Category Archives: Random Thoughts

Al Khwarizmi’s Method of Solving Equations in One Variable

Al Khwarizmi was a Muslim scholar who worked under the patronage of emperor Al Mamun in the 9th century in Baghdad. Al Mamun like his father was interested in supporting learning and formed the House of Wisdom of which Al Khwarizmi was a part. It was here that Greek philosophical and scientific works were translated into Arabic. In recognition of his support Al Khwarizmi dedicated two of his works on algebra and astronomy to the great emperor. His treatise known as Hisab al-jabr w’al-muqabala was the most famous and important one and was translated into several other languages. Over the years al-jabr came to be known as algebra and Al Khwarizmi (Algoritmi  in Latin) became algorithm.

According to Al Khwarizmi equations are linear or quadratic and are composed of units, roots and squares. To Al Khwarizmi a  unit was a number, a root was x and a square was x2. He defined two basic operations to solve equations namely al-jabr (the process of removing negative terms from an equation) and al-muqabala (the process of reducing positive terms of the same power when they occur on both sides of an equation). All Al Khwarizmi mathematics was done entirely in words and no symbol was used e.g. he solves an equation x2+10x=39 as follows:

“… a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square”

Al Khwarizmi Graphical Method for Solving an Equation
Al Khwarizmi Graphical Method for Solving an Equation

One of his masterpieces is his method of solving equations using simple geometry, as shown above. For example to solve the equation x2+10x = 39 he first creates a square with length of each side equal to x and area x2. He then adds 10x to it by creating four rectangles of area 10x/4 = 5x/2 each. Thus the lightly shaded area in figure (c) above represents x2+10x or 39. Now we add the areas of the four corners (darkly shaded) to get  x2+10x+25 or 64 (39+25). So we find out that the side of the larger square is of length 8 (square root of 64). But we already know that the length  of a side is 5/2+x+5/2 = x+5. So we have the final equation x+5 = 8 or x = 3.

Al Beruni’s Method for Calculation of Radius of Earth

Al Beruni was a great Muslim scientist of the eleventh century who had knowledge of many diverse fields such as astronomy, astrology, mineralogy etc. He was member of Mahmud of Ghazni’s court from 1017 to 1030. It is here that he had the opportunity to travel to India (as well as present day Pakistan) and write about it in his books. It is claimed that he even learned Sanskrit during his stay in India. However, Al Beruni is most well known for his experiment to calculate the radius of the earth at a location close to Katas Raj in Kallar Kahar region of Pakistan. Some claim that the location of his measurements was in fact Nandana fort which is about 40 km east of Katas Raj temples.

His method of calculation involved two steps.

Step 1: Calculate the inclination angle of a mountain at two locations with known separation between them. We can then use simple trigonometry to find the height of the mountain. The trick here is to realize that there are two right angled triangles with the same height, which is an unknown, and base lengths which are also unknown but can be factored out (the reader is encouraged to do the math himself).

Calculating the Height of a Mountain

Figure 1: Calculating the Height of a Mountain

Step 2: The second step involves calculating the angle of depression that the horizon makes as viewed from top of the mountain. Then with the height of the mountain already known the radius of the earth can be easily calculated as shown below.

Calculating Angle of Depression

Figure 2: Calculating the Radius of the Earth

As the reader might have noticed there are three angular calculations and one distance calculation involved. Distance is easy to measure but angle is not and an accurate Astrolabe is required for this purpose. The Astrolabe that Al Beruni used was accurate to two decimal places of a degree. The radius that he calculated was within 1% of the accepted radius of the earth today.

PS: The writer has been to Katas Raj temples several times and it his desire to go there again to find more information about the location of the experiment and to possibly re-enact the experiment.

Note:

  1. A polymath is a person whose expertise spans a significant number of different subject areas; such a person is known to draw on complex bodies of knowledge to solve specific problems.
  2. An astrolabe is an elaborate inclinometer, historically used by astronomers and navigators to measure the inclined position in the sky of a celestial body, day or night.
  3. Nandana was a fort built at strategic location on a hilly range on the eastern flanks of the Salt Range in Punjab Pakistan. Its ruins, including those of a town and a temple, are present. It was ruled by the Hindu Shahi kings until, in the early 11th century, Mahmud of Ghazni expelled them from Nandana.

 

Half Wave Dipole Antenna

A dipole antenna is a simple antenna that can be built out of electrical wire. The most common dipole antenna is a half wave dipole which is constructed from a piece of wire half wavelength long. The wire is split in the center to connect the feeding wires. The E-field of the antenna has a circular pattern along a plane which cuts the axis of the antenna perpendicularly and is similar to a figure of 8 in a plane along the axis of the antenna [3D pattern]. The exact E-field can be calculated as:

Expression for E-field of a Dipole Antenna
Expression for E-field of a Dipole Antenna
E-field Pattern of a Dipole Antenna
E-field Pattern of a Dipole Antenna

The MATLAB code for generating the above pattern is given below.

n=377;
Io=1;
r=10;
lambda=0.3;
k=(2*pi)/lambda;
L=lambda/2;

theta=0:0.01:2*pi;
E=j*n*Io*exp(-j*k*r)*(1/(2*pi*r))*((cos(k*L*cos(theta)/2)-cos(k*L/2))./sin(theta));
polar(theta, abs(E))

Note that the above is true within an area at a sufficient distance from the antenna known as the far-field of the antenna. Closer to the antenna i.e. in the near-field the E-field expression is a bit more complex.

Qualcomm In Muddy Waters In India

Remember Qualcomm CEO Paul Jacobs proudly claiming that his company had prevented WiMAX from getting a hold in India by acquiring BWA licenses in four regions of India. Well now Qualcomm is in a bit of bother as the Department of Telecommunication (DoT) in India has raised objections to the license application filed by Qualcomm. According to news circulating on the internet the DoT has objected to Qualcomm filing four separate applications through its nominee companies in the four regions (Delhi, Mumbai, Kerala and Haryana) it had won the licenses on June 12, 2010. Secondly the DoT has also objected to the delay in the filing of application outside the three month period required by the laws.

Qualcomm has rejected these objections saying that it has followed all rules in letter and spirit. According to Qualcomm the license application was filed in August 2010 within the three month period as required by the laws. However this is disputable as Qualcomm also submitted a revised application in December 2010. Qualcomm has also countered the second objection by saying that it plans to merge the four nominee companies so that there is no breach of law. As per the rules “if at any stage the spectrum allocation is revoked, withdrawn, varied or surrendered, no refund will be made”. So if an understanding is not reached between Qualcomm and DoT, Qualcomm is set to lose more than $1 billion that it had paid for the BWA spectrum.

Qualcomm Inc. is a leading wireless chip manufacturing company of the world. It is the pioneer of CDMA technology and its chipsets have been embedded in more than a billion cell phones. Qualcomm has greatly invested in UMTS technology and is a strong proponent of WCMDA, HSPA and LTE standards. It had a paid about a billion dollars for the right to use a 20 MHz chunk of spectrum in the 2.3 GHz band. It plans to bring TDD LTE to India, which is a considered to be a comparatively economical 4G technology.

Solar Analogy

All electromagnetic energy travels in the form of rays. The most obvious example is solar energy that is radiated by the sun in all directions. The further away a body is from the sun the lower the energy that it receives. Objects in the path of these rays cause shadows but not complete darkness as rays reflect from other objects and also diffract around the edges. These rays also have a phase and frequency that determines their behaviour when interacting with objects. The amount of rays that can be collected by a receiver depends upon its size and orientation. Solar energy can be harmful when a body is exposed to it for longer periods.

All these concepts are extendable to wireless communications. Wireless signals decay with distance, suffer from shadowing, reflect, refract, diffract, scatter, have phase and frequency, can be collected by appropriately designed antennas and can be harmful as well. The major difference being that modern transmitters are not isotropic radiators. Practical transmitters are like a sun that radiates solar energy to the earth in a narrow beam while ignoring the other planets.