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You can multiply each side of the multiplicative inverse equation by the other inverse to show that any two multiplicative inverses are equal. Here it is more formally.
Theorem: For all x in R, there exists y in R s.t. x * y = 1. If there is a y' in R such that x * y' = 1, then y = y'.
Proof:
- Start with x * y = 1.
- y * x = 1 (commutative)
- (y * x) * y' = 1 * y' = y'
- y * (x * y') = y' (associative)
- y * 1 = y' (because x*y' = 1)
- y = y'
To prove that the multiplicative inverse is unique, we assume there are two inverses, say (a) and (b), for a given element (x). We then show that (a = b) by using the definition of the multiplicative inverse, which states that (a \times x = b \times x = 1). By multiplying both sides of (a \times x = 1) by the multiplicative inverse of (a), which is (a^{-1}), we get (a^{-1} \times (a \times x) = a^{-1} \times 1). Simplifying the left side gives us (1 \times x = a^{-1}), which is equal to (x = a^{-1}), confirming that the multiplicative inverse is unique.
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What is the multiplicative of -1?
Assuming the question is about the multiplicative inverse, the answer is, -1. It is its own multiplicative inverse.
How do you prove the multiplicative inverse of a real number is unique?
Suppose p and q are inverses of a number x. where x is non-zero. Then, by definition, xp = 1 = xq therefore xp - xq = 0 and, by the distributive property of multiplication over subtraction, x*(p - q) = 0 Then, since x is non-zero, (p - q) = 0. That is, p = q. [If x = 0 then it does not have a multiplicative inverse.]
What is the multiplicative inverse of 4 i?
The multiplicative inverse of 4i is -(1/4)*i.
What is an multiplicative inverse of -.50?
The multiplicative inverse is 1/(-0.50) = -2
What is the multiplicative inverse of −100?
the multiplicative inverse of -100 is 1/-100
Name the additive inverse and the multiplicative inverse for the number 2.5?
Additive inverse: -2.5 Multiplicative inverse: 0.4
What is zero product property?
Multiplicative Inverse of a NumberReciprocal The reciprocal of x is . In other words, a reciprocal is a fraction flipped upside down. Multiplicative inverse means the same thing as reciprocal. For example, the multiplicative inverse (reciprocal) of 12 is and the multiplicative inverse (reciprocal) of is . Note: The product of a number and its multiplicative inverse is 1. Observe that ·= 1. Multiplicative Inverse of a NumberReciprocal The reciprocal of x is . In other words, a reciprocal is a fraction flipped upside down. Multiplicative inverse means the same thing as reciprocal. For example, the multiplicative inverse (reciprocal) of 12 is and the multiplicative inverse (reciprocal) of is . Note: The product of a number and its multiplicative inverse is 1. Observe that ·= 1.
What is the multiplcative inverse of negative 3?
The reciprocal (multiplicative inverse) of -3 is -1/3.The reciprocal (multiplicative inverse) of -3 is -1/3.The reciprocal (multiplicative inverse) of -3 is -1/3.The reciprocal (multiplicative inverse) of -3 is -1/3.
Multiplicative inverse of a negative number?
The multiplicative inverse is the negative of the reciprocal of the positive value. Thus the multiplicative inverse of -7 is -1/7.
What is the multiplicative inverse of 625?
The multiplicative inverse of 625 is 1/625 or 0.0016
What is the multiplicative inverse of negative 9?
-9; the multiplicative inverse: -1/9
What is the multiplicative inverse of -0.25?
The multiplicative inverse of a number is the reciprocal of that number. In this case, the multiplicative inverse of -0.25 is -1 / -0.25, which simplifies to -4. This is because multiplying a number by its multiplicative inverse results in a product of 1, the multiplicative identity.
