The surprising story of the exponential function

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The plot of exponential functions

Understanding math will make you a better engineer.

So, I am writing the best and most comprehensive book about it.

The way you think about the exponential function is (probably) wrong.

Don't think so? Let me convince you. Did you realize that multiplying ee with itself π\pi times doesn't make sense?

Here is what's really behind the most important function of all time.

e raised to the power of pi

First things first: terminologies. The expression aba^b is read "aa raised to the power of bb." aa is called the base, and bb is called the exponent.

The terminology of the exponential function

Let's start with the basics: positive integer exponents. By definition, ana^n is the repeated multiplication of aa by itself nn times.

The definition of positive integer powers

Sounds simple enough. But how can we define exponentials for, say, negative integer exponents? We'll get there soon.

Two special rules will be our guiding lights. First, exponentiation turns addition into multiplication.

Product of powers rule

We'll call this the "product of powers" rule.

Second, the repeated application of exponentiation is, again, exponentiation.

Power of powers rule

We'll call this the "power of powers" rule.

These two identities are the essences of the exponential function. To extend the definition to arbitrary powers, we must ensure that these properties remain true.

The properties of exponentiation with integer powers

So, what about, say, zero exponents? Here, the original interpretation (i.e., repeated multiplication) breaks down immediately. How do you multiply a number with itself zero times?

The "product of powers" property gives the answer. Any number raised to the power of zero should equal to 1.

The definition of zero powers

What about negative integers? We cannot repeat multiplication zero times, let alone negative times.

Negative integer powers

Let's use wishful thinking! (I am not kidding. Wishful thinking is a well-known and extremely powerful technique.)

If powers with negative integer exponents are defined, the "product of powers" rule implies that they must satisfy the following.

The exponential function for negative integer powers

So, we can indeed define the exponential for negative integers accordingly.

The exponential function for negative integer powers

What about rational exponents?

The exponential function for rational powers

You guessed right. Let's use wishful thinking again! The "power of powers" rule yields that it is enough to look at exponents where the numerator is 11.

The exponential function for rational powers

The same rule gives that should they exist, they must be defined by the qq-th root of the base.

The exponential function for rational powers

Following our usual modus operandi, we use this to define the exponential for rational exponents.

The exponential function for rational powers

Now the big finale: what about arbitrary real numbers as exponents?

The exponential function for real powers

Buckle up. We are about to floor the gas pedal. So far, we have defined the exponential function for all rationals. Let's use this to extend it to all real numbers!

Real numbers are weird. Fortunately, they have an exceptionally pleasant property: they can be approximated by rational numbers with arbitrary precision. That is, for all xR x \in \mathbb{R} , there exist a sequence of rationals pn/qnn=1 {p_n/q_n}_{n=1}^{\infty} such that limnpn/qn=x \lim_{n \to \infty} p_n/q_n = x .

For instance, this is how π is approximated by rational numbers. Every time you use π inside a computer, an approximation is used. (Most probably not this one.)

Approximating pi

When the approximating sequence is close to the actual exponent x x , the powers are also close. Closer and closer as n n grows.

The exponential function for real powers

Thus, we can define exponentials for arbitrary real exponents by simply taking the limit of the approximations.

The exponential function for real powers

You have probably seen the exponential functions thousands of times. We use them every day, yet we hardly ever look behind the surface.

Having a deep understanding of math will make you a better engineer.

I want to help you with this, so I am writing a comprehensive book that takes you from high school math to the advanced stuff.
Join me on this journey and let's do this together!