Tag: University of Maryland

To state the obvious, it’s in the numbers!

A wonderful, and educational, item on The Conversation.

I was listening to an item on the BBC yesterday morning and counting came up. The need for counting was incredibly early on in our history. Here is a copy of the part of the introduction to The Universal History of Numbers: From Prehistory to the Invention of the Computer.

A riveting history of counting and calculating from the time of the cave dwellers to the late twentieth century, The Universal History of Numbers is the first complete account of the invention and evolution of numbers the world over. As different cultures around the globe struggled with problems of harvests, constructing buildings, educating their citizens, and exploring the wonders of science, each civilization created its own unique and wonderful mathematical system. 

Dubbed the “Indiana Jones of numbers,” Georges Ifrah traveled all over the world for ten years to uncover the little-known details of this amazing story. From India to China, and from Egypt to Chile, Ifrah talked to mathematicians, historians, archaeologists, and philosophers. He deciphered ancient writing on crumbling walls; scrutinized stones, tools, cylinders, and cones; and examined carved bones, elaborately knotted counting strings, and X-rays of the contents of never-opened ancient clay accounting balls. Conveying all the excitement and joy of the process of discovery, Ifrah writes in a delightful storytelling style, recounting a plethora of intriguing and amusing anecdotes along the way.

Now to that article on The Conversation.

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From thousands to millions to billions to trillions to quadrillions and beyond: Do numbers ever end?

The number zero was a relatively recent and crucial addition − it allows numbers to extend in both directions forever. pixel_dreams/iStock via Getty Images Plus

Manil Suri, University of Maryland, Baltimore County

Curious Kids is a series for children of all ages. If you have a question you’d like an expert to answer, send it to curiouskidsus@theconversation.com.

Why don’t numbers end? – Reyhane, age 7, Tehran, Iran

Here’s a game: Ask a friend to give you any number and you’ll return one that’s bigger. Just add “1” to whatever number they come up with and you’re sure to win.

The reason is that numbers go on forever. There is no highest number. But why? As a professor of mathematics, I can help you find an answer.

First, you need to understand what numbers are and where they come from. You learned about numbers because they enabled you to count. Early humans had similar needs – whether to count animals killed in a hunt or keep track of how many days had passed. That’s why they invented numbers.

But back then, numbers were quite limited and had a very simple form. Often, the “numbers” were just notches on a bone, going up to a couple hundred at most.

How numbers evolved throughout the centuries.

When numbers got bigger

As time went on, people’s needs grew. Herds of livestock had to be counted, goods and services traded, and measurements made for buildings and navigation. This led to the invention of larger numbers and better ways of representing them.

About 5,000 years ago, the Egyptians began using symbols for various numbers, with a final symbol for one million. Since they didn’t usually encounter bigger quantities, they also used this same final symbol to depict “many.”

The Greeks, starting with Pythagoras, were the first to study numbers for their own sake, rather than viewing them as just counting tools. As someone who’s written a book on the importance of numbers, I can’t emphasize enough how crucial this step was for humanity.

By 500 BCE, Pythagoras and his disciples had not only realized that the counting numbers – 1, 2, 3 and so on – were endless, but also that they could be used to explain cool stuff like the sounds made when you pluck a taut string.

Zero is a critical number

But there was a problem. Although the Greeks could mentally think of very large numbers, they had difficulty writing them down. This was because they did not know about the number 0.

Think of how important zero is in expressing big numbers. You can start with 1, then add more and more zeroes at the end to quickly get numbers like a million – 1,000,000, or 1 followed by six zeros – or a billion, with nine zeros, or a trillion, 12 zeros.

It was only around 1200 CE that zero, invented centuries earlier in India, came to Europe. This led to the way we write numbers today.

This brief history makes clear that numbers were developed over thousands of years. And though the Egyptians didn’t have much use for a million, we certainly do. Economists will tell you that government expenditures are commonly measured in millions of dollars.

Also, science has taken us to a point where we need even larger numbers. For instance, there are about 100 billion stars in our galaxy – or 100,000,000,000 – and the number of atoms in our universe may be as high as 1 followed by 82 zeros.

Don’t worry if you find it hard to picture such big numbers. It’s fine to just think of them as “many,” much like the Egyptians treated numbers over a million. These examples point to one reason why numbers must continue endlessly. If we had a maximum, some new use or discovery would surely make us exceed it.

The symbols of math include +, -, x and =.

Exceptions to the rule

But under certain circumstances, sometimes numbers do have a maximum because people design them that way for a practical purpose.

A good example is a clock – or clock arithmetic, where we use only the numbers 1 through 12. There is no 13 o’clock, because after 12 o’clock we just go back to 1 o’clock again. If you played the “bigger number” game with a friend in clock arithmetic, you’d lose if they chose the number 12.

Since numbers are a human invention, how do we construct them so they continue without end? Mathematicians started looking at this question starting in the early 1900s. What they came up with was based on two assumptions: that 0 is the starting number, and when you add 1 to any number you always get a new number.

These assumptions immediately give us the list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, and so on, a progression that continues without end.

You might wonder why these two rules are assumptions. The reason for the first one is that we don’t really know how to define the number 0. For example: Is “0” the same as “nothing,” and if so, what exactly is meant by “nothing”?

The second might seem even more strange. After all, we can easily show that adding 1 to 2 gives us the new number 3, just like adding 1 to 2002 gives us the new number 2003.

But notice that we’re saying this has to hold for any number. We can’t very well verify this for every single case, since there are going to be an endless number of cases. As humans who can perform only a limited number of steps, we have to be careful anytime we make claims about an endless process. And mathematicians, in particular, refuse to take anything for granted.

Here, then, is the answer to why numbers don’t end: It’s because of the way in which we define them.

Now, the negative numbers

How do the negative numbers -1, -2, -3 and more fit into all this? Historically, people were very suspicious about such numbers, since it’s hard to picture a “minus one” apple or orange. As late as 1796, math textbooks warned against using negatives.

The negatives were created to address a calculation issue. The positive numbers are fine when you’re adding them together. But when you get to subtraction, they can’t handle differences like 1 minus 2, or 2 minus 4. If you want to be able to subtract numbers at will, you need negative numbers too.

A simple way to create negatives is to imagine all the numbers – 0, 1, 2, 3 and the rest – drawn equally spaced on a straight line. Now imagine a mirror placed at 0. Then define -1 to be the reflection of +1 on the line, -2 to be the reflection of +2, and so on. You’ll end up with all the negative numbers this way.

As a bonus, you’ll also know that since there are just as many negatives as there are positives, the negative numbers must also go on without end!

Hello, curious kids! Do you have a question you’d like an expert to answer? Ask an adult to send your question to CuriousKidsUS@theconversation.com. Please tell us your name, age and the city where you live.

And since curiosity has no age limit – adults, let us know what you’re wondering, too. We won’t be able to answer every question, but we will do our best.

Manil Suri, Professor of Mathematics and Statistics, University of Maryland, Baltimore County

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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This article was written for those a great deal younger than I am. But, to be honest, it is a fascinating account of something so utterly basic to humans and human cognition.

Enjoy!

Dogs and Noise.

This is very interesting!

Belinda, who lives along Hugo Rd., as we do, sent me late last week a very interesting article on how well dogs can tune out noise.

See you yourself.

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How well do dogs hear their name in the midst of chaos?

Analysis by Dr. Karen Shaw Becker

August 1st, 2019

Humans have the ability of selective hearing, enabling us to tune in to one person speaking, for instance, even in the middle of a noisy room. This phenomenon, dubbed the cocktail party effect, is not unique to humans, however.

Research published in the journal Animal Cognition revealed that not only can dogs recognize their names in noisy conditions, they may do so better than human infants in a similar situation.1 It’s a finding that could be particularly useful for handlers of working or service dogs, who may find themselves needing to attract their dog’s attention in a chaotic environment.

It’s been suggested that hand signals may be best for this, but a vocal command may be preferable, especially since dog’s may miss hand signals as they pay attention to what’s going on in their environment.2

Dogs pick up their names even in noisy environments

For the study, researchers from the University of Maryland used a variety of dog breeds, including pets, service dogs and search-and-rescue dogs, and their owners. The dogs were placed in a booth with their owner, where background noise was played at increasingly loud levels.

Amidst the background noise, a loudspeaker played recordings of a woman speaking the dog’s name or another dog’s similar-sounding name. The dogs listened more intently to the speaker playing their own name and were able to recognize it at varying levels of background noise, up until the noise became louder than the recording of their names.3

“This surpasses the performance of 1-year-old infants,” the researchers noted. Comparatively, adult humans can pick their names out even when background noise is louder than their name. Perhaps not surprisingly, in the study the working dogs performed better at the name recognition than pet dogs.

“I suspect one of the reasons working dogs do better is because people use their names more consistently,” study co-author Rochelle Newman, Ph.D., told National Geographic. “We often end up using nicknames so much.”4 In addition, the researchers concluded:5

“Overall, we find better performance at name recognition in dogs that were trained to do tasks for humans, like service dogs, search-and-rescue dogs, and explosives detection dogs. These dogs were of several different breeds, and their tasks were widely different from one another.

This suggests that their superior performance may be due to generally more training and better attention. In summary, these results demonstrate that dogs can recognize their name even in relatively difficult levels of multitalker babble, and that dogs who work with humans are especially adept at name recognition in comparison with companion dogs.”

Dogs also cue in on other dog and human emotions

Dogs are very in tune with their environments, including the actions and emotions of those around them — both dogs and people. For instance, dogs have been found to display rapid mimicry of the other dogs’ body movements, particularly a play bow and facial expression (a relaxed, open mouth).6

When dogs mimicked each other, their play sessions lasted longer, which suggests it increased the dogs’ motivation to play and possibly strengthened the dogs’ relationship. Given that dogs mimic the emotional states of other dogs, dogs may also be able to mimic their owners’ facial expressions, especially if they’re closely bonded.

“Emotional contagion is a basic form of empathy that makes individuals able to experience others’ emotions. In human and non-human primates, emotional contagion can be linked to facial mimicry, an automatic and fast response (less than 1 second]) in which individuals involuntary mimic others’ expressions,” researchers wrote in Royal Society Open Science. “… All these findings concur in supporting the idea that a possible linkage between rapid mimicry and emotional contagion (a building-block of empathy) exists in dogs.”

The fact that dogs may mimic their owner’s facial expressions and are capable of selective hearing to pick their name out of a host of background noise adds even more understanding of why dogs and humans share such strong bonds.

Dogs associate words with objects

In dog and human communication, it remains a bit of a mystery whether dogs are responding to humans’ words, tone of voice, gestures or other cues — or all of the above.

The featured study suggests dogs do, indeed, respond to their names when spoken verbally, and past research has also shown dogs associate certain words with objects and seem able to form mental pictures that correspond to words they’ve been taught.7 Dogs also tune in to the tone of your voice,8 and may have a heightened response to praise delivered in an upbeat tone. There’s still some debate, though, over whether dogs really understand what you’re saying.

“Some of the old guard say the name is just a bit of noise that is made by the handler, and the dog is familiar with the handler’s voice, so anything the handler says is going to get their attention,” Stanley Coren, Ph.D., professor emeritus of psychology at the University of British Columbia, told National Geographic.9

Yet in the featured study, the dogs responded even though a stranger’s voice said their names, adding more evidence that dogs may understand more than we give them credit for. And, for anyone wondering, there’s evidence that cats also know their names, much like dogs and even when spoken by someone other than their owner.

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That dogs, perhaps not all dogs, understand far more than we give them credit is no real surprise. For a creature that bonds so close to humans and has done for a long time we still don’t really know how they function. Well certainly in the head department!

But that doesn’t reduce by one iota our love for them. They are a very special animal.

Anthropocene era gaining legs

We really may be on the verge of a new geological period.

Just a couple of weeks ago, on the 16th May, I wrote an article called The Anthropocene period.  It was based on both a BBC radio programme and a conference called “The Anthropocene: A New Epoch of Geological Time?”

So imagine my surprise when I collected this week’s copy of The Economist from my mail-box last Saturday.  The cover page boldly illustrated a lead article within, as this picture shows.

US edition, May 28th

The leader is headlined, ‘Humans have changed the way the world works.  Now they have to change the way they think about it, too.’  The first two paragraphs of that leader explain,

THE Earth is a big thing; if you divided it up evenly among its 7 billion inhabitants, they would get almost 1 trillion tonnes each. To think that the workings of so vast an entity could be lastingly changed by a species that has been scampering across its surface for less than 1% of 1% of its history seems, on the face of it, absurd. But it is not. Humans have become a force of nature reshaping the planet on a geological scale—but at a far-faster-than-geological speed.

A single engineering project, the Syncrude mine in the Athabasca tar sands, involves moving 30 billion tonnes of earth—twice the amount of sediment that flows down all the rivers in the world in a year. That sediment flow itself, meanwhile, is shrinking; almost 50,000 large dams have over the past half- century cut the flow by nearly a fifth. That is one reason why the Earth’s deltas, home to hundreds of millions of people, are eroding away faster than they can be replenished.

There’s also a video on The Economist website of an interview with Dr. Erle Ellis, associate professor of geography and environmental systems at the University of Maryland.  That video link is here.

That Economist lead article concludes,

Recycling the planet

How frightened should people be about this? It would be odd not to be worried. The planet’s history contains many less stable and clement eras than the Holocene. Who is to say that human action might not tip the planet into new instability?

Some will want simply to put the clock back. But returning to the way things were is neither realistic nor morally tenable. A planet that could soon be supporting as many as 10 billion human beings has to work differently from the one that held 1 billion people, mostly peasants, 200 years ago. The challenge of the Anthropocene is to use human ingenuity to set things up so that the planet can accomplish its 21st-century task.

Increasing the planet’s resilience will probably involve a few dramatic changes and a lot of fiddling. An example of the former could be geoengineering. Today the copious carbon dioxide emitted to the atmosphere is left for nature to pick up, which it cannot do fast enough. Although the technologies are still nascent, the idea that humans might help remove carbon from the skies as well as put it there is a reasonable Anthropocene expectation; it wouldn’t stop climate change any time soon, but it might shorten its lease, and reduce the changes in ocean chemistry that excess carbon brings about.

More often the answer will be fiddling—finding ways to apply human muscle with the grain of nature, rather than against it, and help it in its inbuilt tendency to recycle things. Human interference in the nitrogen cycle has made far more nitrogen available to plants and animals; it has done much less to help the planet deal with all that nitrogen when they have finished with it. Instead we suffer ever more coastal “dead zones” overrun by nitrogen-fed algal blooms. Quite small things, such as smarter farming and better sewage treatment, could help a lot.

For humans to be intimately involved in many interconnected processes at a planetary scale carries huge risks. But it is possible to add to the planet’s resilience, often through simple and piecemeal actions, if they are well thought through. And one of the messages of the Anthropocene is that piecemeal actions can quickly add up to planetary change.

We are living in interesting times!

Finally, more of Dr. Ellis may be watched on the following YouTube video.