# Interactive Online Tutoring Services

## April 1, 2024

### The foundation of mathematics

Filed under: Mathematics,the foundation of mathematics — Rob burchett @ 1:18 pm

It seems the axiomatic foundation of Mathematics is made of concepts and thus subject to concept sharing. Then the axioms must have other levels. Then there is no irreducible axiomatic system.

A better foundation of Mathematics is seen when we look at the truths that concept sharing allows. Since we must have concept sharing we need to look at the Mathematics that comes from this.

Then this can be a new basis of Mathematics

## March 27, 2024

### The twin prime conjecture

Filed under: Mathematics,the twin prime conjecture — Rob burchett @ 12:09 pm

An interesting notion can come about from something observed in nature, the overlapping of two shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many concepts we wish to be present.

The twin prime conjecture:

There is a lower level of prime numbers using the concept sharing of a number. For examples: (1(1), (2(2), (3(3), (5(5),…I drop the (1), (2) notation for clarity and brevity.

Yet there is now a plane of numbers as shown below. With entries such as (3(5), (5(7), (5(11). These are partially shared prime numbers. We can make a square as shown, with a gap of two on all sides, centered at (4(4). This connects the prime shared numbers.

We might also make copies of these prime’s associated with this square. Yet there is a countable infinity of these possible.

The shared prime pairs can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary prime’s, there can only be one copy of these. The rest of the copies are moved into the infinity that is available.

Since I can make an indefinite amount of copies, the same gap must be possible infinitely as we go further up. Therefore the twin prime conjecture is seen. Also, we see how other gaps must be repeated too.

## January 26, 2024

### Concept sharing and Goldbach’s conjecture

Filed under: goldbach's conjecture,Mathematics — Rob burchett @ 4:26 pm

Abstract: Here I explain and use concept sharing. I take a look at Goldbach’s conjecture.

An interesting notion can come about from something observed in nature, the overlapping of two shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

Goldbach’s conjecture:

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are higher numbers than the naturals, 2 being greater than 1. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from concept sharing with another number. Shown more clearly here: (a(a). For clarity and brevity I drop the (1),(2) designation from now on.

There are other numbers such as (2(3), (3(4), (3(5),… We can think of this as partial sharing. For (2(3) think of three dots colored red. We can have two blue dots overlapping with two red dots forming two purple dots. I say that to give the idea, numbers are not colored dots. Numbers are the exact concept of amount or position only. So that (2(3) can be thought of as two sharing with three and (3(2) can be thought of as three sharing with two. In the first case 1 is left over and in the second case 1 is an extra number. These are the same concept though so (2(3)=(3(2).

Consider the even shared numbers (2(2), (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (2(2)=(1(1)+(1(1) and also (4(4)=(1(3)+(3(1) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the possibilities. They can form products too.

Some numbers have more than one decomposition. Such as (16(16)=(13(3)+(3(13) and (16(16)=(11(5)+(5(11).

Now think of the even shared numbers as being created from the primitives. I designate them (2(2), (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these higher numbers; (2(2), (4(4)…. must come from somewhere else. These even shared numbers must go on indefinitely, as to lead to the numbers 2,4,6,…at the lower level . Then the natural numbers are all at a lower level from these numbers.

For multiplication, in the usual number system we have prime numbers. We can look for the higher level of prime numbers. Suppose I build a higher level of primes from the usual prime numbers by concept sharing two prime numbers. For example (7(3)=(3(7).

But (10(10) = (7(3)+(3(7) =(2(2)(7(3). (7(3) can’t be equal to (7(1)(1(3) as I can change this to (7(1)(3(1)=(21(1). The other part, (3(7) can be changed to (1(21). I can add to obtain (10(10)=(22(22), a contradiction as 10 does not equal 22. This is outside the sharing domain. So (3(1) and (1(7) can’t be factors. So I can’t factorize (a,b) further where a and b are prime.

Then starting at (4(4), we can ask, how is (4(4) created? (4(4)=(2(2)(2(2). (4(4) is an even shared number so there is a division by (2(2) possible. This can be the definition of an even shared number. When I divide both sides by (2(2) I have a shared number on both sides, these are equal but not necessarily identical. For (4(4)=(3(1)+(1(3)=(3(1)+(3(1) =(2(2)(3(1) also. In this case we can think of the shared number (2(2) separating in the number space to form the number (3(1)=(1(3).

Then also (6(6)=(2(2)(3(3). Also (6(6) =(5(1)+(1(5) =(5(1)+(5(1) =(2(2)(5(1) also.

Also (8(8) is not equal to (2(2)(4(4) as (2(2)(4(4)=(2(2)(4(2)(1(2)=(2(2)(2(4)(1(2)=(4(16). This is outside what we could consider the sharing domain as 8+8=16 but 4+16=20. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6)

And so on. Each even shared number must be a multiple of (2(2) and another prime shared number. We consider 1 to be a prime number.

Imagine an extended number line. For example, since (5(5) can separate, lets look for a way (10(10) can be created. (10(10)=(a(b) +(b(a)=(2(2)(a(b). I am taking apart (5(5) so as to create (10(10). Two (5(5)’s are separated and then recombined. ex. (7(3)+(3(7). (10(10) can be created from (a(b) +(b(a) but this is also (a(b)+(a(b) =(2(2)(a(b) so this must work too. As shown in the case above with (8(8), a,b must both be prime. There must be one such factorization.

Every even shared number can be divided by (2(2). The other factor must be a prime shared number as we need this to work in the shared number system. For example (12(12)=(2(2)(6(6). (6(6) must separate into two prime numbers otherwise (12(12) can’t appear (we assumed it was there). It must exist since I need to have 12. I can now take (12(12) out and replace it by 12. So we must have a decomposition with (2(2)(a(b) with a, b both being ordinary prime numbers.

This demonstrates the equality of every primitive decomposition as well (which the lower level was hinting at) since for any even shared number there may be more than one decomposition. We go back to form (16(16) and then can separate once again to find another decomposition. Here we find two, (2(2)(13(3) and (2(2)(11(5)

Then if we look at one half of these binary decompositions, that is, for example, look at 16(16)=(13(3)+(3(13) and see 16 =13+3 we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.

## December 29, 2023

### General Math Concept Sharing

Filed under: general math concept sharing,Mathematics — Rob burchett @ 10:00 am

General Math Concept Sharing

For Concept Sharing we needed to start with sharing numbers. The idea is to form sharing concepts. For this we need these new numbers, to be specific so that we can start with certain mathematical systems.

In mathematics, concepts are mental constructions. They are ideas like shadows with boundaries. They can be thought of like points. The foundation of mathematics is based on concepts. Here then we need to find a new, extended foundation.

We can concept share a point since two items of no extent will still have no extent, but there could be two items here.

So similarly numbers, as we have seen, points, sets, groups, ect. Can all concept share.

In order to do this we must remove the concept which is present initially and replace it with 2 or more sharing concepts. Since it is possible to share concepts there must exist a more underlying concept space.

The two sharing concepts must be different in some way which we can specify based on the nature of the concept itself.

Then an infinite level concept space can form as I can continue into the next level of the concept and so on. One may utilize as many levels as is necessary.

Additionally there can be a finite number, an infinite countable number of an infinite uncountable number of sharing concepts, as this is the understanding of numbers.

## September 5, 2023

### The Riemann hypothesis

Filed under: Mathematics,the riemann hypothesis — Rob burchett @ 12:22 pm

It seems to me there may be an easier way to express the zeta function: Z(s)=1/1^s+1/2^s+1/3^s…..using the ideas of concept sharing as it applies to a new geometry.

Concept Sharing and a new view of the Riemann hypothesis

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding the Riemann hypothesis.

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one.

The objects are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2).

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. The places of places line can be defined in a space of places of places of places. Then a place of place, with a place, can move in the upper half-plane. Make a small jump back to the places of places line. We can remove the places in this small gap then remove other places as motion continues as the line underneath is revealed.

Then some of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

The Riemann Hypothesis:

One may imagine a type of gird with the first square being 1, the next being 1/2^2 the next being 1/3^2… if we use s=2 as an example. See pictures in the notes below. The higher numbers of s can be seen by increasing the dimension. Yet there is always a plane possible with any dimension equal to or higher than 2.

Since with concept sharing geometry there comes a place of places, in which places can vary, we may vary the distance as we choose to always make the zeta function defined. The zeta function can be continued into the extended geometry. Then there is no longer a need for analytic continuation. I can always make the grid into a 1 by 1.

So we can create a grid specific to the Zeta function defined in placement space.

Then we have that there are two types of number involved. A real part and an imaginary part.

I think this can be seen more primitively as a numbers which lead to a square with a positive area and numbers which lead to a square with negative area ie. the negative distance is -i. These can be sharing space.

We can concept share two different numbers in the following way: (-1(-1)*(-i(-i) where * is a concept sharing of a concept sharing= ((-1(-i))((-1(-i)). But -1 and -i have to be different. Let -i be the negative distance and -1 be the other, real distance. Then let this be how the square comes about. We have to expand the zero-dimensionality of the concept sharing. Let -i and -l be numbers at the next level of numbers. That is they are no longer point-like but line like. We can start with a point consisting of an infinite uncountable number of sharing parts and expand it outwards into a line.

The next to last image shows how there are trivial zeros at -2,-4,-6… and how the zeta function could equal -1/12 when s=-1. We are adding an infinite series to get a finite sum. This comes about as we have a addition of positive and negative area. This works for the plane as we can have i and i^2=-1.

If we look at the last image, there are three possible cases. One where s=2, one where s=1/2 and some where s=-2,-4,-6,…These all lead to a plane where we can also share with i, in the last two cases so that we might cause the series to converge.

## January 7, 2008

### Struggling with Math?

Filed under: Tutoring,tutoring articles — Rob @ 6:26 pm

## Are you going to struggle with Math this year?

August 25, 2006

As the new school year is approaching, one thing is certain: Many students will struggle with their courses this year. As always, one of the most prominent areas of difficulty will be Mathematics.

It is not very difficult to understand why math is considered by most Canadian high-school and college students to be such a difficult subject. Math is a discipline that requires a basic foundation and a natural transition from core knowledge to more advanced concepts. However, the reality of North American education is that the transition rarely takes place naturally.

It is often the case that throughout elementary school and early High school, math is taught in a very disorganized fashion. While arithmetic gets more than its fair share of attention, intermediate core concepts are often glossed over by everyone except the very astute students. Only the dedicated students fill in the big education gaps in their spare time. What happens to the less dedicated students?

For other students, the true math difficulties begin in grades 11 and 12. At those levels, math transforms into a serious subject and cracks in the knowledge foundation begin to emerge. Those concepts that are natural extensions of what is considered to be basic mathematical abilities become difficult to grasp for many students.

As a result, many struggling students turn to a private math tutor for additional help, but this is often not a sure-fire path to better understanding. Most tutors have the ability to help students with their immediate areas of difficulty. However, only the more experienced tutors are able to detect true deficiencies in the students’ knowledge and fill in the gaps before concentrating on more complicated topics.

For those looking for a math tutor, it is very important to consider the tutor’s knowledge, experience, and approach, and not just the hourly rate. With math tutoring, like with anything else, you will often get exactly what you pay for.

In order to get the most out of tutoring, it is vitally important to establish specific short and long-term learning objectives early on. A good tutor will be able to use this information effectively in creating a structured learning progression, rather than concentrating on the irrelevant concepts.