The fact that addition is commutative is known as the commutative law of addition or commutative property of addition.

## What Is The Difference Between Inverse And Identity Property

**Identity****Identity****Inverse****Inverse**

Regarding this, what is a inverse property?

We use **inverse properties** to solve equations. **Inverse Property** of Addition says that any number added to its opposite will equal zero. **Inverse Property** of Multiplication says that any number multiplied by its reciprocal is equal to one.

Subsequently, question is, what is the inverse of adding? The **Inverse of Adding** is Subtracting **Adding** moves us one way, subtracting moves us the opposite way. Example: 20 + 9 = 29 can be reversed by 29 9 = 20

Thereof, what is an example of an inverse property?

The **inverse property** of multiplication states that the product of any number and its reciprocal is always 1. To find the reciprocal of a number, express this number as a fraction and flip the fraction. For **example**, the reciprocal of 4 would be 14. **Example** 1 7×17=?

What is the inverse property of subtraction?

The four main mathematical operations are addition, **subtraction**, multiplication, division. The **inverse** of addition is **subtraction** and vice versa. The **inverse** of multiplication is division and vice versa.

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## How To Find Multiplicative Inverse

The multiplicative inverse of a number is also known as its reciprocal. It is very easy to find the multiplicative inverse of a number using the following steps:

**Step 1:**Divide the given number by 1.**Step 2:**Write it in the form of a fraction. Say, the reciprocal of a is 1/a.**Step 3:**Simplify and get the answer.

Let us find the multiplicative inverse of 2/3. The first step is to divide it by 1, which will result in 1/ = 3/2. Therefore, the reciprocal of 2/3 is 3/2.

## Additive Inverse Property Example

**What is the additive inverse of 2/3?**

We know that the given number and its additive inverses sum should be 0.

Let x be the additive inverse.

2/3 + x = 0

Hence, the additive inverse of 2/3 is -2/3.

**Solve for Q using the Additive Inverse Property.**

**Q + 5 = 8**

We have to remove the +5. The inverse property of addition can be employed and -5 can be applied to both sides to get rid of it. -5 is the additive inverse . It is added to each side of the equation.

Therefore,

Q+5 -5 = 8 5

Q = 3.

**What is the multiplicative inverse of 1/5? Also, verify your answer.**

The multiplicative inverse of 1/5 is 5/1.

We know the given number and its multiplicative inverses product should be 1.

To verify it, we will multiply 1/5 with its multiplicative inverse.

× = 1.

Hence, the multiplicative inverse of 1/5 is 5/1.

**Solve for X in the following equation using the inverse property of multiplication.**

**7X = 42**

The multiplicative inverse property of the number positive natural number 7 can be used. This is to eliminate the +7 of the 7X. As we know, x = 1.

Proceed to multiply each side of the equation by the multiplicative inverse = 1/7.

Therefore,

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## Multiplicative Inverse Of A Mixed Fraction

To find the multiplicative inverse of a mixed fraction, convert the mixed fraction into an improper fraction, then determine its reciprocal. For example, let us find the multiplicative inverse of \.

**Step 1:**Convert \ to an improper fraction, that is 7/2.**Step 2:**Find the reciprocal of 7/2, that is 2/7. Thus, the multiplicative inverse of \ is 2/7.

It is interesting to note that the multiplicative inverse of a mixed number is always a proper fraction whose value is less than 1.

## Properties Of Identity Inverses And Zero

- Recognize the identity properties of addition and multiplication
- Use the inverse properties of addition and multiplication
- Use the properties of zero
- Simplify expressions using the properties of identities, inverses, and zero

##### be prepared!

Before you get started, take this readiness quiz.

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## Multiplicative Inverse Of 0

As per the definition of multiplicative inverse, it is the number that when multiplied to the original number results in 1 as the product. But with 0, we know that the product of 0 with any number is always 0. So, the multiplicative inverse of 0 does not exist.

We can also understand this using the properties of division which states that the division of any number by 0 is not defined. The multiplicative inverse of 0 can be written as 1/0, but its value is not defined. So, it does not exist.

## Inverse Property Of Logarithms

It is the inverse of an exponent, according to the definition of a logarithm. As a result, the inverse of an exponential function is a logarithmic function. Recall what it means to be a functions inverse. When two inverses are added together, they equal x. As a result, if f=bx and g=logb x, then: fg=blog bx=x and gf=logb bx=x

These are known as the Inverse Logarithm Properties. Lets tackle the following issues. The Inverse Properties of Logarithms will be used.

**1.Look for 10log56.**

We can see from the first property that the bases cancel each other out.

56 eln6 eln2 = 10log56

Here, e and the natural log cancel out, leaving us with 6.2=12.

**2.Look for log4 16x. **

The second property will be used in this case. Also, change 16 to 4^2.

Log4 16x=log 4 x=log 4 42x=2x

**3.Determine the inverse of f=2e^x-1.**

Replace f with y. Then, reverse x and y.

y=2e^x1

x=2e^y1

We must now isolate the exponent and compute the logarithm of both sides. Also, to begin, divide by two.

x/2=e^y1

ln=ln e^y1

Remember the Inverse Properties of Logarithms we discussed previously in this concept? When we apply logb bx=x to the right side of our equation, we get . Determine the value of y.

ln=y1

As a result, ln+1 is the inverse of 2e^y-1.

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## Is Closure Property True For Division

Closure property is not true for division of rational numbers because of the number 1/2.

**Can you give an example of an inverse property of addition?**

Inverse Property Of Addition. The sum of a number and its negative is always zero. a + = 0. Example: 5 + = 0. Also 5 + = 0.

**What is the definition of the additive inverse property?**

The additive inverse property states that a number is added to its opposite number will result in zero. Learn about the definition of an inverse number, the graphical representation of the property, and see some examples for easier understanding.

## Example: Using Properties Of Real Numbers

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

\begin3\cdot6+3\cdot4 & =3\cdot\left & & \text \\ & =3\cdot10 & & \text \\ & =30 & & \text\end

\begin\left+\left & =5+\left & & \text \\ & =5+0 & & \text \\ & =5 & & \text\end

\begin6-\left & =6+ & & \text \\ & =6+\left & & \text \\ & =-18 & & \text\end

\begin\frac\cdot\left & =\frac \cdot\left & & \text \\ & =\left\cdot\frac & & \text \\ & =1\cdot\frac & & \text \\ & =\frac & & \text\end

\begin100\cdot & =100\cdot0.75+100\cdot\left & & \text \\ & =75+\left & & \text \\ & =-163 & & \text\end

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## What Does Inverse Property Mean In Math

Inverse property of addition tells us that any number + its opposite will = 0. Opposite numbers have different signs , but are the same distance from zero. For example: 6 + its opposite = 0. Or basically, 6 6 = 0. Another example: -8 + its opposite = 0.

**What is the commutative property of addition?**

The commutative property of addition says that changing the order of addends does not change the sum. Heres an example: 4 + 2 = 2 + 4 4 + 2 = 2 + 4 4+2=2+4.

#### What is the distributive property of addition?

The distributive property is a property of multiplication used in addition and subtraction. This property states that two or more terms in addition or subtraction with a number are equal to the addition or subtraction of the product of each of the terms with that number.

#### What is the inverse process of multiplication?

The division is the reverse process of multiplication. Dividing by a number is equivalent to multiplying by the reciprocal of the number. Thus, 7 ÷7=7 × 17 =1.

**Which equation shows the inverse property of multiplication?**

a+=0a is the additive inverse of a. a + = 0 a is the additive inverse of a . Inverse Property of Multiplication for any real number a0 a 0 , a1a=11ais the multiplicative inverse of a.

## Properties Of Real Numbers

Real numbers have unique properties, which makes them particularly useful in everyday life.

**With these three marks in mind, the question is:**

**Associative Property**

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## How To Find The Additive Inverse

The additive inverse of any given number can be found by changing the sign of it. The additive inverse of a positive number will be a negative, whereas the additive inverse of a negative number will be positive. However, there will be no change in the numerical value except the sign. For example, the additive inverse of 8 is -8, whereas the additive inverse of -6 is 6.

The addition of a number and its additive inverse is equal to the additive identity.

Click here to know what is an additive identity and multiplicative identity along with examples.

## Inverse Property Of Numbers Definition Formula And Solved Examples

We will aim to understand the definition and formula of inverse property in this lesson. Its principles can be used to find additive inverse as well as multiplicative inverse in mathematical equations. This is best understood with the help of examples.

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## How To Find The Multiplicative Inverse

Consider that we have seven apples. To make them into groups of 1 each, we need to divide them by 7. Since division is the reverse process of multiplication, dividing by a number is equivalent to multiplying by the reciprocal of the number.

Thus, 7 ÷ 7 = 7 × $\frac$ = 1

Here, $\frac$ is called the multiplicative inverse of 7.

Let us understand the multiplicative inverse of different types of numbers like natural numbers, integers and fractions.

## Is Additive Inverse Same As Additive Identity

No, additive inverse and the additive identity property are not the same. The additive inverse of a given number is obtained by just reversing its sign. This means when the given number and its additive inverse are added we get 0. For example, the additive inverse of 4 is -4 = 0). Whereas, the additive identity of any given number is 0, because when we add any number to zero, it results in the number itself. For example, the additive identity of 4 is 0 .

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## What Is The Inverse Of 12

1/12The multiplicative inverse of 12 is 1/12.

**Which is the inverse of multiplication?**

The same number gets subtracted repeatedly. So, the division is the opposite of multiplication. Hence, multiplication and division are opposite operations. We may say, division is the inverse operation of multiplication.

**What is the formula of inverse property?**

The inverse property of multiplication states that if you multiply a number by its reciprocal, also called the multiplicative inverse, the product will be 1. *=1.

#### What does inverse property mean in math?

Inverse property of addition tells us that any number + its opposite will = 0. Opposite numbers have different signs , but are the same distance from zero. For example: 6 + its opposite = 0. Or basically, 6 6 = 0. Another example: -8 + its opposite = 0.

**Whats the inverse of 3?**

The multiplicative inverse of 3 is 1/3.

**Is division the inverse of multiplication?**

Because division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.