my teacher told me that the inverse of addition was subtraction and that the inverse of subtraction was addition...
could you prove it to me
-x = (-1)x
((-1)x)^-1
I don't see how I'm suppose to get + x by taking the inverse of -x i've always been told in math to just to the opposite of subbtraction which is addition but my teacher is telling me that is a lie and that it's really the inverse of subtraction is addition but I don't see the reasoning behind it
basically can you prove to me that the opposite of subractiion is addition and vise versa??? by taking the inverses????
I don't get it...
like I can prove that the opposite of multiplication is division by taking the inverse and can prove it just by defintion
(5x = 2)5^-1 = x = 5^-1 (2)
that's how you prove that relationship is really just inverses but what about addition and subtraction how are the inverse relationships...???
Thansk
Use numbers.
6 + 2 = 8
Therefore, 8 - 6 = 2 or 8 - 2 = 6
How can you state those relationships in abstract terms?
If x is some number and:
x + y = 0
then y is called an inverse (w.r.t. addition) of x
Then it follows from the same definition that x is an inverse of y. Now, what you need to prove is that inverses are unique. I.e. if for some given x
x + y = 0
and also
x + z = 0
you necessarily have y = z.
So, it then follows that the inverse of the inverse of x is x and it can't be anything else than x.
Then, if we denote the inverse of x by
-x, we can prove that:
-x = (-1)*x
THis is because:
x + (-1)*x =
1*x + (-1)*x =
(1 + (-1))*x =
0*x = 0
Here we have used that -1 is the inverse of 1.
So, (-1)*x satisfies the criterium the inverse of x which we always denote as
-x must satisfy and therefore
-x = (-1)*x
Then the fact that taking twice the inverse yields the same number implies that:
(-1)*(-1) = 1
i agree with all of it but still don't see how
X + B = C
we can simply solve for B by simply multiplying the whole equation by B^-1 which we note as -B because????
(X + B = C)B^-1
B cancels out
X = B^-1 C
what allows us to say that B^-1 is equal to -B
You are confusing terms:
Inverse is not the reciprocal. You are using reciprocal (B^-1) is reciprocal.
Now it is confusing, because the inverse operation to multiplication is division, and the inverse to division is multiplication
http://www.mathsisfun.com/definitions/inverse-operation.html
Watch the usage to "inverse", a lot of folks really mean reciprocal when they use it.
To understand why the inverse of addition is subtraction and vice versa, let's examine the properties of these operations and their inverses.
First, let's define the terms:
- Addition is an operation where you combine two or more numbers to get a total sum. For example, 2 + 3 = 5.
- Subtraction is an operation where you take one number away from another to find the difference. For example, 5 - 3 = 2.
Now, to prove that subtraction is the inverse of addition, and addition is the inverse of subtraction, we need to show that applying one operation undoes the effect of the other.
1. Inverse of Addition (Subtraction):
Let's consider the statement: a + b = c. To find the inverse operation, we want to undo the addition of b. So, subtracting b from both sides would give us: (a + b) - b = c - b.
After simplifying, we have a + (b - b) = c - b. Since b - b equals zero, we are left with a = c - b.
From this, we can see that subtraction undoes the effect of addition, and therefore subtraction is the inverse of addition.
2. Inverse of Subtraction (Addition):
Now, let's consider the statement: a - b = c. To find the inverse operation, we want to undo the subtraction of b. So, adding b to both sides would give us: (a - b) + b = c + b.
After simplifying, we have a - (b - b) = c + b. Since b - b equals zero, we are left with a = c + b.
From this, we can see that addition undoes the effect of subtraction, and therefore addition is the inverse of subtraction.
In summary, the inverse of addition is subtraction, and the inverse of subtraction is addition. These operations undo each other's effects, allowing us to cancel them out and return to the original value.