The post The Quantum Swimming Pool appeared first on Negative Numbers.

]]>Waves crest and trough as opposites, neither positive nor negative. Collapsing down, the deepest depth of the numinous natatorium presents its foamy planckton pixels. Abig, asmall, points without size. Quantum exotic spheres manifest where strings twist orthogonally and enter higher dimensions that stabilise the three. One twist, two twists, three twists , more, off they wriggle to visit their bullish sisters.

Satisfied, stretched, consumed, lengths achieved, leave the pool behind at all ends, to entangle refreshed with the world.

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]]>The post Negative Numbers on a Rotation Ball appeared first on Negative Numbers.

]]>The Covid Pandemic gave me time to study negative numbers. This reignited my frustration with the √-1 = i. The concept of i as a 90^{O} rotation on a circle makes sense. Understanding why multiplying an i with an i results in -1 is more difficult.

I also wanted to make sense of i and negative numbers in 3 dimensions. To do this I needed to study movement in 3 dimensions. I found it difficult to imagine rotations and needed a physical prop to try them out. I went on-line to see if I could buy a 3d sphere that showed rotation but could not find one. I decided to make my own.

During Covid, the Irish government decided not to allow me to use my tennis balls for tennis. Both golf and tennis were forbidden for very questionable reasons. I decided to make my own 3d sphere and I took a Wilson Triniti eco-friendly tennis ball from my dusty sports bag. The only writing on the ball was the Triniti logo which I used to represent the North Pole. I cut off some of the felt with the logo using a scissors.

- At the North Pole, with the Triniti logo the right way around, I wrote Z+ in the centre of the logo to represent the positive end of the z-axis. I wrote the Cartesian coordinates (0, 0, 1) below the logo. See the diagram above.
- I marked Z- in the centre on the opposite side of the tennis ball and (0, 0, -1) below.

By holding the tennis ball at these Z+ and Z- points between my thumb and big finger, I had made the z-axis. I was able to rotate the tennis ball easily with my other hand.

- Holding the ball vertically by the z-axis, I then marked in the X+ pole in the centre facing me and (1, 0, 0) just below. I marked the X- pole in the centre on the opposite side with (-1, 0, 0) just below to get the x-axis.
- Continuing to hold the ball at the z-axis points, I turned it 90
^{O}counterclockwise or to the left and marked in the Y+ pole in the centre with (0, 1, 0) just below and then the Y- pole on the opposite side with (0, -1, 0) just below.

Turning counterclockwise and clockwise on the tennis ball can be confusing as the ball and your viewpoint seem to change. To make sure I was moving correctly I marked the counterclockwise rotation on the ball using the following diagram.

- With Z+ facing me and X+ below, I drew the symbol around the Z+. This was to show the direction of the counterclockwise or positive rotation, no matter what angle I held the tennis ball. I turned the ball and then drew the symbol around the Z- to show clockwise rotation.
- With Y+ facing me and Z+ facing me I drew the symbol around the Y+ and turning the ball drew around the Y-.
- With X+ facing me and Z+ above me I drew the symbol around the X+ and turning the ball drew around the X-.

To check that my arrows were correct, I held each axis parallel to the ground with the positive side towards me. Rotation clockwise and counterclockwise around the axis is then in the same direction as for a clock. i.e. The top of the ball moves left during a counterclockwise rotation and right during a clockwise rotation. I then realised that if I was looking at one side of the ball and it was moving left, the other side was moving right. You cannot really use left and right when describing rotation. If I want the ball to rotate counterclockwise, I need to check that it is moving in the direction of the arrows and vice versa for clockwise.

The Rotation Ball was ready but it needed a nickname. I decided to call it the R-Ball. The next blog describes how to use the R-Ball and continues the search for negative numbers.

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]]>The post Waiting for Quantum Godo – Act 1 appeared first on Negative Numbers.

]]>In the timeless expanse of experience, there existed an electron. A solitary traveller, it ceaselessly roamed the vast reaches of its own universe, an explorer of all it could fathom. Singular and self-assured, it bore the name Oneo, proclaiming itself as the sole entity in its realm.

Through countless eons, Oneo crisscrossed its universe, venturing near and far. Swift and elusive, at times, it seemed to transcend the confines of space, as if ubiquitously present. Yet, a peculiar observation gradually seized its attention — there were limits to its voyages. Its universe harboured a boundary, a finite boundary, and beyond it lay naught.

Oneo, in its solitude, meticulously documented its sojourns, albeit lacking the visual senses to behold its surroundings. Instead, it possessed a unique talent: to gauge distances, the electronic ruler of its universe. Within its digital diary lay an inventory of the spans it traversed between diverse points. With little else to occupy its thoughts, Oneo delved into the analysis of these recorded measurements.

Amongst its contemplations, Oneo stumbled upon a revelation of great significance. A solitary locus within its universe existed, equidistant from every point adorning the periphery of its world. This point held a profound allure for Oneo. Whenever it traversed this spot, a serenity akin to Zen enveloped its being. Its consciousness cleared, granting a transient respite, an interlude of tranquillity.

This, to Oneo, was a nexus of equilibrium, a point of rejuvenation, a departure point for new odysseys. Devoting much of its existence to pondering this phenomenon, Oneo decided to name it “Point 0”, the genesis and epicentre of all its endeavours. The measure from this celestial fulcrum to any edge-bound point became “the 1 distance.” Thus, its entire universe unfolded betwixt the embrace of Point 0 and the reach of the 1 points, a cosmic saga etched in the annals of Oneo’s electronic consciousness.

In a solitary moment, amidst the stillness of Point 0, a singular encounter stirred Oneo’s electronic essence. A photon, darting forth from the abyss of infinity, collided with Oneo’s essence. This collision, a cosmic serendipity, ignited a spark of profound excitement within Oneo. An exhilaration pulsed through Oneo, birthing an impulse that compelled it to release a photon in response, a luminous expression of its euphoria.

As if scripted by the universe itself, another photon descended upon Oneo, bestowing upon it an unexpected encore. Overwhelmed by this mystical dance of photons, Oneo emitted yet another photon, witnessing with bated breath as it transcended the bounds of its universe, vanishing into the uncharted abyss beyond the edge.

With newfound zeal, Oneo embarked on its inaugural experiment, orchestrating a ballet of photons and meticulously documenting each outcome. Like cosmic messengers, every photon emitted was met by the return of another from the mysterious realm beyond.

Next: Waiting for Quantum Godo Act 2.

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]]>The post Waiting for Quantum Godo – Act 2 appeared first on Negative Numbers.

]]>Oneo, the master of measurement, unfurled its digital ruler and scrutinized the messengers of light. An intriguing revelation unfurled before its figurative eyes: not all photons were cut from the same energetic cloth. Their waveforms danced with diversity, a symphony of oscillations.

Yet, Oneo discerned a rhythmic pattern, one that recurred with every pair of photons. The initial photon bore a grand wave, followed by a second with a diminutive wave. Oneo, ever the mimic, adopted this pattern in its photon emissions, a decision that would usher in a remarkable transformation in the incoming photons’ behaviour.

A new symphony of photons, as intricate as a cosmic lullaby, surfaced. Each photon resonated with either a high or low frequency. The rhythmic cadence revealed itself in a cyclic procession of four photons: the first, a low frequency with a modest wave; the second, a low frequency with an expansive wave; the third, a high frequency with a delicate wave; and the last, a high frequency with a majestic wave.

In this cosmic odyssey of discovery, Oneo came to a profound realization—it was not alone in the cosmos. Another intelligence, dwelling beyond the realm of its universe, also engaged in the emission of photons. In homage to their first contact at Point 0, Oneo christened this cosmic counterpart “Zero.”

Time flowed like the currents of the universe. In due course, Oneo and Zero crafted a rudimentary means of communication, forging a unique bond. Zero, like Oneo, traversed its universe ceaselessly, a celestial tennis ball crossing the net of 0 in myriad directions, yearning to explore diverse points. Yet, the edges of its universe posed a challenge, as they all bore striking resemblances. Thus, Zero devised a numeric triad to describe each point, with values capable of being positive or negative based on their relationship to the origin.

Oneo grappled with the concept of positivity and negativity, yet the three-number system struck a chord. It adopted the origin as (0, 0, 0), the North Pole as (0, 0, 1^), and the South Pole as (0, 0, 1v), noting the ^ and v to signify their antipodal nature. Similarly, it coined the East and West axes on the y-axis as (0, 1^, 0) and (0, 1v, 0), and the Near and Far poles on the x-axis as (1^, 0, 0) and (1v, 0, 0).

In this numerically symmetrical realm, Oneo found solace, an elegant solution free from the enigma of negatives. In spite of their cosmic games of tennis, Zero’s negative ideas dwindled and shrank from Oneo’s contemplations. It awaited the day when Zero might elucidate the enigma of positive and negative in the universe, though, much like waiting for Godo, it remained a timeless pursuit.

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]]>The post Zero Usefulness appeared first on Negative Numbers.

]]>There is no doubt about the usefulness of zero. It would be difficult to imagine negative numbers without zero. OK, it is useful but is it really a number? It acts as a place marker to say nothing is there. It really helps when writing numbers, just like the comma. Take the numbers 1,124 and 1,024 for example. When you say them aloud, you say ‘One thousand, one hundred and twenty four’ and ‘One thousand and twenty four’. Saying ‘Zero hundred’ would not add any information to 1,024. Zero is a place marker to tell you that there are not any hundred units in the number 1024. There is 1 x thousand unit, 2 x ten units and 4 x 1 units.

If the primary function of numbers is to count, it could be argued that zero is not really a number as you cannot count zero of anything because there is nothing to count. That is probably going too far, but it is at least fair to say that it is a very special number and unlike the other numbers.

Scientists tend to use BCE (Before Common Era) and CE (Common Era) instead of BC (Before Christ) and AD (Anno Domino). So we know when the years 1 BCE and 1 CE happened but what happened to the year 0? We know where zero is on the number line but when was the year 0? It is not on the Roman calendar. Maybe this is why a person’s age was calculated in Korea as starting at 1 when she was born. This was officially changed to the international system in June 2023.

The past no longer exists. The future has yet to happen. The present is the border between the past and the future. It’s like zero, which is the border between negative and positive numbers. It is the start for counting, the origin.

So adding zero to a number leaves it the same. This sounds sensible. Multiplying a number by zero leaves us with zero. Supposing you have 2 apples and you add 2 more and then add 2 more, you will end up with 6 apples. This is the same as multiplying 2 apples by 3. This makes sense. However if you have 4 apples and you add 4 apples zero times, you still end up with 4 apples. So zero is a bit awkward with multiplication. Of course, multiplying 4 by 0 is interpreted as adding 4 to 0, 0 times giving 0. It seems more a convenience that multiplying by zero results in zero. As mathematics is a system, the operation of multiplication by zero can be defined as required by the system designers. Try dividing by zero and it is clear that zero does not have any usefulness for division.

0^{0} is either 1 or undefined. It helps with some theorems for 0^{0} to equal 1. Now, 0^{1 }is 0. So 0^{1} < 0^{0}. Yes, this is a bit hard to understand.

What is 0^{1/2}? It is √0 = 0. That is OK as 0 x 0 = 0.

How about 0^{-1}? That is 0/0^{1} = 0/0. The answer is not defined. You might think that the answer is infinity. It can’t be because infinity is not a real number. Maybe zero is not a real number as well.

I have been suspicious of the way zero facilitates proofs. The proof that the product of two negative real numbers is a positive real number uses zero as follows:

Given two positive real numbers a, b:

a – a = 0

(a – a)-b = (0 x –b) Multiply by – b. Note use of 0 on RHS

(a x –b) + (-a x –b) = 0

We know that (a x –b) = -(a x b)

-(a x b) + (-a x –b) = 0

-a x –b = a x b

The proof uses a clever mechanism with zero to introduce the term –b on the LHS without changing the RHS. However, if zero is not a true number, is this just a trick? There is no doubt of the usefulness of zero, but is this an abuse of zero?

This contrasts to the logic used by a student teacher in a study in 2009 when he said ‘I would show them that two minus signs can make a plus if you take one minus and put it crosswise over the other one.’

When Descartes designed Cartesian coordinates and the co-ordinate geometry system, he did not have any negative numbers on his x and y-axes- link. He used zero as the start point on each axis. Zero was the origin, the point where the two axes met. Someone had to ask what was left and below the origin.

This is where negative numbers are located. The main feature of the numbers left of the origin is the sense of direction. If you go right a certain number of units from the origin and then go left the same number of units, you end up at the origin. Equal amounts cancel each other out.

The nature of numbers either side of the origin is that equal amounts cancel each other out, just like waves. It is time to develop a system of numbers based on this fundamental feature. It is time to say no to negatives.

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]]>The post Mathematical Constants appeared first on Negative Numbers.

]]>*‘A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g. an alphabet letter), or by mathematicians’ names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, statistics and calculus.’*

Below is a list of some of the Mathematical Constants that includes my favourite Golden Ratio, which Fibonacci made famous.

Generated by wpDataTables

The interesting thing is that all of them are positive except for two of them. The first of the non-positive is 0 and is neither positive nor negative. The second is the imaginary number √-1, which is not just a number but also a rotation of a certain magnitude. So, it’s an imaginary constant.

So are there no other negative Mathematical Constants? You could put a minus sign in front of any of them but that does not explain why they are all positive. They are positive because they emanate from the natural world where negative numbers do not exist. Negative numbers are not real.

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]]>The post What does the Minus Sign Represent? appeared first on Negative Numbers.

]]>Do you think you are familiar with the minus sign? Test yourself by taking the following challenge.

**Express -(-3-5) ^{-2} as an integer.**

Let’s break the equation down.

The symbol before the 3 represents the integer “negative three”.

The subtraction symbol precedes the 5 and results in the subtraction of 5 from -3 to give -8, the value of the number in the brackets.

The symbol before the 2 represents the reciprocal exponent. This is the equivalent of 1 divided by the result of the last step to the power of 2. i.e. 1/(^{–}8*^{–}8) = 1/64. What is negative about a fraction? The minus sign indicates this operation is based on division not multiplication.

The symbol before the expression in the brackets calls for taking the opposite of that expression giving the value of the expression of -1/64. Is that what you calculated?

I asked Chatgpt to solve the equation. I was not able to find a format for the question that got the answer. Please respond with a comment if you can get Chatgpt to solve.

The sine of 30^{o} or sin(30^{o})= ½. The purpose of the sine inverse function is to reverse what the original function does. The minus sign denotes the sine inverse function as in sin^{-1}(½) = 30^{o}. What else does the minus sign represent?

Another example is the limit function. ‘*The limit of a function (if it exists) for some x-value, a, is the height the function gets closer and closer to as x gets closer and closer to a from the left and the right.’** * A one sided limit is the height in the graph of the function that the function gets closer and closer to from the left or the right. A minus sign indicates the one sided limit from the left. The minus sign before the 5 in the equation below means that x has a limit of 5 from the left.

(1)

So, what’s going on with the minus sign? Why is it being used for so many different purposes? It reminds me of the difficulties people have learning English as a foreign language when presented with a sentence such as ‘There is no time like the present to present you with this present’.

No wonder accountants and physicists avoid negative numbers.

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]]>The post Learning Negative Numbers appeared first on Negative Numbers.

]]>The difficulty with learning *negative numbers *is widely acknowledged in the *teaching* profession. In *Making Sense of Negative Numbers *by Cecelia Kilhamn (2011) the goal of the research is ‘*to better understand why the topic of negative numbers is so difficult to teach and to learn.’*

This difficulty does not mean that there is something wrong with the subject. However, it is recognised children do have more difficulty with *learning* the *mathematical concept* of negative numbers than other concepts such as positive numbers, addition, subtraction and multiplication. These concepts exist in the natural world.

Children are great learners. See how they pick up language so easily. I started studying Italian during the lockdown and struggle. I will never develop as good an understanding or accent as a child would.

The question is why do our children struggle so much with the concept of negative numbers.

Many teachers use metaphors to help with the teaching of negative numbers. These include apples and negative apples, money transactions, motion along a path, arrows on a number line, different coloured beads, stairs and thermometers. These are problematic. For instance if the temperature is -3^{o} and gets -8^{o} colder, then the temperature would be -11^{o} or would it be 5^{o} because in arithmetic -3 – (-8) = 5. Explain that to children.

The idea of using metaphors is a good one. The metaphors mentioned earlier work fine for addition and subtraction but not as well for negative numbers. Metaphors are based on the real world, so it is unlikely an adequate metaphor will ever be found to properly represent a concept that is impossible in the real world.

Ask any child which of the following lines A and B is bigger and they will quickly identify A as the biggest.

Now show them the number line and ask them which number is bigger -5 or 3. It is not surprising that children find the concept of negative numbers hard to deal with.

We expect children to understand that in some ways -5 is smaller than 3 and in other ways -5 is bigger than 3.

*Cognitive dissonance* is described as *‘The mental discomfort that results from holding two conflicting beliefs, values, or attitudes. People tend to seek consistency in their attitudes and perceptions, so this conflict causes feelings of unease or discomfort.*’ The use of negative numbers requires our children to hold two *conflicting beliefs*.

Teachers and children are not the only ones to find difficulty with negative numbers.

Negative numbers are difficult to learn and require us to overcome conflicting beliefs. Are they really *fit for purpose* or could we come up with a better system?

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]]>The post Double Entry Bookkeeping and Negatives appeared first on Negative Numbers.

]]>Accountants have realised for a long time that negatives are very unsatisfactory so they invented the double entry bookkeeping system. This is all backed up by the simple equation that

**Assets = Liabilities + Capital**

Assets are debit accounts as debits are used to increase the amount of Assets and credits to decrease them. Liabilities and Capital are on the other side of the equation and are credit accounts as they use credits to increase these accounts and debits to reduce them.

This use of debits and credits reflects their struggle with negative numbers. Liabilities and Capital represent monies owed to other people and Assets represent moneys the company owns. Accountants have recognised that monies owed to other people are still positive in their own way. However, they are different to the positive money a company owns.

Fra Luca Bartolomeo de *Pacioli* (c. 1447 – 1517) is known as “The Father of *Accounting* and Bookkeeping” as it was he who first wrote about *double entry* bookkeeping. He was born in Borgo in Tuscany but spent time in Venice where he published a work describing the Venetian use of the double entry bookkeeping system and the use of debits and credits as opposed to negatives.

I am learning Italian and while speaking with an Italian tennis player, I completely forgot the word for before. Now this is one of the basic words you need in Italian. I remembered afterwards of course that the ‘prima di’ was used for before. Interestingly, ‘la prima’ and ‘il primo’ are the words for first. So, the Italians intuitively considered that there was nothing before the first and used first to represent before. It seems that they did not even consider 0 before 1, let alone consider negative numbers. It is not surprising that they invented double entry accounting without using negative numbers.

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]]>The post The History of Negative Numbers – Part 1 appeared first on Negative Numbers.

]]>The history of negative numbers shows that they have not been widely accepted for long. Only positive rationals interested Euclid (300 BCE) as solutions.

They appeared in China during the Han dynasty (202 BCE to 220 CE) in the *Nine Chapters on the* *Mathematical Art*. They had a system of rods of different colours for positive and negative.

Diophantus lived circa 214 CE and although he did not consider negatives to be numbers, he used subtraction in his workings and did accept that multiplying two negative numbers together would yield a positive number. However in his Greek language work on mathematics, *Arithmetica*, he wrote that the equation 4 = 4x + 20 was absurd because he could not accept negative numbers.

Brahmagupta was an Indian mathematician who lived between 598 and 668 CE. He wrote his work on mathematics, *Brāhmasphuṭasiddhānta*, in verse in the Sanskrit language and did not use any mathematical notation. In his work, he describes positives as ‘fortunes’ and negatives as ‘debt’. He said in one verse that “a fortune subtracted from zero is a debt”. He wrote in another that the “The sum of positive and negative, if they are equal, is zero”. It is quite inspirational that someone could write about mathematics in verse.

Muḥammad ibn Mūsā al-Khwārizmī (c.780 – c850), from Persia which is now Iran, wrote ‘The Compendious Book on the Calculation of Al-jabr and Muqabala’ in arabic. The word ‘Algebra’ comes from the Arabic word ‘al-jabr’, which translates as ‘reunion of broken parts’ or ‘bone setting’. The word muqabala translates as ‘balancing’ and together the words can be better understood as ‘restoring and balancing’. He classified quadratic equations into six classes that allowed him to avoid negative coefficients.

Another Persian, *Omar Khayyam* (1048-1141) wrote his work ‘On the Proofs of the Problems of Algebra and Muqabala’ in Persian. He deals with cubic equations in his work and only looks at positive solutions.

So, there was not much support for negative numbers at this stage of history. Part 2 outlines the development of negative numbers during the second millennium.

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