Big and small currency units

As we have seen earlier in   ,the Vietnamese dong was the world’s least valuable currency but it was recently “superseded” by the Iranian rial.

There are numerous articles on the net about the most valuable and least valuable currencies. These are typical:

As we see, the Kuwaiti Dinar (KWD) is undoubtedly the most valuable currency unit, while the Vietnamese Dong (VND) and Iranian Rial (IRR) bring up the rear.

Let us see what one KWD can get you (based on Yahoo Finance quotes on 19th Jan)

2.31 British Pounds

3.00 Euros

3.28 US Dollars

3.28 Swiss Francs

4.71 Singapore Dollars

222 Indian Rupees

385 Japanese Yen

3,957 South Korean Won

(now into the stratosphere…..)

45,360 Indonesian Rupiah

73,508 Vietnamese Dong

98,954 Iranian Rial

(Full records are not available so it is difficult to say if a KWD has ever been above 100,000 IRR, but in the last few days it has traded above 99,500)

Anyway, if you want to feel like a millionaire or billionaire, you now know where to go. Mourn the passing of the Italian Lira which was had the lowest value among the major European currencies




How far can you see?-continued

So far we have dealt with the distance which you can see while observing from various heights. This has a few more applications.

In general, a sensing device at various heights would also be able to pick up electromagnetic radiation up to the horizon. Even if you cannot fly over enemy territory, you can fly near the border and pick up what you can on the other side. Infra-red cameras can also tell you what lies below the clouds.

As you have seen, the famous U-2 could fly at 70,000 ft which enabled it to observe a radius of over 500 Km. Initially the Soviet forces had nothing to catch them with but by 1960 they had succeeded in shooting down one of them and capturing the pilot. Meanwhile improvements in satellite technology gradually made it easier for the big powers to spy on each other’s vast territories.

For some years India used retired Boeing 707s to spy on our not-so-friendly neighbours. Later there were Mig-25Rs which had a similar ceiling as the U-2s. Thus, if you were flying one of these near Amritsar you could see 500 Km into Pakistan, enough to cover anything of interest (though not their nuclear test sites which are north-west of Quetta).

Spying on Tibet is a bit more challenging as the plateau is at an average height of 15,000 feet. So your effective height is actually 70,000 minus 15,000 or 55,000 feet. That gives you a horizon of 460 Km, which doesn’t get you into the Chinese heartland but enough to see what they are up to in Tibet. As it is a very sparsely populated area, it is relatively easy to spot new constructions. Sooner or later they will link up with Nepal with superhighways and probably railway lines, which would be great news for railfans but not good news for our military.

One last mathematical derivation concerns the visibility of tall objects from a distance.

Objects above the horizon

Geometrical horizon distance

To compute the greatest distance at which an observer can see the top of an object above the horizon, compute the distance to the horizon for a hypothetical observer on top of that object, and add it to the real observer’s distance to the horizon. For example, for an observer with a height of 1.70 m standing on the ground, the horizon is 4.65 km away. For a tower with a height of 100 m, the horizon distance is 35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than 40.35 km away. Conversely, if an observer on a boat (h = 1.7 m) can just see the tops of trees on a nearby shore (h = 10 m), the trees are probably about 16 km away.

Referring to the figure at the right, the top of the lighthouse will be visible to a lookout in a crow’s nest at the top of a mast of the boat if

D_\mathrm{BL} < 3.57\,(\sqrt{h_\mathrm{B}} + \sqrt{h_\mathrm{L}}) \,,

where DBL is in kilometres and hB and hL are in metres.

Application-say you are on a ship off the coast of Dubai and you are at a height of 10 M above sea level. The Burj Khalifa is 828 M. So you can see it at 3.57 ( sqrt (10) + sqrt (828)) which is 114 Km away.

If you are looking for Mukesh Ambani’s Antilia ( a mere 170 M high) and you were at 10 M above sea level, you could see it at 3.57 (sqrt (10) + sqrt (170)) which is 58 km away.

To take something more topical, let us say a burning fishing boat in the Arabian sea has flames which are 10 M high. And your own small boat enables you to observe from 5 M high. Then it is easy to see that you will be able to see it only if you are less than 20 Km away (and less if the weather is bad, but then you should be able to see bright flames through rainy conditions). The point is, if you were more than 20 Km away you would not be able to see it. So it is not as if everyone in the Arabian sea could observe it.

Think of this when you see the news, particularly when military matters are being discussed. Also note that Wikipedia and other websites will give you a quick guide to even super-secret organizations such as RAW and the even less known ARC (Aviation Research Centre). Of course, you cannot expect as much transparency as in the US where the CIA comes for campus recruitment.

All the mathematics you need here is given in

How far can you see?

This begins an occasional series on applied mathematics. Do not disdain your boring old maths classes, though remembering up to 10th grade maths will be useful.

Today we examine a formula which tells you how far you can see on an unobstructed flat surface (land or water).

The entire derivation and formulae can be seen here:

Actually the derivation should be within the capacity of a smart 10th grader, but here we skip straight to the results:

I) Ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth’s surface is about

d \approx 3.57\sqrt{h} \,,

where d is in kilometres and h is height above ground level in metres.


  • For an observer standing on the ground with h = 1.70 metres (5 ft 7 in) (average eye-level height), the horizon is at a distance of 4.7 kilometres (2.9 mi).
  • For an observer standing on the ground with h = 2 metres (6 ft 7 in), the horizon is at a distance of 5 kilometres (3.1 mi).
  • For an observer standing on a hill or tower of 100 metres (330 ft) in height, the horizon is at a distance of 36 kilometres (22 mi).
  • For an observer standing at the top of the Burj Khalifa (828 metres (2,717 ft) in height), the horizon is at a distance of 103 kilometres (64 mi).
  • For an observer atop Mount Everest (8,848 metres (29,029 ft) in altitude), the horizon is at a distance of 336 kilometres (209 mi).

II)  With d in miles and h in feet,

d \approx 1.22\sqrt{h} \,.

Examples, assuming no refraction:

  • For an observer on the ground with eye level at h = 5 ft 7 in (1.70 m), the horizon is at a distance of 2.9 miles (4.7 km).
  • For an observer standing on a hill or tower 100 feet (30 m) in height, the horizon is at a distance of 12.2 miles (19.6 km).
  • For an observer on the summit of Aconcagua (22,841 feet (6,962 m) in height), the sea-level horizon to the west is at a distance of 184 miles (296 km).
  • For an U-2 pilot, whilst flying at its service ceiling 70,000 feet (21,000 m), the horizon is at a distance of 324 miles (521 km)

III)  Also, if d is in nautical miles, (i.e. 6080 feet or 1.15 statute miles or 1.85 km) and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:

d \approx \sqrt h

  Now let us refine this a little further. Current estimates are that an average Indian male adult is 5’5″ or 1.65 M, and that his female counterpart is 5’0″ or 1.52 M. (This may seem low, but remember that it is an average from all over the country). However the height of their eyes above the ground would be about 0.12 M less. Neglecting the thickness of footwear, the average height to be considered is 1.65-0.12 = 1.53 M for a male and 1.52-0.12 = 1.40 M for a female.

Using formula I) above, assuming there is nothing to obstruct your view:

Mr X can see for a radius of 4.42 Km and Ms Y can see for 4.22 Km.

On land, this would generally be true only on a featureless plain such as a desert. More realistically, you would be interested in knowing how far you could see from a building or a vessel at sea. Some examples are given above. More examples are:

From the top of the Qutab Minar (if you were allowed to go there) and neglecting your own height, take h = 73 M which gives the horizon at 30.5 Km. From the top of the Eiffel Tower (301 M) it will be 61.9 Km.

A more practical example would be on a small fishing boat where your eyes may be 2.5 M above the water. Then you could see for 5.6 Km.

A typical jet airliner flies at 30,000 to 35,000 feet. Let us take a round figure of 10,000 M for this. This gives a horizon of 357 Km, though it is somewhat meaningless as there will usually be clouds below it.

So far we have dealt with objects on the surface of the horizon. For objects at a height a small refinement is needed. To be continued.

Some material has been taken from Wikipedia.