What Is Zero Raised To The Power Of Zero?

The exponential function , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC is a self-multiplication operation where , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC is multiplied by itself , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC number of times. However, things get peculiar when we try to determine what is , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC. What is nothing multiplied by itself zero times? Does that sentence even make sense? Surely, zero multiplied by itself any number of times is zero, right? However, isn’t any number raised to zero equal to one?

Until the 19th century, most mathematicians agreed, for reasons we will discuss below, that , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC is equal to 1. However, doubt seeped in when Cauchy, a gem of a mathematician, grouped , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC and , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC in the same column, which he referred to as “indeterminate forms”. So, what then is its value?

Augustin Cauchy

French mathematician Augustine Louis Cauchy (Photo Credit : public domain/Wikimedia Commons)

Is it equal to 1?

A clever student may cite this proof to prove this claim:

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

For , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC.

The student has proved it very slyly, for while the proof appears to be consistent, only someone scrupulous will tell you that it actually isn’t: after the substitution, we are dividing zero by zero, an operation that is meaningless or undefined. While it cannot be proved algebraically, can it be proved analytically?

Consider this trend:

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

The trend limns the function , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC. Surely, the trend suggests that , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC must be equal to 1?

However, this is not true. The function approaches , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC as the value of , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC approaches , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC, but approaching a value is different from reaching and being exactly it. This does not “prove” that , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC  In fact, the minute we substitute , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC for , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC, the proof becomes self-defeating.

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

, What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC

This, in fact, represents a neat proof for why , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC does not equal , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC, but , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC!

It is undefined

There is no consensus today about whether the value is , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC or , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC; one can prove both with equally compelling arguments. Some mathematicians have accepted that it has no definite value, but instead depends on the context in which it is being used. However, there are other mathematicians who repudiate this convenience and are adamant that the value is 1. This is because certain rules require the equation to be necessarily assigned to , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC. For instance, the derivative of the exponential function becomes invalid if , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC isn’t considered to be equal to , What Is Zero Raised To The Power Of Zero?, Science ABC, Science ABC.

There seems to be no answer either algebraically or analytically. We’ve only run into contradictions, just like one does when trying to determine what is zero divided by zero. The essential contradiction being: zero raised to any positive number must be zero, but any positive number raised to zero must be one; we cannot have it both ways. Mathematicians describe these problems as either meaningless or undefined.

References

  1. Harvey Mudd College, Claremont, California
  2. The University of Utah
  3. Wikipedia

 

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Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

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