# The Theory of Effort

The Theory of Effort forms a part of our new concept of 4D maths. Using this powerful technology, we are able to track the number of numerical iterations, and quantify calculations that would otherwise appear to produce the same results.

## Overview

The Theory of Effort is a key concept of 4D maths. It appears in the 4D calculator as the length of a number line, tracking the result of specific settings as the value dissolves towards its limit. Using the effort value we can qualify calculations in a completely revolutionary way, that has deep implications for mathematics, science, and computer technology.

## KEy Points

• The Theory of effort allows us to quantiy reiterative calculations
• This allows us to explore the temporal nature of calculations
• It finds application in the realm of computer techology as it can track changes in the system

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# Concept

## Counting Calculations

The Theory of Effort is found in the construction of the 4D calculator. Whenever as value is entered into the time, space and dimension values, the 4D calculator reiterated an equation. As it does, the results dissolve towards a specific value. Each iteration is counted as a single effort value. By noting the effort value, we can identify a specific value for any calculator setting. This is important as more often than not all space values will dissolve towards the same number. Only by tracking the effort values are we able to differentiate one number from another.

This is quite different from standard mathematical conventions, that tend to reiterate values to infinity, without quantifying the process at each step. Additionally, we can quantify calculations that repeatedly exhibit the same answer, such as 1×1 = 1 (effort 1), or 1x1x1= 1 (effort 2), which can be used to explore certain calculations in greater detail.

## Quantifying an equation

The Theory of Effort is a very simple idea, yet is also very powerful when it comes to qualifying a mathematical equation. Let’s take a simple example:

#### -1 x -1 x -1 = -1

The two equations above produce two kinds of result, one positive and one negative. In the first case -1 x -1 = 1 as two negatives make a positive. As the first number is multiplied by a second one, we can say it goes through a single effort value. In the second example the result is -1. As the original number goes through two calculations its has an effort value of 2. We could extend the calculation following the same pattern.

##### -1 x -1 x -1 x -1 x-1 = -1

We notice there is a very simple pattern. If the effort value is odd then the result will be positive, and if the effort value is even, the the result will be negative. This will be true regardless of how many times we multiply by -1. Notice that we do not count the first number in the effort value, we only count the number of times a multiplicative action is performed.

## Where the fraction ends

Next we can consider a slightly more advance use of the Theory of Effort. Let us consider the equation that generates the number e. The number e is found by adding together the reciprocal values of all factorial numbers plus 1.

###### Here we can see that the number e can only be represented as a fraction up to effort 7. After this it can only be represented as a decimal fraction.

We can see from the image above that there is a limit to the number e in terms of its representation as a fraction comprised of two numbers. After effort 7 all subsequent values can only be expressed as a decimal fraction. This notion is strangely absent from traditional mathematical thinking. The importance of being able to express a number in terms of a rational fraction is normally not recognised. However, as the universe operates in a quantised manner, being able to qualify numerical constants to a certain degree does play a significant role. You might notice that effort 0 to 3 are musical ratios. The atomic structure is limited to 7 S-orbitals, all of which are important clues in recognising the nature of reality, and the periodic table.

The final fraction 685/252 is expressed as decimal 2.71825396825396825396, which exhibits a recursive series (highlighted in bold), that excludes the numbers 1, 4 and 7. As a side note 825/369 is a very close approximation to √5.

As numbers like e are so important in mathematics, it is quite incredible that no one has taken the time to evaluate them in terms of effort values, and make any attempt to qualify them over time. It is worth noting that recursive functions play an important role in a vast amount of mathematical solutions, including the field of fractal geometry.

## Effort Values and Computer science

The Theory of Effort also plays a role in the realm of computer science. Computers operate on the basis of binary code. Binary can only be expressed by the numbers 0 and 1. This is because computer hardware is made of small transistor switches that can either allow an electrical signal to pass or to accept a lower voltage which prevents the electrical flow. In simple terms, this is akin to a switch that can either be turned off or on. As a computer operates over time, a binary switch changes position, which processes information, allowing programs to be executed.

Let’s take the example of a single switch. Starting in the off position, the switch can be changed into the on position, which is an effort value of 1. When the switch changes back to a 0 this will be effort 2. When switched again into the on position, this will be effort 3. What we notice is that any ON position will be an odd effort number, whereas any OFF position will be an even effort value.

Notice that this is very similar to our previous example of -1 x -1, where all odd effort values equal a positive and all even effort values equal a negative number. This example is the simplest possible using a single binary switch. However, when we increase the number of switches to 2 we find that the complexity of each state increases exponentially.

We can demonstrate this with two switches we can call left and right, both in the off position. Now we have two options. We can switch either the left or the right switch. This means that at effort 1, the switches will be opposite to each other. Either the left is on and the right is off, or the right is on, and the left is off. Next, we can change the state of the switches a second time. If the left switch was changed for the first effort value, then switching the left on again will return both switches to their original position. On the other hand, if the right switch is changed, then both switches will be in the on positions. It takes effort 2 to turn on two switches. The same can be said of the right switch. This means that whilst there are only two possible states, either both on or both off, the pathway by which the state is reached can be different. Either the left or the right will be first. This means that there are four possible states once we consider the pathway at effort 2.

The point here is that only if we track the changes over time can we know which pathway is the correct one to arrive at that particular state. If we choose to change the position of the switch a third time, (effort 3), we can be guaranteed that one switch will be off, whilst the other switch will be on. There is no other option. However, the pathway doubles in number to eight. Only by tracking the effort values can we be sure which switch was change and in which order.

What we notice is that odd effort values will result in the switches being in opposite positions, whereas even effort values will always result in the switches being in the same orientation, either both on or both off. However, only by tracking the effort at each step can we be assured of the correct pathway by which the state has arrived at is current condition.

In this simple example, we have only examined the nature of two binary switches. As the number of switches increases, so the number of variations and pathways also grows exponentially. The Theory of Effort is a methodology of tracking changes over time, which has a variety of uses in computer science, particularly in the field of cybersecurity. By developing computer systems that can distinguish the pathway by which bits of code alter states, we can envision a methodology to use Effort Values to create advanced digital locks that can only be correctly decoded once the effort of the final state is known.

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# Conclusion

## So why is the theory of effort important?

More often than not, mathematics is concerned with the result of an equation. However, the Theory of Effort allows us to quantify each step that a calculation performs. In terms of the 4D calculator, we can identify numbers that are devolved from a particular setting, as the value dissolves towards it limit.

## What are the consequences of the theory of effort?

As we are able to track calculations through time we have a novel new way to examine many mathematical functions. This includes any reiterative equation, fractal geometry, and binary systems.

### Mathematics without algebra

Central to mathematics and science is algebra. Yet 4D Maths suggests that x + y = z might not be such an immutable principle as we first thought.

###### Question?

This idea seems to be relatively simple. Are you sure no one has thought of it before?