mathematically, how to prove we cant divide by zero. and why multiplication of zero with any number leads to zero itself . can we mathematically prove that

I can prove it logically.

If a cake sits on the table and no one (0) eat it, the cake is still whole.

If you have $100.00 saved and it doesn't earn interest, it's multiplied by 0 and doesn't change.

Look at these cases:

12/3 = 4 because 3x4 = 12
20/10 = 2 because 2x10 = 20
5/0 = ?? because ?? x 0 = 5 , but that is a contradiction, since anything times zero is zero.

or

5/.1 = 50
5/.01 = 500
5/.001 = 5000
5/.000000000001 = 5000000000000
notice I am dividing by a smaller and smaller number and my answer is getting bigger and bigger, or "bigly" as Trump would say.

so as the divisor is getting closer and closer to zero, my answer is approaching infinity.
But mathematically we consider such an infinitely large number as "undefined"

here is another explanation:

if multiplication is repeated addition,
e.g. 3x4 = 3+3+3+3 = 12
and division is repeated subtraction:
e.g.
20 ÷ 4

20-4 = 16
16-4 = 12
12-4 = 8
8-4 = 4
4-4 = 0
I subtracted the 4 five times to get to zero
so 20÷4 = 5

so how about 20 ÷ 0
20-0 = 20
20-0 = 20
20-0 = 20
...
how many times would I have to subtract the zero to get to zero ???

As long as we're dividing by zero, what about 0/0? Why is it undefined?

Suppose there is a value x such that

0/0 = x
Then, multiplying by zero, we get
0 = 0*x
But that is true for any value of x. So there is no particular value we can say is equal to 0/0.

I always taught that 0/0 was indeterminate and did not fall into the "undefined" category.

Such as in limits that result in 0/0 situations,
e.g.
lim (x^2 - 4)/(x-2) , as x ---> 2
you would get 0/0

= lim (x-2)(x+2)/(x-2) as x ---> 2
= limi x+2 , as x --->2
= 4
so in this case 0/0 yields a result of 4

But Steve knows all that , I am sure.

To understand why we cannot divide by zero and why multiplying zero with any number results in zero, let's delve into the mathematical principles involved.

1. Why division by zero is undefined:
Division is essentially the inverse operation of multiplication. When dividing a number by another, we are finding the number that, when multiplied by the divisor, gives the dividend. For instance, dividing 10 by 2 means finding a number x such that 2 times x equals 10 (2 * x = 10), which gives x = 5.

However, dividing any number by zero leads to a contradiction because there exists no number that, when multiplied by zero, can yield any non-zero number as a result. Let's suppose we can divide by zero and find x such that 0 times x is equal to a non-zero number, say 10 (0 * x = 10).

To solve this equation, we would need to divide both sides by zero: x = 10 / 0. But this leads to an undefined result because division by zero does not follow the rules of arithmetic. It violates the fundamental concept that multiplication is the process of combining equal groups, and dividing by zero has no clear interpretation in this context.

Hence, we mathematically prove that division by zero is undefined and not possible within the current number system.

2. Why multiplication with zero yields zero:
When we multiply any number by zero, the result is always zero. We can illustrate this using the distributive property of multiplication over addition.

Let's consider a number "a" multiplied by zero: a * 0. We can rewrite this expression as (a * 1 - a * 1) since any number minus itself is zero. Using the distributive property, we get: a * 1 - a * 1 = a * (1 - 1).

Now, since anything multiplied by zero is zero, we can simplify the equation further: a * (1 - 1) = a * 0 = 0.

Therefore, mathematically, we prove that multiplying any number by zero results in zero based on the properties of arithmetic and algebraic manipulation.

In conclusion, we mathematically verify that division by zero is undefined, and multiplying any number by zero yields zero based on the principles of arithmetic and algebra.