apologies if this is a repeat but i’m new and don’t understand how to search a board. Ive gotten really invested in math for my second chance at college and this is really piqued my interest, but I haven’t found much on it. Like I under Archimedes found pi and it’s first few decimals but I don’t get how or why
In: 14
We tend to forget that for ancient mathematicians, math was shapes not numbers. Something squared wasn’t an exponent, but rather an actual square of sides equal to the original value. Simply by drawing shapes and using string, it would have been easy to notice that every circle shares the same relationship between its diameter and its circumference
Edit: typos
[removed]
Pi is a circle’s circumference divided by the diameter. For any size circle, c/d = pi. The early guys doing geometry and measurements of any round objects (wheels, pillars, etc) noticed this.
You can “discover” the first few decimals of pi with a string and a ruler. Measure a few round objects’ diameters with a ruler, and circumference with a string (that you then hold up to the ruler). Divide c by d. The bigger the round objects the better (measurement error is relatively smaller), and the more different round objects you measure and average them together the more accurate your pi gets.
Wait till you get to the shit where mathematically, the force between charged particles like electrons has the same math formula as the force of gravity between 2 planets.
Or how the equations for water flowing down a pipe align with the formulas for electricity propagating down a wire.
As people already said Pi is what you get, when you divide circumference by diameter on a circle.
pi = circumference / diameter
pi * diameter = circumference
2 * pi * radius = circumference
As people already said as well, you can get pi by measuring the circumference and the diameter of a circle.
You couldn’ be sure about the exact digits this way though. You’d know it’s reasonably accurate for the size of circles you have measured it with, but, maybe you wouldn’t want to base big bridge construction or spaceflight on it.
One way of getting accurate digits to a certain depth is by approximating a circle with angular shapes. It is easy to calculate the circumference and the “width” of squares and triangles.
You can form an approximation for a circle out of squares (like in Minecraft buildings) or out of triangles (like in uh… Tomb Raider). The more squares and triangles you use, the more the ratio of circumference to width will be similar to pi.
If you build a circle out of 1000 squares slightly bigger than a circle and one that is slightly smaller than a circle you will get two different numbers, but they will have the same digits up to a certain point. Those digits will be the same that pi has at those places.
—-
Thinking about it again, I think it doesn’t work easily with squares, because a plus-sign or a jagged circle has the same edge-length as a square. So you’d *have* to use triangles (like pizza pieces). You could still use squares, if you calculated the *area* of a circle and then get pi by calculating “area / square-root of radius”.
We did both those things at school (circle out of triangle circumference and circle out of squares/rectangles area).
This gets very deep, very quickly. Can you even say that maths was discovered? Or was it invented?
Perfect squares or circles do not exist in the real world, maths is just a “model world” where you can ignore imperfections to make computations easier.
One day someone wondered: “what if we could make a shape such that all points are at the exact same distance from the center?”
And that allowed to start making computations with circles.
In the end a mathematician plays around with the things he knows, and sometimes “discovers” something interesting ( e.g. that in all circles, the ratio area/circumference is the same).
Then those things get a name (constants, theorems…) so that they can be easily used by other people.
I will add to the already awesome answers, and that is how things become well known or agreed upon scientifically.
Let’s say 50 years ago you have two guys on different sides of the Earth studying gravity. They never worked together but basically studied the same things.
And then, they come to a conclusion about something. Maybe *this constant value that keeps on popping up* whenever they model two masses having gravity, and neither of them have a name for it but the models don’t work exactly if I don’t apply it.
So, they give it a name for themselves and keep testing it.
Then, that work get’s published with technology and writing progress, and people across the Earth start saying, “Yes, I had the same results”, basically corroborating the findings. “Was your number ‘X’? “Yes, it was!” “Mine was off by two!?” “Oh, that’s because you forgot to by divide by two, right here…” “Oh, of course!”
Basically, confirming for each other, and the most important part is: others can test their theories themselves, confirming it for themselves using their papers and number,
Vaunted Greek big brain math boy Archimedes decided he wanted to approximate the area of a circle. And he worked from first principles by first drawing a hexagon such that corners of the square touched the edge of the circle and solving for area.
Then he did a 12 sided polygon.
Then a 24 sided polygon.
Then he repeated, doubling the number of sides to solving the area of a 96 sided polygon.
His work put pi as between 3 and 1/7 as the upper bound (3.1428) and 3 and 10/71 (3.1408)
As better formulas/methods were developed, that number became even more precise.
The most accurate for about a thousand years came from Zu Chongshi, a Chinese mathematician who used the algorithm developed by Liu Hui (which was built from the same first principles as Archimedes) with a 12288 sided polygon, and got Pi down to between 3.1415926 and 3.1415927
It’s also worth noting that these are just the people who sat down and tried to formalize the ratio that we know the method of. Ancient Babylonians, who basically defined math as it relates to circles (to the point that they are the reason circles have 360 degrees, why there are 60 seconds in a minute and 60 minutes in an hour) were aware that the area of a circle was slightly more than 3 times the square of it’s radius, and so were ancient Egyptians who used circles as measuring sticks and for a lot of their architecture, and similarly got the ratio to good enough for the practical purposes they needed it for
Latest Answers