Inputs
01111
+ 00101
Base 10
15
+ 5
20
Base 2 (Binary)
01111
+ 00101
10100
Mathematics and Equations
Number Systems
People choose to count using a base 10 number system. But we could actually be using any other base system. Base 8, base 12, base 15. We most likely use base 10 because we have 10 fingers so its easy to count on our hands
But our number system could have easily looked like 0 1 2 3 4 5 6 7 8 9 ❀ ⁍
Here you can see what other base systems would have looked like:
0123456789
Base 10 Quantity Base 10
1 ● 1
5 ●●●●● 5
10 ●●●●●●●●●● 10
16 ●●●●●●●●●●●●●●●● 16
25 ●●●●●●●●●●●●●●●●●●●●●●●●● 25
42 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 42
100 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 100
255 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 255
Decimal (Base 10) — Our everyday number system, likely because we have 10 fingers.
Around 700 CE the arabic world invented arabic numerals. Around 1100 the European world adopted these and by 1500 the number system was standardized.
Read More About Number Systems Here: LibreTexts MathBinary
Around 1703 Leibniz showed that all nubmers could be represented in the binary system (only using 0 and 1). A base 2 system.
Then around 1850 boole showed that all calculations could be reduced to using the binary system.
Later in the 1900s binary math was expanded and additional mathematical operators were found that were only possible with a binary system. These bitwise operators (AND, OR, XOR) became fundamental to computer logic gates.
Binary Addition — Works like decimal but carries at 2 instead of 10. 1 + 1 = 10 (carry the 1)
Instead of 4×3 we had operations like 100 + 11 leading to 111. This drastically simplified how math could be represented as there were only 2 states (on or off).
Equations
Basic Equations
People started to see that they could model the world around them using mathematics.
They could use math to predict how the world around them would behave.
Basic algebra can be used to determine the total amount of food needed for a voyage given different amount of days. Or the total cost of a loan given a certain rate of interest over a period of time. Or physical movements like the speed at which a trebuchet will launch an object.
These modelings of the world showed that math could predict how the world around us acts.
Parameters
Parabolic Motion Equations
x(t) = v₀ · cos(θ) · t y(t) = v₀ · sin(θ) · t - ½gt²
Range: 206.6m Max Height: 51.7m Flight Time: 6.49s
How it works: Ancient mathematicians discovered that thrown objects follow a predictable parabolic path. By knowing the initial velocity and angle, we can calculate exactly where it will land—enabling accurate catapult aim, ballistic calculations, and modern physics.
To Learn more about equations and algebra there are free online textbooks- LibreTexts Algebra Bookshelve
Building Equations Based on Data
People eventually learned that they did not always need the exact physical equation behind something. By measuring how something behaved and plotting those measurements, we could construct equations directly from data.
The least squares method (Legendre 1805, Gauss 1809) gave a formal way to fit lines to a set of points with minimal error. This let us take recorded data and produce a function that matched it well enough to predict future values or reconstruct past behavior.
As more phenomena were studied, it became clear that many patterns were not just linear. Curves such as exponentials, polynomials, logarithmic functions, and logistic curves were used because they matched different types of real-world data better than a straight line.
Mapping Planetary Orbits
r = a(1 - e²) / (1 + e·cos(θ))
How it works: Astronomers like Kepler discovered that planets follow elliptical paths. By recording positions over time, we can fit the data to an ellipse equation and predict future positions.
By 1847, Cauchy introduced what would become gradient descent—an iterative method for improving a curve’s fit when a direct solution was too difficult to compute. Instead of solving everything at once, the parameters are adjusted step-by-step to reduce error.
This shift meant equations were no longer only derived from known laws. They could be built and refined directly from measured data.
Gradient Descent: Finding the Lowest Point
x_{new} = x - α · ∇f(x)
Steps: 0 Position: 50.0 Loss: 0.223
How it works: The rover descends by following the gradient (slope). Momentum helps
it build speed and roll past shallow valleys. Exploration adds random jumps to escape local minima.
Try high momentum + exploration to find the global minimum (deepest valley on the right)!
