Final answer:
In mathematics, multiplying any number by 0 always results in 0, which is evident in multiplication tables or probability products. This principle is also seen in vector algebra with the null vector and in simplifying calculations by shifting the decimal point when multiplying by powers of ten.
Step-by-step explanation:
When dealing with multiplication, any number multiplied by 0 results in 0. This is a fundamental property in math which states that the product of any number and 0 is always 0.
In the context of a table where a column represents the product of a value and its probability xP(x), if one of the factors is 0, the result in the third column will also be 0. This aligns with what we observe in mathematics generally - if x is 0 or P(x) is 0, the product xP(x) will be 0.
The same idea can be applied to other mathematical concepts. For example, in vector algebra, a null vector is composed of all zeros and has no direction or length. Similarly, when multiplying by powers of ten, you can use the number of zeros in the power to shift the decimal point, thus simplifying calculation. For instance, multiplying by 100 (which has two zeros) moves the decimal point two places to the right. This pattern of zeros in various mathematical contexts allows for shortcuts and simplifications in calculations.