Introduction

The new vector ratio data type is a new addition to FL64.

This tool helps you solve complex numbers that sequentially grow and change forever, 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.

A value like 0.12345 can be expressed as a set of fractions that approach the value: 1÷10=0.1, 12÷100=0.12, 123÷1000=0.123. In this way, we can express both sides of the fraction as a growing, changing pattern. We say this ratio is raised to the 10th (or 10D ratio) since 1×10=10, 10×10=100, and 100×10=1000; the sequence increases by 10 each time we add one to the pattern and move by 10 for the base sequence.

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, 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 numbers digits 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 smaller and smaller lines 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 a 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.

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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 and are 2D ratios.

An vector of 1,0,0,23 is the cube root of 23 and are 3D ratios.

A vector such as 1,0,0,0,2 is 2 to the root of 4 (4D ratio), 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 and cannot be calculated as a singular square root.

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). Is the center of all 2D ratios.

A vector of 1,1,1,1 is the Tribonacci constant. Is the center of all 3D ratios.

A vector of 1,1,1,1,1 is raised to the fourth term and is the Quatronacci constant. Is the center of all 4D ratios.

This can go as high as you want in higher and higher dimensions.