Tag Archives: education

Why does -1 * -1 = +1 ?

I have been asked this by perplexed teachers, parents and kids:

Why did them silly mathematicians decide that the multiplication of two negative numbers is positive? What’s the logic?

The first thing is to acknowledge that this is a convention. We could define -1*-1 to be whatever we want. But there is good reason to decide it should be +1.

Let’s try to calculate:

-1 * ( 1 + -1)

Well, -1 is the negative of +1, which mean, by definition:

-1 + 1 = 0

So, -1 * (1+ -1) = -1 * 0

and multiplication by zero must be zero, so we have:

-1 * (1+ -1) = -1 * 0 = 0

But, if we open the parenthesis, we have:

-1 * (1 + -1) = -1*1 + -1*-1

now, multiplication by 1 is doing nothing, so -1*1 = -1, so we get:

-1 + -1*-1 = -1*1 + -1*-1 = -1*(1+ -1) = -1*0 = 0

hence, -1 is the negative of -1*-1, so the latter has to be +1.

What have we used? We used the definition of zero and 1, we used the definition of the negative of a number, and we used the law of distributivity (by which we can open the parenthesis).

We could decide, say, that -1*-1 = -1, but then we’d have to abandon one of the principles mentioned above, and that would make arithmetic rather unpleasant.

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EyezMaze

I not the gamer I used to be, now that I have a toddler to take care of. So finding a computer game which is both fun for my 3-year-old boy and interesting enough for me is not easy. We tried the incredible machine – which is a very very cool game. But my toddler is a tad to young for it, and he enjoys the much more after I show him how to finish the levels.

Another great download is phun – basically a Master’s degree work by Emil Ernerfeldt, which enables you to build all kinds of stuff – from bouncing balls to rockets and strange machines. If only I had Phun when I was a boy!

But both Phun – great fun that it is – and the incredible machine were made for children – not for toddlers. Toddler games are usually unimaginative, silly, and utterly dull. Toddlers need large buttons with an easy operation. They need compensation when they get stuck or do the wrong thing. If your character just dies or explodes, they cry. They need something more emotionally- friendly, where a wrong move still gives an interesting outcome, or better some insight on what the right thing to do was. Where do you find such a game?

Then comes ON.

I don’t know his name, but his creations are just what I sought for. They are witty, hilarious, ingenious, interesting, sometimes hard – but always trivial to function. These flash games are all that is good in computers and can’t be made without one. Most of his games are played by pressing on large, well defined buttons. It always does something – your job is to discover the correct order in which to press the buttons.

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Towards a new theory of Democracy – general framework

In the first entry I tried to show that many of us have some intuitive notion of Democracy that goes beyond the “majority rules” principle. In this and the following few entries I intend to dismantle that principle altogether.
In this post I’ll give the essentials – a short description of the claims which I intend to convince you of.
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Notes towards a new theory of Democracy – I

A few years ago, a friend of mine was working as a teacher, filling in for a sick colleague on a “citizenship” class. The teenagers in class had very little interest in the subject, as expected in normal schools. The substitute teacher found the class impossible to teach. “We know everything there is to know” they told him. “Israel is a Democratic state. That’s all there is to it, really”.
“And do you know what democracy is”? He demanded.
“Sure we do: it’s the rule of the majority”.

“Ok”. said my friend. “As you already know it all, we won’t study today. However, your teacher asked me to make sure you don’t leave the classroom a mess. So we thought maybe we’ll make a simple arrangement. I propose this: as of today, and throughout this year, all pupils sitting in rows 1-2 will stay a few minutes after school is over, to clean the classroom. Next year it will be rows 3-4, and the year after that it will be rows 5-6”.

Naturally, rows 1-2 were vehemently resentful. So he proposed a vote. Democracy, right? The class agreed, and naturally enough rows 3-6 were for, rows 1-2 against, so the proposal won by 2/3.
Rows 1-2 were now very upset. They declared the result “undemocratic”. “But the result passed by a majority!” claimed the teacher. So a discussion ensued, in which rows 1-2 spoke and the others had very little to say. Eventually they agreed that the vote violated the right for free speech, because they didn’t have a chance to discuss it properly.
He gave them 10 minutes. The second vote was identical to the first. Rows 1-2 were baffled. It seemed awfully undemocratic to them, but they couldn’t spell out the reason.
Neither could most people.

This blog is mainly about that reason. What exactly is Democracy? What is it good for? What can it do for you and how can you use it and improve it?