Uncategorized – Page 15

Can we find the inverse of a function which is not one-to-one? (part two)

So, in the last post we had seen that while the sin function is not one-to-one and thus doesn’t have an inverse, so long as we restrict it to a given domain, you will find that it is invertible. The domain that we found (indeed chose), was between $[-\frac{\pi}{2},\frac{\pi}{2}]$ . It’s inverse was a function with domain $[-1,1]$ . The name of the inverse is arcsin(x). How can we use this to help us to solve problems?

Well, what if I asked you to solve:

$sin(x)=\frac{1}{2}$

You might think that because we have found the inverse of sin, that we can simply say that the solution to this is:

$x=arcsin\frac{1}{2}$

Well, because arcsin is itself a one-to-one function, restricted to the domain $[-1,1]$ this will clearly give us a single number (the answer is about 0.52):

Is that it then? Well, let’s look at the graph of sin(x) and see if this is the only solution to $sin(x)=\frac{1}{2}$ :

In fact, clearly there are an infinite number of solutions to the equation $sin(x)=\frac{1}{2}$ and we have just caught the one within the region $[-\frac{pi}{2},\frac{\pi}{2}]$ .…

By Jonathan Shock| March 16th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Can we find the inverse of a function which is not one-to-one? (part one)

Asking what the inverse of a function is, is the same as asking what is the function that will undo this function?

What is the inverse of the function $f(x)=x+3$ ? That is asking the question, if I put a number into this function (call that number a), it will give me another number (call it b). What is the function which, whatever number I put in, when applied to the number that comes out from the first function will be the original number. That is to say:

If f(a)=b. Then what is the function g for which g(b)=a. That would give you g(f(a))=a. g Is then the inverse of f and we can write $g(x)=f^{-1}(x)$ . g(x) is the thing that undoes f. Put simply, composing the inverse of a function, with the function will, on the appropriate domain, return the identity (ie. not do anything to the number you put in).…

By Jonathan Shock| March 15th, 2016|Uncategorized|3 Comments

Mathemafrica and the NEF #2

On Monday we worked hard to bring our Maths exhibition exhibition to the Next Einstein Forum venue, and to setup everything. Find below a few pictures! It was hard, time was short, logistics was tricky, but: the venue is amazing! We got the prime spot of the venue, just right at the main entrance – and the whole NEF organisation is very professional!

Find below some pictures of our “setup day”:

Packing all items (inside and on top of the car).

Mounting the exhibition (roll-ups, images on the wall, screens, touch-screens). A big part of the exhibition is about “Mathematics of Planet Earth”, we will also launch a new competition about open exhibits – so everybody can contribute with own exhibits.

The local NEF tech team help us to fix the images on the wall.

This is the final setup (the computers already turned off). On Tuesday, the exhibition start around 2 pm, in the morning the presidents of Senegal and Rwanda are expected to be at the NEF (and might also visit the exhibition)

This could be a good spot to take pictures… today many people already took selfies in front of our images.…

By Andreas Daniel Matt| March 8th, 2016|Uncategorized|0 Comments

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Mathemafrica and the Next Einstein Forum (NEF) #1

English, Event, News, Uncategorized

Mathemafrica and the Next Einstein Forum (NEF) #1

Dear Mathemafrica readers,

I am sitting at the AIMS-Senegal institute in Mbour (about 1.5 hours drive South of Dakar) and together with my team member Sebastian and the AIMS-Senegal staff and students, we are preparing an interactive mathematics exhibition (as part of the IMAGINARY – open mathematics project).. It will be shown as of March 8 at the Next Einstein Forum (NEF), to be held at a huge conference venue just outside Dakar.

There will be many ministers, scientists, politicians, even presidents from (apparently all fifty-four) African countries – and also many international guests – joining the NEF, with the goal to discuss about scientific innovations, collaborations and solutions in Africa!

We have to plan to blog live from the NEF, with insights and views from participants. And of course, we will let you know about our exhibition, about a new competition, we will launch and everything happening around!

Prepare yourself for the NEF at:

gg2016.nef.org/

iameinstein.org

www.facebook.com/NextEinsteinForum

And the Twitter hashtag: #AFRICASEINSTEINS

Please find a picture from our first technical setup yesterday at the AIMS Institute.…

By Andreas Daniel Matt| March 6th, 2016|English, Event, News, Uncategorized|0 Comments

Computational Complexity; A Soft Approach

Motivation: Mathematics for the Masses

It is my firm conviction, and I preach it when ever I can, that one day in
the near future, mathematics shall save us all. A ”grand claim,” I hear you
say; but not at all, mathematics is believed by many to be the language of
Mother Universe, and indeed, those who have adopted it as a native tongue
have been granted glimpses into her secrets. Intuitively, my claim is not hard
to defend, given the pervasive influence that technology has over our lives; from
health to communication, entertainment, art and culture; science has become an
indispensable companion. Amongst the sciences, mathematics is the common
denominator that binds them all. It is the life blood of all other scientific inquiry.
As the world faces seemingly intractable challenges, be it global health, world
peace or universal prosperity, it has become imperative that more of us engage
in scientific exploration and innovation, for, if history is anything to go by, this
is where we shall find the answers we seek.…

By Tapiwa Chadenga| March 3rd, 2016|Uncategorized|0 Comments

Ishango, Nyakubereka Svomhu

Tikaverenga nezve matangiro nemakuriro akaita ruzivo resvomhu pasi rose, tino katyamadzwa kuti hakuna zvakawanda zvakanyorwa pamusoro wezvakaitwa nevanhu Africa yevatema. Kushaikwa kwezvinyorwa uku kunopa kuti tifunge kuti hapana ruzivo rwesvomhu rwakakosha runobva munzvimbo iyi. Asi ichi ichokwadi here? Tichitarisa matangiro akaita svomhu tinoona kuti yaive mbesa yekuti vanhu vakwanise kubudirira mutsinhana nemukurima. Makore churu apfuura Africa yevatema yekawona kukwira kwemarudzi avanhu kwakawanda; kukwirira uku kuchibva mutsinhanha nemukurima; Kwakaita vanonzi vaNok vekuma dokero kweAfrica, kwoitawo vaBuganda kumabvazuva, koitawo vainzi vanhu veGreat Kongo vachiri kuwanikwa pakati peAfrica; tisinga kanganwi rudzi rwekwaMutapa vaiva vakavaka Dzimba Dzemabwe; pamarudzi ose awa nemamwewo akakwira nekudonha hapana here rudzi rwakaumba ruzivo rwesvomu, sokuti mamwe acho akanga akabudikira zvakanyanya tichitarisa pasi rose nguva iyoyo. Mubvunzo uyu wakandipa kuti ndiite tsvagiridzo munyaya iyi, izvi ndizvo zvimwe zvakakosha zvandakawana.

Pano ganhurana nyika dzeD.R. Congo neUganda ndipo pane muromo werwizi rweNaire (Nile) unonzi Lake Edward. Makore makumi maviri ezviuru apfuura (25,000 years ago), pane musha wevanhu wakadzika midzi pamuganhu uyu.…

By Tapiwa Chadenga| March 1st, 2016|Uncategorized|1 Comment

Mathematics and Science are the keys to unlocking Africa’s potential

Angelina Lutambi was born into a peasant family in Tanzania’s Dodoma region, where HIV/AIDS has decimated much of the population. Her future could easily have been bleak – but Angelina had a keen aptitude for maths. She financed her own schooling by selling cold drinks with her siblings and was awarded a grant to study at the University of Dar Es Salaam.

How to reduce the fear of mathematics

I sat this morning reading a little of The Book of Life, by Krishnamurti – something which I like to browse through and ponder from time to time. This morning’s meditation somehow felt very apt as I attempt to get almost 800 students to enjoy mathematics, and learn its techniques as well as its beauty. The meditation was the following:

How is the state of attention to be brought about? It cannot be cultivated through persuasion, comparison, reward or punishment, all of which are forms of coercion. The elimination of fear is the beginning of attention. Fear must exist as long as there is an urge to be or to become, which is the pursuit of success, with all its frustrations and tortuous contradictions. You can’t teach concentration, but attention cannot be taught just as you cannot possibly teach freedom from fear; but we can begin to discover the causes that produce fear, and in understanding these causes there is the elimination of fear.

…

By Jonathan Shock| February 20th, 2016|Uncategorized|3 Comments

First week of lectures

So the first week of lectures has ended. In MAM1000 we have only dealt with sets and functions thus far, but in great detail using set builder and interval notation. In our first tutorial we have even started using parametric equations. The Modulus(Absolute value) Function had been added by the end of the week as well. Modulus function is nice to work with as the answer coming out of it must always be positive. If a variable (x) is shown in modulus it must be its non-negative version for example: if x in itself has a negative value then the value of x after modulus has been applied will be -x as this will then be a positive number. Similarly if x is positive then the output will be x. |x| is how modulus is written. We have now also learned that |x+y|<=|x|+|y|, this is called the Triangle inequality and is very important for future use.…

By Francois Kruger| February 20th, 2016|Uncategorized|2 Comments

Absolute values and inequalities

Things that I learnt today. Emphasis on the I, I couldn’t make for MAM1000W today.

Absolute value definition
Properties of absolute values
Rules for inequalities

Absolute value

The absolute value of a number represents the distance between that number and $0$ on the real number line. Absolute value of a number $n$ is denoted by $|n|$ which is equal to $\sqrt{n^{2}}$ which is from the distance formula. Since it is the distance between $0$ and $n$ Hence $|n|=n$ if $n \geq 0$ and $|n|=-n$ if $n < 0$