As an expert in the field of mathematics, I can provide an in-depth explanation of why any number multiplied by zero equals zero. This fundamental principle is rooted in the properties of multiplication and the definition of zero within the number system.

Multiplication is essentially a repeated addition. When you multiply a number by another, you are adding the first number to itself the second number of times. For instance, \(3 \times 4\) means you are adding the number 3 to itself four times: \(3 + 3 + 3 + 3 = 12\).

Now, let's consider the number zero. Zero is unique in the number system because it represents the absence of quantity. It is the neutral element in addition, meaning that adding zero to any number does not change the value of that number. Mathematically, this is expressed as \(a + 0 = a\) for any number \(a\).

When we combine these two concepts, we can understand why multiplying any number by zero results in zero. If you are told to add a number to itself zero times, you are essentially not adding anything at all. There are no repetitions of the addition, so the initial quantity remains unchanged, which is zero times the number.

Let's illustrate this with an example. Suppose you want to calculate \(5 \times 0\). According to the definition of multiplication, you would be adding the number 5 to itself zero times. Since zero repetitions of any action result in no action at all, you are not adding 5 to anything. Thus, the result is \(0\).

This principle holds true for all numbers, not just integers. Whether you are dealing with fractions, decimals, or even complex numbers, the multiplication with zero will always yield zero. This is a direct consequence of the properties of the number zero and the nature of multiplication.

In summary, the reason any number times zero is zero is because multiplication by zero is equivalent to not performing any additions at all, which aligns with the definition of zero as the additive identity.

Now, let's move on to the next step as per your instructions.

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The equation 0 x 2 = 0 is read as 'Zero times Two equals Zero'. That means, you are taking '2' zero times, effectively taking nothing. So, multiplying any number by zero will always be zero.read more >>