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## 'Twizzle Arithmetic' printed from http://nrich.maths.org/

In

Arrow Arithmetic 2 and

Arrow Arithmetic 3 (published in February 2007), the problems
were about finding a way to do arithmetic using geometry where
numbers are represented by a double arrow. You might need to refer
back to these problems to remind yourself of the idea.

Now take a look at these four audio/visual clips to see the four
arithmetic operations in action:

And, in case you'd prefer it in text, here's a summary of the
geometrical technique:

Addition |
Make all the unit arrows agree in length and direction. Then
arrange the number arrows in a nose-to-tail chain. The answer is
represented by a number arrow that stretches from the tail of the
first arrow in the chain to the head of the last arrow in the
chain. |

Subtraction |
Rotate the number arrow of the the number to subtract by a half
turn with respect to its unit arrow. Then add. |

Multiplication |
Rotate, translate, and scale until the unit arrow of one number
is the same as the number arrow of the other. Ignore these two
equated arrows. The product is formed from the remaining number and
unit arrows. |

Division |
Take the reciprocal of the divisor by swapping its number and
unit arrows. Then multiply. |

Now let's add a twist!
Try adding the green arrow-pair (which I'll call a

twizzle ) to the blue twizzle using
the addition rule. Here's an animation in which to experiment - you
should find that the grey twizzle represents the answer. Check
you're on the right track by looking at

this a/v
clip .

Full Screen
Version
This text is usually replaced by the Flash movie.

Now let's try multiplication. In the next animation, the grey
twizzle is the product of the blue twizzle and the green
twizzle:

Full Screen
Version
This text is usually replaced by the Flash movie.

Check you can do the multiplication geometrically yourself.

Click on the twizzle spots and notice the names that these numbers
have. These are shown when you roll the mouse over the number
arrow. Each number has two names (click the spot again to see the
second 'polar' name).

Notice that a number arrow of length

r turned through an angle

d degrees is called

r cis (d) .

Notice that

1 cis (90) has
the special name

i .

What is the value of

i
^{2} ?

Here's the problem :

Calculate (1+i)(1+i)
geometrically and by multiplying out the brackets. Use Pythagoras'
theorem to show that both methods give the same answer.