Introduction
The new vector ratio data type is a new addition to FL64.
This tool helps you solve complex numbers that sequentially grow and never repeat, such as 0.123456, which we know will only get larger as the next digit is added by one, extending to a new digit, causing the chance of a repeat never to happen and for the value to be impossible to be expressed as a finite value or by a single divide between two numbers (fraction).
A division produces a fixed number value, and a recurring pattern occurs when we end up with the same remainder while dividing, which repeats; for example, 11 ÷ 8 = 0.727272. The following page provides a detailed explanation of why this happens:
Periodic number patterns. To express values that grow per place value, we do the following: 123 ÷ 1000 = 0.123, then say it continues as follows: 1234 ÷ 10000 = 0.1234. The two numbers we divide by expand on both sides as we move further down the decimal place, adding the growing changing sequence that does not repeat or recur. A single divide is just a fixed value that terminates or has recurring digits.
Some values have no visible pattern in their digits, but they grow sequentially the same way, progressing with a slight expanding change that can not be found by looking at the numbers down the decimal point, such as the ratio 3.1415926535, which we call PI.
We refer to these complex number pattern expansions as irrational numbers. Unravelling complex values takes a lot of work when the expanding ratio down the decimal point is invisible.
Originally, we would look for equations that match the number ratio we are looking at through a lot of calculation and proof work, such as comparing a circle with triangles to find the expansion ratio for the outside distance of a circle called PI
Using this tool
Enter a custom vector combination, such as 1,8,99, then hit calc to see the steps for computing each part. It then shows how the parts are added to an irrational number.
The resulting irrational number is broken back into parts step by step.
It then shows how the parts are added back into a vector ratio, and proves the algorithm operations and steps.
All operations are simple: fundamental divides, multiplies, additions, and subtractions.
You can also use an irrational number, such as 1.414213562, instead of entering a vector combination.
This tool will guide you through breaking the number into its parts.
Once the value is broken down into parts, it will show you how to add the parts to a vector ratio per column and solve your numbers by expanding the ratio sequentially.
Vector ratios
A vector that looks like this: 1,0,77 is the square root of 77 as a vector ratio.
An vector of 1,0,0,23 is the cube root of 23.
A vector such as 1,0,0,0,2 is 2 to the root of 4, which is the value 1.18920711500272. Multiplying the value 1.18920711500272 four times by itself is 2.
Not all irrational number ratios can be computed as the root of a number.
A vector ratio of 1,8,99 is close to the square root of 99, but can not be calculated as the root of any number.
The last column of a vector is to the root of a number as long as the first column is one and the rest are zero, with the last value being the value of the root.
The vector computes each of the whole parts of a number ratio sequentially.
Important note
A vector of 1,1,1 is the Fibonacci constant (golden ratio).
A Vector of 1,1,1,1 is the Tribonacci constant.
A vector of 1,1,1,1,1 is raised to the fourth term and is the Quatronacci constant.
This can go as high as you want in higher and higher dimensions.