The inverse property of addition states that any number + its inverse will equal 0. Opposite numbers have opposite signs (they are on opposite sides of 0), but they are the same distance from zero. For instance, 6 + its inverse (-6) = 0. In other words, 6 – 6 = 0. According to the inverse property of multiplication, if you multiply a number by its reciprocal, also known as the multiplicative inverse, the product is 1. (a/b)*(b/a)=1. In this article, we are talking about this topic. So, keep reading to know more about it.

**Inverse Property definition**

If A is a non-singular square matrix, there exists a n x n matrix A-1, known as the inverse of a matrix A, that satisfies the property:

Where I is the Identity matrix, AA-1 = A-1A = I. The 2 x 2 matrix’s identity matrix is given by I = [1 0 0 1]. It should be noted that in order to find the matrix inverse, the square matrix must be non-singular and have a determinant value that is not zero. Consider the square matrix A.

A = [a c b d]

Where a, b, c, and d are the numbers. The matrix A’s determinant is written as ad-bc, where the value is not equal to zero. The additive inverse is the number’s inverse. Any integer multiplied by its additive inverse equals zero.

a+(-a)=0

The multiplicative inverse is the number’s reciprocal. Any nonzero integer multiplied by its inverse equals one.

a*1/a=1

**Inverse Property formula**

When you express a number as a fraction, its reciprocal is a fraction with the numerator and denominator reversed. For instance, the reciprocal of 4/5 is 5/4. Some general guidelines to remember:

- The reciprocal is undefined if the fraction has a 0 in the numerator.
- If the denominator of the fraction is 0, the fraction is undefined. As a result, it cannot have a reciprocal.
- Every other fraction has one reciprocal.

Finally, the original fraction is the reciprocal of a reciprocal. So, this indicates that if you flip the numerator and denominator of a fraction and then flip them back, you obtain the original value. 4/5→5/4→4/5

According to the inverse property of multiplication, if you multiply a number by its reciprocal, also known as the multiplicative inverse, the product is 1. (a/b)*(b/a)=1.

**Inverse property of multiplication**

A reciprocal is a multiplicative inverse. What exactly is a reciprocal? A reciprocal is one of two integers that, when multiplied together, equal the number 1. For example, if we have the number 7, the multiplicative inverse, or reciprocal, is 1/7 since multiplying 7 and 1/7 yields 1.

**Inverse property of multiplication examples**

**Example 1**

What is 15’s multiplicative inverse? In other words, whatever number, when multiplied by 15, produces the number 1? Let us solve this algebraically, with x representing the unknown multiplicative inverse.

15 * x = 1 x = 1/15

That’s all! It was actually that easy! A number’s multiplicative inverse is that number as the denominator and 1 as the numerator. We get 1 when we multiply 15 by 1/15.

**Example 2**

What is 1/4’s multiplicative inverse? This example is a little different because we start with a fraction. Let us solve this algebraically once more, with x being the unknown multiplicative inverse of 1/4.

1/4 * x = 1

x = 1 / (1/4)

(1/1) / (1/4) = (1/1) * (4/1) = 4

When dividing fractions, remember to reverse the numerator and denominator of the second fraction before multiplying. As the multiplicative inverse of 1/4, we obtained 4. Doesn’t that make sense?

Read Also: Rationalize the denominator: Meaning, Examples, Problems and More

So, from these two instances, we can conclude that when you have a whole number, the multiplicative inverse of that number is that number in fraction form, with the whole number as the denominator and 1 as the numerator. When you have a fraction with 1 as the numerator, the multiplicative inverse is just the fraction’s denominator.

**Example 3**

Example 1: A pizza is cut into eight pieces. Also, tom keeps three slices of pizza on the counter and the rest on the table for his three buddies to enjoy. What percentage does each of his pals receive? Is the multiplicative inverse used here?

Solution:

Tom ate three pieces out of eight, implying he consumed one-eighth of the pizza. The pizza that was left out = 1 – 3/8 = ⅝. 5/8 to be split among three pals ⇒ 5/8 ÷ 3.

To simplify the division, we use the multiplicative inverse of the divisor.

5/8 ÷ 3/1

= 5/8 × 1/3

= 5/24

Each of Tom’s buddies will receive a 5/24 share of the leftover pizza.

**Example 4**

The total distance between Mark’s house and school is 3/4 kilometer. He can pedal a third of a kilometer in one minute. How long will it take him to get to school from home?

Solution:

3/4 km is the total distance from house to school.

A minute’s travel equals 1/3 kilometer.

Total distance/distance covered in a minute = total distance/distance covered in a minute

= 3/4 ÷ 1/3

3 is the multiplicative inverse of 1/3.

3/4 x 3 = 9/4 x 2.25 = 2.25 minutes

Mark’s total time to travel the complete distance is 2.25 minutes.

**Example 5**

Find the multiplicative inverse of -9/10 in Example 3. Also, double-check your response.

Solution:

-10/9 is the multiplicative inverse of -9/10. Also, to check the result, multiply -9/10 by its reciprocal and see if the product is 1.

(-9/10) × (-10/9) = 1.

-10/9 is the multiplicative inverse of -9/10.

**Inverse property of addition**

Consider the concept of items canceling out to understand what an inverse attribute is. Consider a board game in which a game piece is placed on the board’s initial position. If the player rolls a die and it comes up two, the game piece travels two spaces forward. However, any player that lands on that place is required to relocate two spaces back. How far did that gaming piece eventually travel? Zero!

According to the definition of the inverse property, when combined together with its additive inverse, the total of the two integers is zero. The result of a number multiplied by its multiplicative inverse is one.

a+(−a)=0.

This indicates that the inverse of a number is the same number with the opposite sign for the inverse feature of addition. The inverse property of addition asserts that when adding a number and its inverse, the sum will always be zero.

It is critical to practise discovering and applying the inverse to isolate a variable using an inverse property of addition example.

**Inverse property of addition example**

**Example 1**

To solve for x in the case 37+x=14, use the following formula:

- Determine the additive inverse of 37, which is (-37).
- To get x by itself, add the inverse to both sides of the equation.

−37+37+x=−37+14

- Reduce both sides of the equation to obtain x=23.
- 13=3+c is another example.
- To obtain c on its own, perform the following steps: 1) Determine the inverse of the number or term added or removed from the isolating variable. Because the number in this situation is -3, the inverse is 3.
- Multiply both sides of the equation by the inverse.

3+13=3−3+c.

- To get 16=c, simplify both sides of the equation.

**Example 2**

Liz ran 1.96 kilometers less than Sonya at the track event on Saturday. Liz completed 1.258 kilometers of running. Sonya ran how many kilometers?

First, the important phrase “less than” indicates that this might be a subtraction equation. Next, create a variable. You’re attempting to find out how many kilometers Sonya ran in this problem. Let the variable equal the amount of kilometers run by Sonya. Create an equation now. You know that if you take Sonya’s amount of kilometers and subtract 1.96, you get Liz’s number of kilometers. Liz also ran 1.258 kilometers, as you are aware.

x-1.96=1.258

Finally, work out the equation. Also, to isolate, multiply both sides of the equation by 1.96.

x-1.96+1.96=1.258+1.96

Then, combine like terms to simplify the left side of the equation.

x=1.258+1.96

Now, combine like terms to simplify the right side of the equation.

x=3.218

Sonya completed 3.218 kilometers.

**Inverse property of rational numbers**

According to the multiplicative inverse feature of rational numbers, for any rational number a/b, b ≠ 0, there exists a rational number b/a for which a/b b/a = 1. A rational number b/a is the multiplicative inverse of a rational number a/b in this example. The multiplicative inverse of 7/3, for example, is 3/7. (7/3 X 3/7 = 1).

Every rational number multiplied by 0 equals 0. If a/b is any rational integer, a/b × 0 = 0 × a/b = 0. For example, 7/2 × 0 = 0 × 7/2 = 0.

**Example 1**

Fill in the blanks with rational number attributes.

- a) 2/3 + 1/6 = _ + 2/3
- b) 21 × 23 × 32 = 32 × _ × 23

Solution: We can fill in the blanks using the commutative property of rational numbers.

- a) 2/3 + 1/6 = 1/6 + 2/3
- b) 21 × 23 × 32 = 32 × 21 × 23

**Example 2**

Using the distributive property of rational numbers, solve 7/2(1/6 + 1/4).

Solution:

Let us write the preceding statement using the distributive principle of rational numbers as A (B + C) = A × (B + C) = AB + AC

= 7/2(1/6 + 1/4)

= 7/2 × (1/6 + 1/4)

= (7/2 × 1/6) + (7/2 × 1/4)

= (7/12) + (7/8) = 35/24

**Example 3**

If 8/3 (7/6 5/4) = 35/9, determine the product of (8/3 7/6) 5/4.

Solution:

According to the associative feature of rational numbers, any three rational numbers (A, B, and C) may be represented as (A B) C = A (B C).

Given = 8/3 × (7/6 × 5/4) = 35/9

Using the associative feature of rational numbers, we can deduce that (8/3 × 7/6) × 5/4 = 35/9.

Let us first solve the equations inside the brackets to verify this.

(8/3 × 7/6) × 5/4 = 56/18 × 5/4\s= 35/9

Hence, 8/3 × (7/6 × 5/4) = (8/3 × 7/6) × 5/4 = 35/9.

**Inverse property of subtraction**

Subtraction and addition are inverse operations. For example, if you add 5 to any number and then deduct 5 from the total, you will get back to the original number. The addition was reversed by the subtraction. The opposite of a number is referred to as its additive inverse.

**Formula**

We deal with equations a lot in math. Equations are mathematical expressions that contain an equals sign. The two sides must be balanced. It’s like having two bowls with the same number of chocolate candies in them. The subtraction property of equality states that if we subtract from one side of an equation, we must likewise subtract from the other side to maintain the equation the same.

So, instead of having two bowls of chocolate candies, consider consuming a few candies from one dish. You’d have to eat a few candies from the other bowl to keep the two bowls the same. The same is true for equations. Also, to keep them the same, apply the same logic to both sides of the equation. When you deduct four from one side, you must also subtract four from the other.

If a = b, then a – c Equals b – c, according to the formula. So, this means that if we have two bowls with the same number of chocolate candies in each, if we take away from one, we must take away the same quantity from the other to maintain the two bowls the same. Let’s look at some examples of how this subtraction characteristic of equality is used.

**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 function’s inverse. When two inverses are added together, they equal x. As a result, if f(x)=bx and g(x)=logb x, then: f∘g=blog bx=x and g∘f=logb bx=x

These are known as the Inverse Logarithm Properties. Let’s 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 (42)x=log 4 42x=2x

** **

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

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

y=2e^x−1

x=2e^y−1

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

x/2=e^y−1

ln(x/2)=ln e^y−1

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 (ln e^y-1=y1). Determine the value of y.

ln(x/2)=y−1

ln(x/2)+1=y

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

**Inverse property additive**

The number that is added to a given number to make the total zero is known as the additive inverse. For instance, if we take the number 3 and multiply it by -3, the outcome is zero. As a result, the additive inverse of 3 is -3. In everyday life, we encounter circumstances in which we nullify the value of a number by taking its additive inverse.

A number’s additive inverse is its inverse number. When a number is multiplied by its additive inverse, the sum of both numbers equals zero. The straightforward approach is to convert a positive number to a negative number and vice versa. We already know that 7+ (-7) Equals 0. Thus, -7 is the additive inverse of 7, and 7 is the additive inverse of -7.

When the sum of two real numbers is zero, each is said to be the additive inverse of the other. As a result, R + (-R) = 0, where R is a positive integer. R and -R are additive inverses of one another. For instance, 3/4 + (-3/4) Equals 0. 3/4 is the additive inverse of -3/4 in this case, and vice versa. So, this is an example of a fraction’s additive inverse.

Assume you have a pail of room temperature water. You pour in a liter of hot water, raising the total temperature of the bucket by a given amount. Pour in another liter of cold water. The different temperatures of the water put to the bucket will balance each other out, resulting in a bucket of room temperature water. The same approach applies for determining a number’s additive inverse. The additive inverse property applies to both real and complex numbers.

**Formula**

The generic formula for the additive inverse of a number can be written in the number’s own form. When a positive integer is added to its inverse, the result is a zero sum. We must find the inverse of the provided integer N. In other words, we must find -1. (N). As a result, we may say:

N = -1 × Additive Inverse (N)

**Inverse property examples**

**Example 1 **

What is the additive inverse of -6/14?

Solution:

We already know that the sum of the given integer and its additive inverse equals zero.

Assume x is the additive inverse.

-6/14 + x = 0

x = 6/14

6/14 is the additive inverse of – 6/14.

**Example 2**

What is the additive inverse of 13x + 5y – 9z?

To obtain the solution, we must first determine the additive inverse of the entire equation.

It may be found simply multiplying the entire equation by -1.

-1(13x + 5y – 9z) = -13x – 5y + 9z

Answer: -13x – 5y + 9z is the additive inverse of the given expression.

**Example 3**

Find the additive inverse of the fraction -6/5.

Solution: We may use the additive inverse formula, -1 R, to determine the solution. So, using -6/5 in the calculation, -1 R = -1 (-6/5) = 6/5 As a result, the additive inverse of -6/5 is 6/5. 6/5 is the additive inverse of the fraction -6/5.

**Some frequently asked questions**

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

According to the inverse property of multiplication, if you multiply a number by its reciprocal, also known as the multiplicative inverse, the product is 1. (a/b)*(b/a)=1.

**What is an example of inverse property in math?**

To solve for x in the case 37+x=14, use the following formula:

- Determine the additive inverse of 37, which is (-37).
- To get x by itself, add the inverse to both sides of the equation.

−37+37+x=−37+14

- Reduce both sides of the equation to obtain x=23.
- 13=3+c is another example.
- To obtain c on its own, perform the following steps: Determine the inverse of the number or term added or removed from the isolating variable. Because the number in this situation is -3, the inverse is 3.
- Multiply both sides of the equation by the inverse.

3+13=3−3+c.

- To get 16=c, simplify both sides of the equation.

**Is the inverse property of addition?**

Subtraction and addition are inverse operations. For example, if you add 5 to any number and then deduct 5 from the total, you will get back to the original number.

**What is the inverse property of the multiplication example?**

The simple premise is that multiplying a number by its multiplicative inverse returns you to one. 5 × 1/5 Equals 1.

**What is an inverse in math?**

Inverse operations are mathematical procedures in which one operation reverses the action of the other, such as addition and subtraction, multiplication and division. In most cases, the inverse of a number is its reciprocal, i.e. x – 1 = 1 / x. A number with its inverse (reciprocal) product equals one.

**What is the inverse of 3x 4?**

(x+4)/3 is the inverse function of 3x – 4. Do the inverse function test to see if the examples above are inverses of each other. If f o g = g o f, two functions are said to be inverse of each other. They are diametrically opposed.

**What are the two inverse properties?**

The additive inverse property simply asserts that adding a number and its inverse results in a sum of zero. According to the multiplicative inverse property, multiplying a nonzero integer by its inverse yields a product of one.

**What is identity property?**

According to the identity property of 1, every integer multiplied by 1 retains its identity. In other words, every number multiplied by one remains unchanged. The number remains constant since multiplication by one produces one duplicate of the number.

**Which is an example of associative property?**

The associative feature of addition states that changing the order of the addends has no effect on the sum. For instance, (2 + 3) + 4 = 2 + (3 + 4) (2 + 3) + 4 = 2 + (3 + 4) +4=2+(3+4) equals, 2, plus, left parenthesis, 3, plus, 4, right parenthesis, plus, 4 equals, 2, plus, left parenthesis, 3, plus, 4, right parenthesis.

**What is associative property and distributive property?**

According to the associative property, while adding or multiplying, the grouping symbols can be altered without affecting the result. The formula is (a+b)+c=a+(b+c). The distributive property is a multiplication strategy that includes multiplying one integer by each of its independent addends.