The multiplicative inverse is defined as:
For every number a ≠ 0 there is a number, denoted by a⁻¹ such that
a . a⁻¹ = a⁻¹ . a = 1
First we need to prove that any number times zero is zero:
Theorem:
For any number a the value of a . 0 = 0
Proof:
Consider any number a, then:
a . 0 + a . 0 = a . (0 + 0) {distributive law)
= a . 0 {existence of additive identity}
(a . 0 + a . 0) + (-a . 0) = (a . 0) + (-a . 0)
= 0 {existence of additive inverse}
a . 0 + (a . 0 + (-a . 0)) = 0 {Associative law for addition}
a . 0 + 0 = 0 {existence of additive inverse}
a . 0 = 0 {existence of additive identity}
QED
Thus any number times 0 is 0.
Proof of no multiplicative inverse of 0:
Suppose that a multiplicative inverse of 0, denoted by 0⁻¹, exists.
Then 0 . 0⁻¹ = 0⁻¹ . 0 = 1
But we have just proved that any number times 0 is 0; thus:
0⁻¹ . 0 = 0
Contradiction as 0 ≠ 1
Therefore our original assumption that there exists a multiplicative inverse of 0 must be false.
Thus there is no multiplicative inverse of 0.
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That's the mathematical proof.
Logically, the multiplicative inverse undoes multiplication - it is the value to multiply a result by to get back to the original number.
eg 2 × 3 = 6, so the multiplicative inverse is to multiply by 1/3 so that 6 × 1/3 = 2.
Now consider 2 × 0 = 0, and 3 × 0 = 0
There is more than one number which when multiplied by 0 gives the result of 0.
How can the multiplicative inverse of multiplying by 0 get back to the original number when 0 is multiplied by it?
In the example, it needs to be able to give both 2 and 3, and not only that, distinguish which 0 was formed from which, even though 0 is a single "number".
The multiplicative inverse of a number (other than zero) is the number such that the product of the two is 1. Thus, the multiplicative inverse of x is 1/x.
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.
yes
1/4m
Every non zero number has a multiplicative inverse, which is 1 divided by that number. This stands for both real and complex numbers. This can be proved by letting x=some non zero number. x*(1/x)=x/x=1, therefore the multiplicative inverse of x is 1/x.
No.
The multiplicative inverse of a number is : 1/number i.e., one divided by the number. This doesn't apply to zero. Zero has no multiplicative inverse.
The multiplicative inverse of any non-zero integer, N is 1/N.
The multiplicative inverse of a number (other than zero) is the number such that the product of the two is 1. Thus, the multiplicative inverse of x is 1/x.
The statement is true only for non-zero fractions and it follows from the definition of a multiplicative inverse.
Yes, and for any non-zero rational x, the multiplicative inverse is 1/x.
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.
The multiplicative inverse of a number is any number that will multiply by it to make zero. Here, the multiplicative inverse of -6 is -(1/6), or negative one sixth.
All rational numbers, with the exception of zero (0), have a multiplicative inverse. In fact, all real numbers (again, except for zero) have multiplicative inverses, though the inverses of irrational numbers are themselves irrational. Even imaginary numbers have multiplicative inverses (the multiplicative inverse of 5i is -0.2i - as you can see the inverse itself is also imaginary). Even complex numbers (the sum of an imaginary number and a real number) have multiplicative inverses (the inverse of [5i + 2] is [-5i/29 + 2/29] - similar to irrational and imaginary numbers, the inverse of a complex number is itself complex). The onlynumber, in any set of numbers, that does not have a multiplicative inverse is zero.
yes
1/4m
There is no such thing.However, the multiplicative inverse of any non zero number, x, is (1/x).The multiplicative inverse has the property that x*(1/x) = 1 = (1/x)*x.