What happens when a number is added to its opposite?

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Additive inverse is the number that is added to a given number to make the sum zero. For example, if we take the number 3 and add -3 to it, the result is zero. Hence, the additive inverse of 3 is -3. We come across such situations in our daily life where we nullify the value of a quantity by taking its additive inverse. Let us learn the additive inverse property of real and complex numbers in this article.

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+ (-7) =0. Thus -7 is the additive inverse of 7 and 7 is the additive inverse of -7.

Additive Inverse Property

When the sum of two real numbers is zero, then each real number is said to be the additive inverse of the other. So, we have R + (-R) = 0, where R is a real number. R and -R are the additive inverses of each other. For example: 3/4 + (-3/4) = 0. Here 3/4 is the additive inverse of -3/4 and vice-versa. This is an example of the additive inverse of a fraction.

Let’s say you have a bucket of water at room temperature. You add a liter of hot water to it which makes the overall temperature of the bucket rise by a certain amount. Now, add another liter of cold water to it. The contrasting temperatures of water added to the bucket will cancel out each other, and the result will be a bucket of water at room temperature. The same rule applies while finding the additive inverse of a number. The additive inverse property holds good for both real numbers and complex numbers.

Additive Inverse of Real Numbers

The given number can be a whole number, a natural number, an integer, a fraction, a decimal, or any real number. The additive inverse of real numbers is just the negative of the given number.

Additive Inverse of Complex Numbers

The algebraic property of complex numbers states the existence of additive inverse. Given any complex number z ∈ C, there is a unique complex number, denoted by -z, such that z + (-z) = 0. Moreover, if z = (x,y) with x,y ∈ R, then -z = (-x,-y).

Let Z = x + iy be the given complex number. Then its inverse is -Z = -x - iy. For example, the additive inverse of - i - 1 = - (- i - 1) = i + 1.

Additive Inverse of a Fraction

The additive inverse of a fraction a/b is -a/b, and vice-versa. It is because a/b + (-a/b) = 0. The additive inverse of a positive fraction is the same fraction with the negative sign, while for a negative fraction, its additive inverse is the same fraction without the negative sign.

Additive Inverse Formula

The general formula for the additive inverse of a number can be given in the form of the number itself. Any number when added to its negative will cancel out each other and give the overall sum as zero. We need to find the negative of the given number N. In other words, we need to find -1 × (N). Hence, we can say that:

Additive Inverse of N = -1 × (N)

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 in Algebraic Expressions

The property of additive inverse can be extended to algebraic expressions. Following the same rule as stated above, the additive inverse of an algebraic expression is one that makes the sum of all the terms zero. Note: The additive inverse of the expression is -(expression). The additive inverse of x2 + 1 is - (x2 + 1) = -x2 - 1.

For example, the additive inverse of 2x + 3y is -2x - 3y, making the sum of all the elements zero.

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1. Example 1: What is the additive inverse of -6/14?

   Solution:

   We know that the sum of the given number and its additive inverse = 0

   Let x be the additive inverse.

   -6/14 + x = 0

   x = 6/14

   Answer: The additive inverse of - 6/14 is 6/14.

2. Example 2: What is the additive inverse of the expression 13x + 5y - 9z?

   To find the answer, we need to find the additive inverse of the whole expression.

   It can be calculated by multiplying the whole equation by -1.

   -1(13x + 5y - 9z) = -13x - 5y + 9z

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

3. Example 3: Find the additive inverse of the fraction -6/5.

   Solution: To find the answer, we can apply the additive inverse formula, -1 × R. So, let us substitute -6/5 in the formula, -1 × R = -1 × (-6/5) = 6/5 Therefore, the additive inverse of -6/5 is 6/5.

   Answer: The additive inverse of the fraction -6/5 is 6/5.

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FAQs on Additive Inverse

Additive inverse is what you add to a number to make the sum zero. For example, the additive inverse of 4 is -4 because their sum is zero. When two numbers are added together to get 0, then we say both the numbers are additive inverses of each other.

How do you Find the Additive Inverse of a Given Number?

In order to find the additive inverse of a given number, we just change the sign of the given number to get its additive inverse. For example, the additive inverse of 8 is -8 and the additive inverse of -35 is 35.

What is the Additive Inverse Formula?

The additive inverse formula is -1 × R, where R is any real number. This formula can be applied to any number to get its additive inverse. For example, if we need to get the additive inverse of 7, let us substitute 7 in the formula, -1 × R = -1 × 7 = -7. Therefore, the additive inverse of 7 is -7.

What is the Additive Inverse of Zero?

Since zero does not have a positive or negative sign associated with it, the additive inverse of zero is zero.

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 → ( 4 + (- 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 → (4 + 0 = 4).

What is the Difference between Additive Inverse and Multiplicative Inverse?

Additive inverse is what we add to a number to make the sum zero, whereas, the multiplicative inverse is the reciprocal of the given number, which when multiplied together, gives the product as 1.

Does 0 have an Additive Inverse?

Yes, since zero does not have a positive or negative sign associated with it, the additive inverse of 0 is 0.

What is the Additive Inverse of 7?

The additive inverse of 7 is -7. This can be verified as: 7 + (-7) = 0.

What is the Additive Inverse of 12?

The additive inverse of 12 is -12. This can be verified by: 12 + (-12) = 0.

What is the Additive Inverse of 2/3?

The additive inverse of 2/3 is -2/3. This can be verified by: 2/3 + (-2/3) = 0.

A mathematical “operation” refers to calculating a value using operands and a math operator. The numbers used for an operation are called operands. Based on the type of operation, different terms are assigned to the operands. Operators are the symbols indicating a math operation, for example:

• + for addition
• − for subtraction
• × for multiplication
• ÷ for division
• = for equal to, indicates the equality, that is, the left hand side value is equal to the right hand side value. 

What are Inverse Operations?

Inverse means the opposite. So in math, an inverse operation can be defined as the operation that undoes what was done by the previous operation. The set of two opposite operations is called inverse operations. 

For example: If we add 5 and 2 pens, we get 7 pens. Now subtract 7 pens and 2 pens and we get 5 back. Here, addition and subtraction are inverse operations.

Examples of inverse operations are given below: 

Addition and subtraction are inverse operations, which means addition undoes subtraction and subtraction undoes addition. We can rearrange the numbers given in the addition equation and then we can use the addition equation to get two different subtraction equations.

For example: $15 + 6 = 21$

The two subtraction equations formed are: $21$ $–$ $6 = 15$ and $21$ $–$ $15 = 6$

We can even show after this that when you take a number for example 15, add and subtract the same number to 15, we get 15 again.

i.e., $15 + 6$ $–$ $6 = 0$

Similarly, we can rearrange the numbers that are given in the subtraction equation and then we can make two addition equations.

For example: $30$ $–$ $14 = 16$

The addition equations formed are: $14 + 16 = 30$ and $16 + 14 = 30$

When we add a number and subtract the same number later on, the effect is reversed.

$16 – 4 + 4 = 0$

Multiplication and Division

Multiplication and division are the inverse operations, which means multiplication undoes division and division undoes multiplication. We can rearrange the numbers given in the multiplication equation and then we can do two different division equations. 

For example: $7 \times 4 = 28$

The two division equations formed are: $28\div4=7$ and $28\div7=4$

Also, when we multiply a number and then divide the same number later on, the effect is reversed.

$12\times4=48; 48\div4=12$

Similarly, we can rearrange the numbers that are given in the division equation and then we can make two multiplication equations.

For example: $45\div9=5$

The multiplication equations formed are: $9 \times 5 = 45$ and $5 \times 9 = 45$

Also, when we divide a number and then multiply the same number later on, the effect is reversed.

$1\div24=3; 3\times4=12$

Properties of Inverse Operations

1. 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{x}$, such that:

$\text{x + ( – x ) = x – x} = 0$

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

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

1. 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 $\text{x}$, i.e.,$\frac{1}{\text{x}}$, such that:

$\text{x}\times\frac{1}{\text{x}}=1$

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

Suppose, $\frac{1}{x}$ is the original number, then its multiplicative inverse will be reciprocal of  $\frac{1}{x}$, i.e., $x$. 

            For example: multiplicative inverse of $\frac{3}{4}$ is $\frac{4}{3}$. 

Solved Examples

Example 1: Form the subtraction equations from $24 + 13 = 37$.  

Solution: $37$ $–$ $24 = 13$ and $37$ $-$ $13 = 24$

2. What is the additive inverse of –10?

Solution: Since –10 is a negative number, its additive inverse will be a positive number. So, the additive inverse of –10 will be 10.

3. What is the multiplicative inverse of $(3-\frac{1}{4})$?

Answer: $(3-\frac{1}{4})=\frac{12-1}{4}=\frac{11}{4}$

Multiplicative inverse of $\frac{11}{4}$ is $\frac{4}{11}$.

Practice Problems

Attend this quiz & Test your knowledge.

Correct answer is: $12 + 3 = 15$
The addition equation for $15$ $-$ $12 = 3$ is $12 + 3 = 15$.

Correct answer is: Both A and B
The division equations will be $12\div3=4$ and $12\div4=3$.

Correct answer is: $-18$
$3\times6=18$. Additive inverse of 18 is $-$ $18$.

Frequently Asked Questions

What is the inverse operation for the square of the number?

The inverse operation for the square of a number is the square root. If $8^2=64$, then $\sqrt{64}=8$.

What is the purpose of inverse operation?

Inverse operations are used to reverse the effect of one operation on the other. The purpose of inverse operations is to understand the relation between the basic math operators $+$, $−$, $\times$, $\div$ so that solving an equation becomes easier and time saving.

How is inverse operation used in solving the equations?

The inverse operations are used in solving the equations for isolating the variables by applying inverse operations on both the sides.