Category Archives: Maths & Science

Megabytes, microSD cards and typing

We take for granted how much ‘real world’ data our storage devices hold. A quick example:

One byte of storage is 1 typed character on the screen/page.

  • 1 megabyte (1 million bytes) is 1 million characters; approx 500 typed pages (roughly!)
  • 1 gigabyte (GB) is 1,000 megabytes:  about 500,000 typed pages
  • 1 terabyte is 1,000 gigabytes: about 500,000,000 typed pages (500 million)

My phone’s microSD card is tiny (11mm x 15mm) and yet holds 32 GB.  That is 16 million typed pages.

The poor old Encyclopedia Britannica was about 33,000 printed pages.  So the little phone memory card holds about 484 Britannicas worth of data.

Contrails Fun #2–Angles and Distances

After last years first episode with contrails, another one popped up today.  During a late afternoon walk, on a sunny Winter’s day, I saw a great contrail right over the Melbourne area:

 

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[click on all pics to zoom in]

The FlightRadar24 app on the phone said NZ to South Africa and at about 34,000 ft (~ 10,000 m). So I continued the walk and a bit later I looked left (~West) and the day was so clear, I could still see it ‘miles’ away: I mean it making fresh trails. I firstly snapped a – zoomed in – picture:

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Then I checked FlightRadar24 again. It was way over near Skipton:

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I guessed its angle above the horizon, using the cool outstretched hand-fist rule. I’d estimated between 5 and 10 degrees.

So later on at home a bit of high-school maths. And some help from Google as I’d forgotten the basics! :

Skipton is about 130 km from me. The plane is 10 km off the ground. What is the angle (theta) from the horizon?

So, Tan(theta) = 10/130  [“opposite over adjacent”]

Tan(theta) = 0.0769

theta = 4.4 degrees

Very close to my 5 to 10 degrees. Yay me.

How many credit cards per person?

Another quick and fun example of everyday deceptively LARGE numbers action. I recently used my first ‘virtual’ or temporary credit card, for a USA purchase. This site generated a single-use card, pre-loaded with $n USA (but it couldn’t be recharged).  Mmm, says the mathematician in me, how many cards could each person on Earth have?

Some in the head, approximate numerical pondering followed:

(a) 16 numerical digit card number.  1o16 combinations (a 1 followed by 16 zeros)

(b) 7 billion people on Earth. 7x(109).  That 7 is near 10, so I’ll round the whole thing  up to 1010

Thusly dividing (a)/(b) we can have 10 (16-10)  each (remember your exponential maths?). Which is 106       That is 1 million cards each.

32 digits and the joy of logarithmic remembering

I only recently learned about GUID’s, that is Globally Unique Identifiers, usually meaning the Microsoft one. It’s a ‘number’, but made up of the digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).  The computer dudes refer to this as Hex or Base16. We humans normally use Decimal or Base 10.

Anyway GUID’s are 32 Hex numbers long.  It wouldn’t take that long to write one out.  In fact, like this:  21EC2020-3AEA-1069-A2DD-08002B30309D.  Their main purpose is to provide a unique number for things like serial numbers. For example each Ipod has one. I’m not sure if it’s a GUID but it’s similar. You can also see them inside your Windows Registry.

So how many unique ones are there?

The answer is 16 raised to the 32nd power. That is 1632. Or roughly 1038. That is a 1 followed by 38 zeros.

Now, on this pale blue dot of ours – planet Earth – there are about 6 billion people.  Something like 109

So how many GUID’s per person?  If you remember your maths it’s quite easy: 1038 divided by 109.  That is: 10(38 – 9) or 1029. So we can each have a 1 followed by 29 zeros worth of GUIDs . Well beyond a billion billon billon each. Probably enough.

But that’s not enough maths for today. Oh no. We still have our own galaxy to think about.  The milky way has about 100 billion stars.  That is 1011 stars. If each of those stars supported a population like ours, there would be 10(11 + 9) or 1020 beings.

So with our humble 32 hex GUIDs, we could give each being 10(38 – 20) or 1018 GUIDs each. Hey, isn’t that a billion billion each? Nice.

Why no Melways of the Bush?

I remember years ago wondering why there wasn’t a Melways (street directory) of the entire State of Victoria, including the bush areas.

And here’s one reason….

This is going to be just a rough calculation, so don’t get upset about decimal places and rounding off.

An A4 sheet of paper is 21×29 cm. Leave 1cm margin (each side) for bindings etc, so we’ve got 19x27cm of printable area.

A bushwalking map is usually in the scale of 1:25,000. Hence 1cm on the map is 25,000 cm in real life, or 250 m.  It follows that 4cm of map = 1 km etc.

So an A4 map of this scale would be about 5km x 7km, that is 35 km²

Right, so how big is Victoria? According to the government it’s about 227,400 km².  So do a simple division and you’ll need about 6,500 A4s. Printing double sided means ‘only’ 3,250 pages.

And if printed in the main Melway’s scale it would be worse. The majority of their maps are in a smaller scale, 1:20,000.  So even more that 3,250 pages would be needed.