Have you ever questioned our basic number system that we use? I don't mean mathematics I mean the fundamental numbers, the 0, 1, 2, 3, 4,....
You may be wondering what there is to question about the basic numbers. But that is the point, we learn the numbers in early childhood but then don't give them much more thought.
Two things made me question our basic number system.
The first was fundamental mathematical constants like pi, the ratio of a circle's circumference to it's diameter. By mathematical constant I mean one that is dimensionless, not involving our artificial units of measurement.
At first glance pi seems to be very simple. But yet it requires an infinite number of digits in our number system to express it. This didn't seem to make sense. Most physicists agree that everything is really numbers being manifested, which is why mathematics is so useful. But then everything must somehow ultimately be expressible as either whole numbers, or as a ratio of whole numbers.
It just didn't make sense that, if we express reality in numbers, that something as simple as the ratio of a circle's circumference to it's diameter should require an infinite number of digits, and thus an infinite amount of information, to express.
The second thing that made me question our basic number system was biology. Everything is really numbers being manifested. Inanimate sciences, like chemistry and physics, are very math-intensive, which is what we would expect.
But why then is biology, which involves more complex processes than inanimate chemistry and physics, so much less math-intensive? There is mathematics in biology, but not to the same extent as chemistry and physics.
The conclusion I came to is that it is our number system that cannot handle the complexity of biology, so we describe it more with words than with numbers.
Here is a question. The value of pi is the ratio of the circumference of a circle to it's diameter. A circle and the line which forms it's diameter are the simplest of geometric forms. But yet the value of pi actually contains an infinite amount of information.
Pi is an irrational number, meaning that it can be expressed neither as a whole number or a rational number, or ratio. Pi is equal to 3.1415927..... It goes on and on to an infinite number of digits. Computers have calculated the value of pi to quadrillions of digits, and there is no end in sight. The fraction of 1 / 3 can also be calculated to an infinite number of digits, .3333... But it does not contain an infinite amount of information because it is repeating.
The fraction of 22 / 7 is often used as a close approximation of pi, and is good for most purposes where extreme accuracy is not necessary. But it is not exactly correct.
This really requires a special explanation. How can an infinity of information come from the simplest of geometric forms? Why can such a basic concept as pi not be expressed in finite numbers?
Pi is far from the only irrational number, which can be calculated to an infinite number of decimal places. Another common one is e. e is the function of exponential growth, such as compound interest. The value of e is ( 1 + 1 / x ) raised to the x power, with x being any large number. The larger the value of x, the more accurate will be the calculation of e. The true value of e is 2.718... but, like pi, it can be calculated to an infinite number of digits.
The universe is basically a simple place. So then why can the mathematics which are used to describe the workings of the universe get so complex? Why do we need pages and pages of mathematics to understand what is really a simple universe?
The way that our numbers work is that 1, 2 and, 3 can be described as the fundamental numbers, and the rest are multiples of these. If we bring addition into the picture, we can add 1 to certain even numbers to create numbers outside the factor tree, known as prime numbers. A prime number is one that has no multiples other than 1 and itself. So these are the fundamental numbers, and their factors and additives begin with 4.
Suppose that there were beings which reproduced and spread by dividing in half, and then going off in opposite directions. If they were aware of numbers, they would only be aware of the number 2 and it's multiples. Their number system would be 1, 2, 4, 8, 16... We use numbers to describe the world around us and they would not be aware of any other numbers.
Now suppose that the beings began to study the world around them. They would encounter situations that involved the number 3 and it's multiples. But they only knew the numbers that they had seen before, and built their number system around multiples of 2.
When they expressed multiples of 3 in their number system, they would not be able to express it as either a whole or a rational (ratio) number. They would only be able to express it as an irrational number, which could be taken to an infinite number of digits. What we express as 3 would be a missing number.
The same could be said of a young student who was taught about 1 and 2 and multiplication, but not about addition. The student might surmise that there were multiples of 2, such as 4 and 8. But all of the other numbers that we know would be missing numbers. It was not that the other numbers did not exist, it was just that the student had no knowledge or experience of them and did not include them in his number system. The student would only be able to express other numbers as irrational numbers with an infinite number of digits, just like we express pi.
3 could not be described simply as halfway between 4 and 2 because it's relationship to other numbers would not be correct, and the student would not know how many other "missing numbers" there might be. Any such number would appear as an infinite, non-repeating decimal. The outside number would not be able to be expressed as a ratio of existing numbers, but could only be an irrational number that could be calculated to an infinite number of digits.
Now, here is what we have to consider. What if we have missed some numbers in our number system? Like the beings who multiply by dividing in two, we use numbers to describe the world around us that we know. That is why numbers are useful.
Why couldn't there have been numbers outside our experience that we missed? This is not as unusual or unprecedented as it may seem. As we saw in "The Zero Hypothesis", on the progress blog, humans could not do complex calculations until the importance of zero was understood. The ancients had various counting devices, such as the abacus, but could not do complex calculations yet because they did not understand the importance of zero. You may have noticed that there are no references to complex arithmetical calculations in any ancient texts, although there was geometry.
That is because zero was once a missing number. It had not been included because, when the number system was developed, there seemed to be no reason to count zero of anything. Zero is vital because matter does not fill the universe and we have to deal with empty space, and we cannot do complex calculations without it.
The entire universe actually had it's own missing numbers. We saw in "The Even Number Bias" how even numbers must have come before odd numbers, and we can still see the "ghost" of that today. The fusion process that takes place in stars favors atoms with even numbers of nuclei in the nucleus. The most stable and common elements in the universe are those with even numbers of nuclei, the original hydrogen being the exception.
The 25% of original atoms that were helium, and heavier than hydrogen, were pulled by gravity into fusion in stars, and this formed a "factor tree" of 2 x 2 x 2... The lighter hydrogen atoms could be added to form odd numbers, but this was originally the exception rather than the rule. Odd numbers thus started as universal missing numbers.
It is important to understand that we cannot get to missing numbers, expressing them as whole or rational numbers, by any kind of operations using existing numbers. If we could, then they would not be missing numbers.
But how would we have missed any numbers?
We use numbers because they are useful in describing the world around us. Our world is made of matter. The reason that humans missed the importance of zero for so long is that it is the number of empty space.
We see how the number two began, because there are two electric charges that make up the universe. 3 was introduced into the universe of matter because of the three quarks that make up nucleons. A quark cannot have a partial charge based on only the numbers 1 and 2, 3 is required. Dimensions of space is what brings in multiplication.
Notice that the mass of a proton is 1836 times that of an electron, which involves multiples of both 3 and 2 and also 1 because it brings us down to the prime number of 17. This is what brought our fundamental numbers of 1, 2, and, 3 into being, and other numbers are multiples and additives of those.
A number has no real existence in itself. It exists only when it is manifested in some way. Our presumption is that all possible numbers have been somehow manifested and that we have included them in our number system but, as we can see here, we cannot know that for sure.
WE ARE NOT MAKING FULL USE OF NUMBERS
But these numbers that we have are the numbers of matter. The vast majority of the universe consists of space, and matter exists in space and not the other way around. In my cosmology theory, space existed first and matter is a relative "newcomer" to the universe.
Notice that it is pure mathematical constants, such as pi or e, which tend to be irrational numbers, and not the constants of chemistry and physics. These sciences are the science mostly of matter. The stoichiometry of chemistry revolves around whole numbers. The constants of physics revolve around our artificial units of measure, such as seconds, meters and, kilograms.
It is the dimensionless constants of pure mathematics which tend to be irrational, meaning that they cannot be expressed without an infinite number of digits, this shows that they were part of the universe before matter and our numbers are the numbers of matter.
We learn the numbers at age 5 or 6 and then never question them. To really understand the universe, we must "get outside ourselves". Just as we know that there are planets outside our solar system, so there are numbers outside our number system. When we encounter them, we cannot fully express them or their multiples as part of our number system.
But even though we seem to be missing numbers, that is only because we are not making full use of the number system that we have. This makes it so that many mathematical constants cannot be expressed in finite form.
THERE ARE REALLY NO IRRATIONAL NUMBERS
Pi does not fit into our number system, but the fault is with the system. There should not be any such thing as irrational numbers. If basic mathematical constants, like pi, can only be expressed as irrational numbers then the only possible explanation is that we must somehow be missing numbers in our number system.
Imagine someone making the universe from the beginning, putting it together. Everything must somehow be expressible in whole numbers. It has to be this way. There has to be numbers that we missed because we did not notice or need them. We see that our factor numbers begin with 4, and notice that basic mathematical constants are virtually all below 4, which is where they could have most easily have been missed without being noticed or needed.
What if common irrational numbers, that are at the center of so many of our formula, such as pi and e, are really based on whole numbers that we missed because they were somehow not manifested within our experience?
The whole idea of such basic mathematical constants not being able to be expressed in whole numbers doesn't really make sense. My conclusion is that irrational numbers cannot really exist, we must be missing some whole numbers.
Everything should be able to be expressed as a whole number or ratio, as long as we are not dealing with anything that humans have created artificially, such as units of measurement. This is not the case with pi because it is dimensionless, meaning without units.
THE IMPORTANCE OF RATIOS
A number by itself means essentially nothing. The first step in using numbers in meaningful expression is in the form of ratios, or rational numbers. A ratio is not just a number, but a number in relation to another number.
This is how reality really operates, in the form of ratios rather than of whole numbers. Remember, as we saw in "The Lowest Information Point", that the point of least information is a "favored point" because the universe seeks the Lowest Information Point just as it seeks the lowest energy state, because energy and information is really the same thing.
I define the Lowest Information Point as the interaction of two ratios, where A / B = B / C, so that "A is to B as B is to C". This the Lowest Information Point because it involves only three points of information, as opposed to the four in A / B = C / D.
The trigonometric functions are also ratios. If we have a right triangle, or a radius from an intersection of a X and Y axis, the sine of the angle between the radius and the X-axis is defined as Y / R, or the length of Y-axis involved over the length of the radius. The cosine is defined as X / R. The tangent is defined as Y / X. Then there are the other three functions that are the inverse of these. The co-secant is the inverse of the sine. The secant is the inverse of the cosine. The cotangent is the inverse of the tangent. In the functions beginning with co-, the value gets smaller as the number gets larger.
What we refer to as "diminishing returns" is also based on ratios. If we were given a large amount of money, we would be very grateful. But if we were then given the same amount of money again, we would still be grateful but not quite as grateful as with the first amount of money. While the second amount of money would be numerically the same as the first, it's value as a ratio to the money that we already had would be less.
Any comparison is based on ratios. If there were two children, aged 5 and 10, we would say that there is a big age difference between them. But if they were 45 and 50, we would not say that there was much of an age difference, even though the difference is numerically the same.
The concepts of "far" and "near" have no exact numerical definition, but are based on ratios. If two atoms were 5 km apart, we would say that they were very far away from each other. But if two towns were 5 km apart, we would say that they were near each other.
So much of nature is based on ratios. A very important number is how reality operates is the so-called "Golden Ratio". The Golden Ratio is defined as the sum of two unequal numbers being of the same ratio to the larger of the two numbers as the larger of the two numbers is to the smaller. Expressed in decimal terms, the Golden Ration is an irrational number, 1.618034...
So not only is the Golden Ratio an important ratio it is also another basic constant, like pi and e, that is so important to us but our number system is unable to express it as either a whole number or a ratio of whole numbers.
Much of nature, such as the construction of plant leaves, operates by the Golden Ratio. It is also very important to humans because it enhances our perception if something is in the Golden Ratio. Notice how television and computer screens, pages in books, maps and, billboards almost always have the approximate dimensions of the Golden Ratio.
Even when we express something in whole numbers we are still effectively using ratios. The number is the numerator and the "thing" is the denominator. To say "5 apples" is actually 5 / apples.
Most physicists agree that everything is really numbers being manifested. But we can see how the universe really operates not by whole numbers but by ratios, one number in relation to another. We could say that whole numbers are "simple numbers" while ratios are "complex numbers". A number doesn't really mean much by itself, unless it is contrasted with another number, or a word that is used as a substitute for a number, as the denominator of a ratio.
IMAGINARY AND COMPLEX NUMBERS
In algebra class, many of us were mystified by what "imaginary numbers" were supposed to be useful for. An "imaginary number", denoted as "i", is defined as the square root of negative one, -1. This means that i squared = -1.
The trouble is that a negative number cannot have a square root because two negative numbers multiplied equals a positive number. But yet we have to learn this to pass algebra class.
But later, after I came up with my cosmology theory, it suddenly made sense. In fact, the reason that it at first doesn't seem to make sense is the limited use of our system of "real" numbers.
The "imaginary numbers" are actually defined as another line of numbers that is perpendicular to our usual line of numbers. "i" is defined as being one point away from our numbers if we picture them as being on a straight line. Any of our "real numbers", which are ordinary whole numbers, has an "i" component of zero.
I do not think that "imaginary numbers" is a good term to use, since these numbers turn out to not be "imaginary" at all. A better term is "complex numbers". The difference between complex numbers and our common "simple numbers" is that two complex numbers can be equal or equivalent, without being the same thing. That is not possible with our usual "simple numbers". In addition, every complex number has more than one square root.
The "i" of "imaginary numbers", defined as the square root of -1, is useful for differentiating a complex number from the addition of two "real" numbers. A complex number thus looks like this:
( 5 + 7i )
When we see the "i", it tells us that this is a complex number and we do not add the 5 and 7 as in an addition operation. If we just express the whole number 5, the i component would be zero: ( 5 + 0i ).
The number system thus becomes a set of points on a two-dimensional grid, rather than on a one-dimensional line. This is necessary to express spatial concepts like pi.
Next, considering that the universe really operates by ratios, a number by itself is essentially meaningless until it is in relation to another number, a complex number then takes this form:
( A / B + C / Di )
The "i", once again, reminds us that this is a complex number and the two ratios are not fractions to be added together.
This is how the universe really operates, as ratios and complex numbers. The reason that we cannot express such important numbers as pi in the form of whole or "real" numbers is the fault of our use of simple or "dumb" numbers. These complex numbers are really the "smart numbers" which can express how the universe really operates.
Mathematical formulae, not those involving chemistry or physics which tend to have our artificial units, such as meters, kg, seconds, etc., but "pure" mathematical formulae, such as pi, are expressible in finite form but only if we use complex numbers as I am describing here. If we have to use an infinite number of digits, or an infinite series of fractions, to express pi, then we are using only the limited simple numbers.
Formulae for curves, hyperbolas and, parabolas are also actually complex numbers when we add the variables to the equation.
"Real" numbers should be able to express a relatively simple concept like pi in finite form, and here it is: (source-Wikihow)
pi = 2 ( arcsin ( square root of ( 1 - X squared) ) + absolute value ( arcsin ( X ) ) )
Absolute value means the value of something, regardless of whether it is positive or negative. Remembering that positive and negative, unlike in our "real" numbers, is really interchangeable.
Arcsin is the inverse of the sine function. If the sine of 30 degrees is 0.5, then the arcsin of 0.5 is 30 degrees.
So here is the answer to where our missing numbers are, as described above. It is really that we are still using the same old simple numbers when we should be using the complex numbers that truly describe the operation of the universe. Two numbers must be able to be equivalent without being the same thing, and that is possible only with these complex numbers.
THERE ARE REALLY NO NEGATIVE NUMBERS
Numbers are traditionally taught as operating in a line beginning at zero. On the other side of zero is a mirror-image line of corresponding negative numbers.
The arithmetical rules of dealing with negative numbers are as follows:
A positive number multiplied by a negative number gives a negative number.
Two negative numbers, or two positive numbers, multiplied gives a positive number.
But what I have concluded is that negative numbers do not really exist outside of the world of human beings. We only see negative numbers because of our perspective on the universe.
Negative numbers can be compared to color. We know that color does not really exist in the universe of inanimate matter. Color is just how our eyes and brains interpret different wavelengths of electromagnetic radiation.
The only examples of negative numbers that I can think of are in expressing temperature, debt and, exponents. But our temperature units are our artificial creations. There are no negative temperatures in the Kelvin Scale, which begins at Absolute Zero. Debt is a function of our artificially-created money system and economics. Exponents are part of our artificially-created units and number system.
Negative numbers thus only exist in systems that we have created ourselves, and not in the universe of inanimate matter outside of ourselves. Negative numbers are an artificial creation that does not really exist.
What do you know? That "imaginary number", denoted as "i" for imaginary, is so baffling to algebra students because it is supposed to be the square root of -1. But yet we are taught that negative numbers cannot have square roots because two numbers multiplied together always equals a positive number.
Well as we can see here negative numbers do not really exist. But yet this concept of i must have some value or algebra students wouldn't be forced to learn it. What it really is is the step toward the complex numbers by which the universe really operates.
BIOLOGY AND COMPLEX NUMBERS
Along with why something as basic as pi, the ratio of the circumference of a circle to it's diameter, can never be expressed as either a whole number or a ratio, the other thing that always got me wondering about our number system is biology.
Sciences like physics and chemistry are really mathematically intensive. There is mathematics in biology but it is quite a bit less intensive than chemistry and physics. Has anyone ever wondered why?
The answer is that our mathematics cannot as easily represent biology, which is more complex than inanimate sciences like chemistry and physics. We saw in the compound posting on this blog, "How Biology And Human Life Fits Into Cosmology" June 2016, that biology contains four points of information, whereas inanimate matter contains only two.
The two points of information in inanimate matter is, of course, the two electric charges of negative and positive. The DNA of all living things contains four points of information. These are abbreviated as the A, T, G and, C. With twice as many points of information this means that there is a second dimension of information in DNA.
Physicists usually agree that everything is really numbers being manifested. We can see this with inanimate sciences like chemistry and physics. It is less easy to see with the more complex biology, but if everything is really numbers then that must include biology also.
But what do you notice here? The complex numbers also have an extra dimension, and this extra dimension would fit with the extra dimension of biology. By using complex numbers we could use mathematics to describe biology in the same way that we do now with chemistry and physics.
COSMOLOGY AND SMART NUMBERS
A grid of negative and positive numbers is a model of the alternating electric charges comprising space. Each charge is a point on the number grid. But it must be a two-dimensional grid in order to represent our spatial universe. That is where these complex numbers come in, the i represents a dimension of numbers that is perpendicular to the traditional whole numbers.
There are two opposite directions in every spatial dimension because, from any given point, there are two permutations of charges, one beginning with a negative in one direction and the other beginning with a positive in the other direction. That means that we had to create a line of numbers, with higher numbers in one direction and lower in the other, as a useful tool to represent reality. But the number system must have perpendicular lines of numbers to accurately represent our spatial universe.
The two possible solutions involved in complex numbers are inverses of one another, just like the perpendicular directions in space.
The two square roots of complex numbers, actually there is one square root per dimension, can be expressed as diagonal lines at 45 degree angles on a grid, just like the dimensions of space in the cosmology theory.
