Tivadar Danka
@tivadardanka.bsky.social
I make math accessible for everyone. Mathematician with an INTJ personality. Chaotic good. Writing https://thepalindrome.org
Thus, we can define exponentials for arbitrary real exponents by simply taking the limit of the approximations.
And we are done!
And we are done!
November 13, 2025 at 1:00 PM
Thus, we can define exponentials for arbitrary real exponents by simply taking the limit of the approximations.
And we are done!
And we are done!
When the approximating sequence is close to the actual exponent x, the powers are also close.
Closer and closer as n grows.
Closer and closer as n grows.
November 13, 2025 at 1:00 PM
When the approximating sequence is close to the actual exponent x, the powers are also close.
Closer and closer as n grows.
Closer and closer as n grows.
Real numbers are weird.
Fortunately, they have an exceptionally pleasant property: they can be approximated by rational numbers with arbitrary precision.
This is because no matter how close we get to a real number, we can find a rational number there.
Fortunately, they have an exceptionally pleasant property: they can be approximated by rational numbers with arbitrary precision.
This is because no matter how close we get to a real number, we can find a rational number there.
November 13, 2025 at 1:00 PM
Real numbers are weird.
Fortunately, they have an exceptionally pleasant property: they can be approximated by rational numbers with arbitrary precision.
This is because no matter how close we get to a real number, we can find a rational number there.
Fortunately, they have an exceptionally pleasant property: they can be approximated by rational numbers with arbitrary precision.
This is because no matter how close we get to a real number, we can find a rational number there.
Thus, we finally see how to make sense of rational exponents.
November 13, 2025 at 1:00 PM
Thus, we finally see how to make sense of rational exponents.
The same rule gives that rational exponents with numerator 1 must be defined in terms of roots.
November 13, 2025 at 1:00 PM
The same rule gives that rational exponents with numerator 1 must be defined in terms of roots.
What about rational exponents?
You guessed right. Wishful thinking!
The "power of powers" rule yields that it is enough to look at exponents where the numerator is 1.
You guessed right. Wishful thinking!
The "power of powers" rule yields that it is enough to look at exponents where the numerator is 1.
November 13, 2025 at 1:00 PM
What about rational exponents?
You guessed right. Wishful thinking!
The "power of powers" rule yields that it is enough to look at exponents where the numerator is 1.
You guessed right. Wishful thinking!
The "power of powers" rule yields that it is enough to look at exponents where the numerator is 1.
What about negative integers?
We cannot repeat multiplication zero times, let alone negative times. Again, let's use wishful thinking.
If powers with negative integer exponents are indeed defined, the "product of powers" tells us what they must be.
We cannot repeat multiplication zero times, let alone negative times. Again, let's use wishful thinking.
If powers with negative integer exponents are indeed defined, the "product of powers" tells us what they must be.
November 13, 2025 at 1:00 PM
What about negative integers?
We cannot repeat multiplication zero times, let alone negative times. Again, let's use wishful thinking.
If powers with negative integer exponents are indeed defined, the "product of powers" tells us what they must be.
We cannot repeat multiplication zero times, let alone negative times. Again, let's use wishful thinking.
If powers with negative integer exponents are indeed defined, the "product of powers" tells us what they must be.
In this case, the "product of powers" property gives the answer: any number raised to the power of zero should equal to 1.
November 13, 2025 at 1:00 PM
In this case, the "product of powers" property gives the answer: any number raised to the power of zero should equal to 1.
These two identities form the essence of the exponential function.
November 13, 2025 at 1:00 PM
These two identities form the essence of the exponential function.
Second, the repeated application of exponentiation is, again, exponentiation.
We'll call this the "power of powers" rule.
We'll call this the "power of powers" rule.
November 13, 2025 at 1:00 PM
Second, the repeated application of exponentiation is, again, exponentiation.
We'll call this the "power of powers" rule.
We'll call this the "power of powers" rule.
But how can we define exponentials for, say, negative integer exponents? We'll get there soon.
For that, two special rules will be our guiding lights. First, exponentiation turns addition into multiplication.
We'll call this the "product of powers" rule.
For that, two special rules will be our guiding lights. First, exponentiation turns addition into multiplication.
We'll call this the "product of powers" rule.
November 13, 2025 at 1:00 PM
But how can we define exponentials for, say, negative integer exponents? We'll get there soon.
For that, two special rules will be our guiding lights. First, exponentiation turns addition into multiplication.
We'll call this the "product of powers" rule.
For that, two special rules will be our guiding lights. First, exponentiation turns addition into multiplication.
We'll call this the "product of powers" rule.
The number a is called the base, and b is called the exponent.
Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.
Sounds simple enough.
Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.
Sounds simple enough.
November 13, 2025 at 1:00 PM
The number a is called the base, and b is called the exponent.
Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.
Sounds simple enough.
Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.
Sounds simple enough.
First things first: terminologies.
The expression aᵇ is read "a raised to the power of b."
(Or a to the b in short.)
The expression aᵇ is read "a raised to the power of b."
(Or a to the b in short.)
November 13, 2025 at 1:00 PM
First things first: terminologies.
The expression aᵇ is read "a raised to the power of b."
(Or a to the b in short.)
The expression aᵇ is read "a raised to the power of b."
(Or a to the b in short.)
The way you think about the exponential function is wrong.
Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?
Here is what's really behind the most important function of all time:
Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?
Here is what's really behind the most important function of all time:
November 13, 2025 at 1:00 PM
The way you think about the exponential function is wrong.
Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?
Here is what's really behind the most important function of all time:
Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?
Here is what's really behind the most important function of all time:
The single most undervalued fact of mathematics: mathematical expressions are graphs, and graphs are matrices.
Yes, I know. You already heard this from me, but hear me out.
Viewing neural networks as graphs is the idea that led to their success.
Yes, I know. You already heard this from me, but hear me out.
Viewing neural networks as graphs is the idea that led to their success.
November 12, 2025 at 1:00 PM
The single most undervalued fact of mathematics: mathematical expressions are graphs, and graphs are matrices.
Yes, I know. You already heard this from me, but hear me out.
Viewing neural networks as graphs is the idea that led to their success.
Yes, I know. You already heard this from me, but hear me out.
Viewing neural networks as graphs is the idea that led to their success.
The same logic can be applied, thus giving an explicit formula to calculate the elements of a matrix product.
November 11, 2025 at 1:00 PM
The same logic can be applied, thus giving an explicit formula to calculate the elements of a matrix product.
We can collapse the linear combination into a single vector, resulting in a formula for the first column of AB.
This is straight from the mysterious matrix product formula.
This is straight from the mysterious matrix product formula.
November 11, 2025 at 1:00 PM
We can collapse the linear combination into a single vector, resulting in a formula for the first column of AB.
This is straight from the mysterious matrix product formula.
This is straight from the mysterious matrix product formula.
Recall that matrix-vector products are linear combinations of column vectors.
With this in mind, we see that the first column of AB is the linear combination of A's columns. (With coefficients from the first column of B.)
With this in mind, we see that the first column of AB is the linear combination of A's columns. (With coefficients from the first column of B.)
November 11, 2025 at 1:00 PM
Recall that matrix-vector products are linear combinations of column vectors.
With this in mind, we see that the first column of AB is the linear combination of A's columns. (With coefficients from the first column of B.)
With this in mind, we see that the first column of AB is the linear combination of A's columns. (With coefficients from the first column of B.)
Now, about the matrix product formula.
From a geometric perspective, the product AB is the same as first applying B, then A to our underlying space.
From a geometric perspective, the product AB is the same as first applying B, then A to our underlying space.
November 11, 2025 at 1:00 PM
Now, about the matrix product formula.
From a geometric perspective, the product AB is the same as first applying B, then A to our underlying space.
From a geometric perspective, the product AB is the same as first applying B, then A to our underlying space.
(If unwrapping the matrix-vector product seems too complex, I got you.
The computation below is the same as in the above tweet, only in vectorized form.)
The computation below is the same as in the above tweet, only in vectorized form.)
November 11, 2025 at 1:00 PM
(If unwrapping the matrix-vector product seems too complex, I got you.
The computation below is the same as in the above tweet, only in vectorized form.)
The computation below is the same as in the above tweet, only in vectorized form.)
Moreover, we can look at a matrix-vector product as a linear combination of the column vectors.
Make a mental note of this, because it is important.
Make a mental note of this, because it is important.
November 11, 2025 at 1:00 PM
Moreover, we can look at a matrix-vector product as a linear combination of the column vectors.
Make a mental note of this, because it is important.
Make a mental note of this, because it is important.
Matrices represent linear transformations. You know, those that stretch, skew, rotate, flip, or otherwise linearly distort the space.
The images of basis vectors form the columns of the matrix.
We can visualize this in two dimensions.
The images of basis vectors form the columns of the matrix.
We can visualize this in two dimensions.
November 11, 2025 at 1:00 PM
Matrices represent linear transformations. You know, those that stretch, skew, rotate, flip, or otherwise linearly distort the space.
The images of basis vectors form the columns of the matrix.
We can visualize this in two dimensions.
The images of basis vectors form the columns of the matrix.
We can visualize this in two dimensions.
By the same logic, we conclude that A times eₖ equals the k-th column of A.
This sounds a bit algebra-y, so let's see this idea in geometric terms.
Yes, you heard right: geometric terms.
This sounds a bit algebra-y, so let's see this idea in geometric terms.
Yes, you heard right: geometric terms.
November 11, 2025 at 1:00 PM
By the same logic, we conclude that A times eₖ equals the k-th column of A.
This sounds a bit algebra-y, so let's see this idea in geometric terms.
Yes, you heard right: geometric terms.
This sounds a bit algebra-y, so let's see this idea in geometric terms.
Yes, you heard right: geometric terms.
Similarly, multiplying A with a (column) vector whose second component is 1 and the rest is 0 yields the second column of A.
That's a pattern!
That's a pattern!
November 11, 2025 at 1:00 PM
Similarly, multiplying A with a (column) vector whose second component is 1 and the rest is 0 yields the second column of A.
That's a pattern!
That's a pattern!
Now, let's look at a special case: multiplying the matrix A with a (column) vector whose first component is 1, and the rest is 0.
Let's name this special vector e₁.
Turns out that the product of A and e₁ is the first column of A.
Let's name this special vector e₁.
Turns out that the product of A and e₁ is the first column of A.
November 11, 2025 at 1:00 PM
Now, let's look at a special case: multiplying the matrix A with a (column) vector whose first component is 1, and the rest is 0.
Let's name this special vector e₁.
Turns out that the product of A and e₁ is the first column of A.
Let's name this special vector e₁.
Turns out that the product of A and e₁ is the first column of A.