Multiplicative Absorbing Element
The absorbing element of multiplication is the number that, when multiplied by any other number $n$, always results in itself. In the set of real numbers, this absorbing element is zero. $$ n \times 0 = 0 \times n = 0 $$
In other words, multiplying any number by zero always gives zero.
It doesn’t matter if you’re multiplying large numbers or very small ones; the presence of zero in the multiplication "absorbs" everything, and the result disappears.
In other terms, zero cancels out any other number, making it “disappear” from the final outcome. As we’ll see, this seemingly simple concept is at the core of many mathematical properties.
An Example
The absorbing element nullifies any number it’s multiplied by.
No matter the number you choose, when multiplied by 0, the result is always 0:
For instance, if you multiply 5 by 0, the result is 0:
$$ 5 \times 0 = 0 $$
If you multiply 100 by 0, the result is still 0:
$$ 100 \times 0 = 0 $$
This holds true for very large numbers and very small ones alike:
$$ 100000000 \times 0 = 0 $$
$$ 0.0000001 \times 0 = 0 $$
The result is always zero.
Why Is the Absorbing Element Important?
The concept of an absorbing element in multiplication may seem simple, but it plays a crucial role in many areas of mathematics.
For example, in algebra, the absorbing element is used to simplify expressions and solve equations. This leads to the zero product property: if a product equals zero, then at least one of the factors must be zero. $$ a \times b = 0 $$
The reason you cannot divide any number by zero is directly linked to the absorbing element of multiplication.
When you attempt to divide a number $n$ by zero, you’re essentially asking: "What number $m$, when multiplied by zero, gives $n$?"
$$ \frac{n}{0} = m $$
But since any number $m$ multiplied by zero will always result in zero, there is no valid answer.
$$ m \times 0 = 0 \ne n $$
This is why dividing by zero is considered undefined in mathematics.
For example, if you try to divide 5 by zero: $$ \frac{5}{0} = m $$ Now, try to find a number $m$ that, when multiplied by zero, equals the dividend (5). Such a number doesn’t exist, because multiplying any number by zero always gives zero. $$ 1 \times 0 = 0 \\ 2 \times 0 = 0 \\ 3 \times 0 = 0 \\ \vdots $$
The Absorbing Element in Other Operations
We’ve focused on multiplication, but other mathematical operations also have their own absorbing elements.
For example, in Boolean logic, which is widely used in computer science, the absorbing element for conjunction (AND) is false. $$ \text{False AND True = False } $$ $$ \text{False AND False = False } $$ Generally, if one of the values in a logical operation is false, the result will always be false, regardless of the other value. This mirrors the role of zero in multiplication.
