1.0

Convert Percent to Decimal Take the number and divide it by 100. A simple way to do this is by moving the decimal point two places to the left. For example, in the expression 75%, move the decimal two points to the left to get 0.75

1.1

Exponents Exponents tell us how many time to times a number by itself. For example, in the expression 8², 8 is the base and 2 is the exponent, which means 8 X 8. It is a shorthand for repeated multiplication and is also known as raising a number to a power. It can also be expressed as 8^2.

1.2

Finding the Square Root of any number. This tutorial builds on the Finding Square Root Using Long Division tutorial. Start at the first digit that is closest to the decimal point and group pairs of digits.

$$\sqrt{2025}$$
$$\sqrt{2025}$$
Now we move to the first group of numbers which is 20 and compute a diviser that will be the root that is closest to the number 20 without going over. We know that 4x4=16 and 5x5=25. So since 5x5 is greater than 20 we use 4.
4 x 4 = 16. Subtract for the remainder and pull down the next pair of numbers.

$$\begin{array}{c}4\\ 4\sqrt{2025}\\ 16\\ \mathrm{\_\_\_\_\_\_\_\_\_}\\ 425\end{array}$$
Next we have to create a new diviser. We need to take the diviser and multiply it by two. The diviser is the number in front of the square root. That will be 4 x 2 = 8. We take this number 8 and ask "What number times 8 equals 425?" We do this in the same way as we did it the first time and get the answer that equals the number closest to 425 but not over. This require us to estimate the answer somewhat. If we take 80 x 1 = 80, and then take 80 x 10 = 800 we can assume that 80 x 5 would get us pretty close to the right answer. If we then substitute the 0 for 5 and try 85 x 5 we get 425.

$$\begin{array}{c}45\\ 4\sqrt{2025}\\ 16\\ \mathrm{\_\_\_\_\_\_\_\_\_}\\ 425\\ 85x5=425\end{array}$$
We then double check by testing if 45 x 45 equals 2025.
Thus the square root of 2025 is 45.

Estimation and Approximation (Babylonian Method) For a manual method that provides increasingly accurate rational approximations, you can use the Babylonian method (also known as Heron's method or the estimation method).$$\sqrt{2}$$

The Definition of the Logarithm states that the logarithmic function is the inverse of the exponential function. For the exponential equation b^y=x, the equivalent logarithmic form is log_b(x)=y. This means that the logarithm log_b asks, "to what power y must the base b be raised to get the number x?."

Exponential form: $b^y=x$

Logarithmic form: ${\log }_b\left(x\right)=y$

Product Rule states that the logarithm of a product is the sum of the logarithms of its factors. This rule allows you to expand a logarithm of a product into a sum or to combine a sum of logarithms into a single logarithm of a product. For example, log(5*10) can be rewritten as log(5)+log(10).

$${\log }_b\left(\mathrm{xy}\right)={\log }_b\left(x\right)+{\log }_b\left(y\right)$$

Quotient Rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This rule is based on the quotient rule for exponents, which states that when dividing powers with the same base, you subtract the exponents x^a/x^b=x^a-b. The quotient rule for logarithms mirrors this by converting division inside the logarithm to subtraction outside of it.

$${\log }_b(x/y)={\log }_b\left(x\right)-{\log }_b\left(y\right)$$

Power Rule states that the logarithm of a number raised to a power is equal to that power multiplied by the logarithm of the number. This rule is useful for simplifying logarithmic expressions and solving equations by allowing you to move the exponent to the front of the logarithm.

$${\log }_b\left(x^n\right)=n\left({\log }_b\right(x\left)\right)$$

Change of Base Rule The change of base formula for logarithms is used to rewrite a logarithm in any base as a ratio of logarithms in a new, chosen base.

$${\log }_{a}b=\frac{{\log }_c\left(b\right)}{{\log }_c\left(a\right)}$$

Log of One Rule The logarithm of one is always zero, regardless of the base, because any number raised to the power of zero equals one b⁰=1. This is a fundamental property of logarithms.

$${\log }_b\left(1\right)=0$$

Log of Base Rule Log of base to itself will equal one.

$${\log }_b\left(b\right)=1$$

Inverse Properties Rule Exponent and Log cancel each other out.

$${\log }_b\left(b^x\right)=x$$
$$b^{{\log }_{x}n}=x$$

1.0

Logarithms The log() function, or logarithm tells us what the exponent is. 2^x=8 becomes log₂8=x which is log₂8=3 It answers the question: "To what power must a given base be raised to obtain a specific number?"

Consider the equation 10³=1000. In this case, the base b is 10, the exponent y is 3, and the result x is 1000. The logarithmic form of this equation is log₁₀(1000)=3. This means "the power to which 10 must be raised to get 1000 is 3."

Key Elements:
Base: The number that is being raised to a power. Common bases include 10 common logarithm, often written as log(x) without a subscript and e natural logarithm, written as ln(x).
Argument: The number for which the logarithm is being calculated. This number must always be positive.
Result: The exponent to which the base must be raised to equal the argument.

Example: log₂(8).
This asks: "To what power must 2 be raised to get 8?"
We know that 2^1 = 2, We know that 2^2 = 4, and We know that 2^3 = 8.
Therefore log₂(8)=3.

1.0

Summation Notation or Sigma is used in mathematics to represent a summation. This notation is a shorthand for adding up a sequence of numbers or expressions that follow a certain pattern.

Example: Consider the equation $$\sum _{i=1}^4{i}^2$$
$$\sum _{i=1}^4{i}^2=1^2+2^2+3^2+4^2$$
$$\sum _{i=1}^4{i}^2=1+4+9+16$$
$$\sum _{i=1}^4{i}^2=30$$
In this case, the Upper Limit is 4, the Lower Limit is i=1 so we start at 1. The expression is i to the second power so this equation says "Starting at 1 iterate through i to the second power 4 times and add them all together."

Key Elements:
The Summation Symbol (∑): A stylized Greek capital letter "sigma" that means "sum".
The Index of Summation (i): variable, often i, j, or k, that acts as a counter. It represents the position of each term in the series.
The Lower Limit (i=1): The starting value for the index of summation. The summation begins by evaluating the expression for this value.
The Upper Limit (4): The ending value for the index of summation. The summation stops once the expression has been evaluated and added for this value.
The Expression (i=1): The formula that generates the terms to be added. The index (i) is substituted into this formula for each step from the lower to the upper limit. 

1.0

Entropy calculates the degree of randomness or disorder within a system. In the context of information theory, the entropy of a random variable is the average uncertainty, surprise, or information inherent to the possible outcomes. To put things simply, it measures the uncertainty of an event.

$$E=-\sum P\left(X\right)*\log P\left(X\right)$$

The summation represents the combining of the different calculations. If we calculated the 50% chance, the first calculation would be the same as the second calculation so we could just add the first calculation to itself. If we are calculation 25% and 75% though we get both calculations and at the end add them together hence summation.

To calculate the entropy of a specific event X with probability P(X) you calculate this:

$$-P\left(X\right)*{\log }_2\left(P\right(X\left)\right)$$

As an example, let's calculate the entropy of a fair coin.
The probability of heads is 50%. Here's the entropy we get when plugging that 0.5 into the equation:

$$E\mathrm{heads}=-0.5*{\log }_2\left(0.5\right)$$
$$E\mathrm{heads}=-0.5*-1$$
$$E\mathrm{heads}=0.5$$$$E\mathrm{tails}=0.5$$
$$E=0.5+0.5$$
$$E=1$$

Now lets plug in .25 and .75:

$$E\mathrm{twenty-five}=-0.25*{\log }_2\left(0.25\right)$$
$$E\mathrm{twenty-five}=-0.25*-2$$
$$E\mathrm{twenty-five}=0.5$$
$$E\mathrm{seventy-five}=-0.75*{\log }_2\left(0.75\right)$$
$$E\mathrm{seventy-five}=-0.75*-0.415$$
$$E\mathrm{seventy-five}=0.311$$
$$E=0.5+0.311$$
$$E=0.811$$

1.1

Cross Entropy is a measure of the difference between two probability distributions for a given random variable or set of events.

Information quantifies the number of bits required to encode and transmit an event. Lower probability events have more information, higher probability events have less information.

Lower Probability events have more information. They are more surprizing.
Higher Probability events have less information. They are less surprizing.

$$\mathrm{hx}=-\log \left(\mathrm{Px}\right)$$
Entropy in the context of AI it measures the uncertainty of an AI model's predictions or the disorder in a dataset. It is the number of bits required to transmit a randomly selected event from a probability distribution. A skewed distribution has a low entropy, whereas a distribution where events have equal probability has a larger entropy.

A skewed probability distribution has less “surprise” and in turn a low entropy because likely events dominate. Balanced distribution are more surprising and turn have higher entropy because events are equally likely.
Skewed Distribution has low entropy. It is unsurprizing.
Uniform Distribution has higher entropy. It is surprizing.

Mean is the average of a data set, calculated by adding all the numbers together and then dividing the sum by the count of numbers in the set

Latent Space is a compressed, abstract representation of data that captures its essential features and underlying patterns in a lower-dimensional space. In an array of vectors that represent the small starting of a smile to a full on smile it represents the mean of the smile.

Interpolate is the process of generating new data points or content that fit smoothly within a set of existing, known data points.

Scalar A single value with magnitude but no direction.

Vector A quantity with both magnitude and direction, represented as a 1D array.

Matrix A 2D grid of values, which can be thought of as a 2nd-rank tensor.

Tensor A generalization that can have more than two dimensions, used to describe complex data like images or video.