A student wants to know why, if we can define 0! as 1, we cannot define 1 over 0
suppose 1/0 = x
That means that 1 = 0*x
But zero times anything is zero. There is no number x which can be defined as 1/0.
0! is useful because n! is a recursive definition.
That is,
n! = n * (n-1)!
3! = 3 * 2!
2! = 2 * 1!
1! = 1 * 0!
But, n! is also defined as the product of all the numbers from 1 to n. Since 1! is clearly 1, that means we must have 0! = 1.
The factorial of a non-negative integer n, denoted as n!, is defined as the product of all positive integers less than or equal to n.
By convention, 0! is defined to be equal to 1. This definition is useful in many mathematical calculations and also makes sense when considering combinatorics, where 0! represents the number of ways to arrange zero objects.
On the other hand, dividing any number by zero, including 1 over 0, is undefined in mathematics. This is because division is the inverse operation of multiplication, and we cannot find a number that, when multiplied by zero, will give us a non-zero result.
In other words, division by zero leads to contradictions and mathematically inconsistent results. It breaks the fundamental rules and properties of arithmetic. Therefore, dividing by zero is not allowed in mathematics and does not have a defined value.
To summarize, while 0! is defined as 1 based on mathematical conventions and principles, dividing by zero is undefined due to the inconsistency and contradictions it leads to in mathematics.
The reason why we can define 0! (0 factorial) as 1 but cannot define 1/0 is because of the fundamental differences between the two concepts.
First, let's understand what factorial means. Factorial is a mathematical operation that multiplies a given number by all positive integers smaller than it. For example, 4! (read as "4 factorial") is calculated as 4 x 3 x 2 x 1 = 24. By convention, mathematicians define 0! as 1, even though it might seem counterintuitive at first.
Now, let's move on to division by zero (1/0). Division is the process of splitting a quantity into equal parts. However, division by zero is undefined in mathematics. This is because when dividing a number by zero, there is no meaningful way to distribute the quantity equally among the non-existent parts. In other words, division by zero leads to contradictory or nonsensical results.
To understand this concept more intuitively, let's consider an example. Suppose we have 10 apples, and we want to divide them equally among 0 people. Mathematically, we would write this as 10/0. The question now is, how many apples will each person get? Since there are no people to distribute the apples to, it becomes impossible to determine the answer. Any answer we could propose would be arbitrary and inconsistent with the principles of division.
Division by zero also leads to contradictions when considering limits in calculus or solving equations. Therefore, division by zero is undefined in mathematics, unlike defining 0! as 1, which is consistent with the factorial concept and has specific mathematical reasoning behind its definition.