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There is a hideout…

The cozy abode of the buzzing bees

At the far end, stands an avocado tree

Laden with flowers adorning its leaves

Every morning, I step into the yard

And gaze at the tree; it’s a sight to behold

The buzzing bees dancing with glee

Around the tree’s blossoms,

Sipping on nectar; O busy bees…

And there follows a thought:

Did the tree blossom to feed the bees,

Or, did the bees come to serve the tree?

Ah! I can’t help but marvel in awe

Of this nature’s unspoken synergy!

She has a lot to say, but she speaks to none

Her feelings are only for her to keep

The safe trove unto which none gets to peek

The louder one plays along

Dances with the crowd for many a song

While she tumbles and rumbles, never to cease

The silent one quietly revels in her peace

*Adapted somewhat from :*

ಎರಡು ಕೋಣೆಗಳ ನೀಂ ಮಾಡು ಮನದಾಲಯದಿ |

ಹೊರಕೋಣೆಯಲಿ ಲೋಗರಾಟಗಳನಾಡು ||

ವಿರಮಿಸೊಬ್ಬನೆ ಮೌನದೊಳಮನೆಯ ಶಾಂತಿಯಲಿ |

ವರಯೋಗಸೂತ್ರವಿದು – ಮಂಕುತಿಮ್ಮ ||

– ಮಂಕುತಿಮ್ಮನ ಕಗ್ಗ

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What distorts the perfect spherical shape of the Earth? Let’s find out.

The Earth is nearly, but not a perfect sphere. That’s because it spins. The rotational velocity of Earth is about 1,674.4 km/hr. This is high, though it is not tangible to us on Earth! The spin exerts a centrifugal force on the sphere of Earth acting perpendicular to and away from its axis of rotation, much like the force acting on a spinning ball. The centrifugal force is maximum at the Earth’s Equator. As a result, Earth’s surface is flatter at the poles by some amount and bulged out at the Equator by about the same amount. This shape is called an ** oblate spheroid**. The pole to pole diameter of the Earth is 12,713.6 km. The diameter across the Earth’s Equator is 12,756.2 km – a difference of about 42.6 km.

So, can we call Earth an *oblate spheroid*? No, not yet. For, Earth’s spin is not the only deciding factor of Earth’s shape. There is the pull of gravity exerted by the nearest object to the Earth – the Moon. This results in what we call ‘tides’. A tide is a bulge on the Earth’s surface caused by the Moon’s gravitational force. The *Earth’s surface* here refers to both the ocean and the solid land. More often, we associate tides with oceans. That’s because water is more flexible than the land. So, the effect on the oceans is strongly visible compared to that on the solid surface. The land does however bulge, albeit to a lesser extent than the ocean does. The tidal pull from the Moon have an amplitude of roughly a meter in the water, and may be 30 cm on the solid Earth.

Is that all? Not yet. There is the Sun too. More massive but far more distant than the Moon is to the Earth, the Sun also causes tides, but only a little less than half as high as that caused by the Moon. This makes the Moon the dominant tidal source. However, something worth noting is when the Sun, the Earth and the Moon are in perfect alignment, the gravitational pull of the Sun and the Moon act together on the Earth, resulting in tides higher than usual. These are known as **Spring tides**. Similarly, when the geometry of the Moon, the Earth and the Sun is such that these three objects form a right-angle, the tides are lower than usual, resulting in **Neap tides**. For the scope of this article, it should suffice to know that both the Sun and the Moon contribute to the Earth’s tides, with the Moon wielding a dominating effect. And that the tidal effect varies across the surface of the Earth based on the relative position of both the Moon and the Sun.

There are other forces at work as well, such as tectonic forces, weight of the continents etc. However, the shape deviations due to these are less than the distortion due to Earth’s spin.

Now that we know the Earth is not a perfect sphere, how does it matter to us? Does the difference in the Earth’s polar and equatorial diameters bear any significance at all? Yes, it does. It has been important in the context of planning space launches, the orbits of satellites, and in circumnavigating the globe. The less time taken to pass over the Arctic or Antarctica sometimes makes it the preferred path.

Lastly, it indicates that the Earth’s diameter varies depending on where you measure it from. So, how do we calculate the Earth’s diameter then? Astronomers and scientists often employ the averages i.e. **the mean diameter** which stands at 12,742 km. It’s the average of the Earth’s polar and equatorial diameters.

*If this reading was useful, watch out this space My blogs on Astronomy for more such related articles that I plan to add. *

A series of sessions on “Indian contribution to Mathematics and Astronomy” by **Rangarathnam Gopu** early this year and the subsequent follow-up conversations with him on various subjects; all that only a true mentor would do.

The curiosity arising out of the content from his blogs on **Ancient Indian Astronomy** is sure to be the source of several of my blog topics in the days to come, until I explore and exhaust them all!

This article is a compilation from various sources as listed down here:

]]>*Why is minus into plus a minus?*

For answers, read on…

Arithmetic is mathematics in its most nascent form. The operations of addition, subtraction, multiplication and division that are useful to man in day to day life come under this elementary branch of mathematics.

The algorithm underlying the arithmetic operations is best understood using the Number line. The relative arrangement of negative numbers, zero and positive numbers on the number line matters in this context more than any other. With the help of a series of lateral movements and flipping actions on the number line, we can actually calculate the result of some of the otherwise inconceivable arithmetic operations involving negative numbers.

For the ease of understanding, in this article, we discuss the arithmetic operations involving integers only. However, the concept is easily extendible to the real numbers.

Also, the various techniques and rules of arithmetic operations have been well laid out already. This article does not aim at finding new avenues. Rather, it is an attempt to reinvent the wheel with the goal to understand the logic behind the whole process. Be prepared – it may get messy at times!

The operations of addition and subtraction become easy when we pictorially project the number line as a series of infinite stairs with negative numbers at the lower end and positive numbers at the higher end.

We can afford to paint such a picture of the numbers because the number line has its markings gradated in the ascending order of numerical value from left to right. An analogy with stairs is therefore easier to relate to than a flat number line. The ascent takes you to higher numbers and the descent takes you to lower numbers relative to your current position on the stairs.

**Addition**

Say, you’re at step 3. You climb up by 5 stairs. You’ll find yourself at step 8. What you inadvertently did here is addition. 3 + 5 = 8

**Subtraction**

Say, you’re at step 3. You climb down by 5 stairs. You’ll find yourself at step -2. This in effect is subtraction. 3 – 5 = -2

Does it mean, we can declare that addition is about ascent (rightward movement on the number line) and subtraction is about descent (leftward movement)?

Not exactly. Here’s why.

**The signs of the numbers involved also matter, not just the operations**. The signs determine the direction of travel – upward or downward.

Say, we’re at step 3. We’re asked to climb up by -2 steps. How do we do it? Physically, it does not mean anything to climb up by -2 steps. But conceptually, it implies moving in the opposite direction i.e. relatively lower by 2 steps. When we do so, we find ourselves at step 1. In effect, what we did here is subtraction by 2 i.e. 3 + (-2) = 3 – 2 = 1.

**Climbing up by -x steps = Climbing down by +x steps
**In other words, addition by -x is equivalent to subtraction by x, where x is a positive number.

Now, say we’re at step -1. We’re asked to climb down by -3 steps. As before, it doesn’t make sense to go down by -3 steps. It means moving in the relatively opposite direction by 3 steps. So, we actually climb up by 3 steps and find ourselves at step 2. In effect, what we did is addition by 3. i.e. -1 – (-3) = -1 + 3 = 2.

**Climbing down by -x steps = Climbing up by +x steps**

In other words, subtraction by -x is equivalent to addition by x, where x is a positive number.

Effectually, we can say, **ascent is an addition process and descent is a subtraction process.**

The table in Fig. 4 shows how the addition and subtraction operations work for different combinations of positive and negative integers :

**Multiplication**

Multiplication is equivalent to repetitive addition process. When we multiply ‘a’ by ‘b’, we actually add ‘a’ repeatedly ‘b’ number of times, considering ‘b’ is positive. Physically, multiplication can be perceived as being given a method to reach some destination. Here’s how it is so :

**Multiplication along the positive axis**

(2 x 3) = 2 + 2 + 2 = (3 x 2) = 3 + 3 = 6

(2 x 3) is about making either 3 hops of length 2 OR two hops of length 3, to find yourself at 6. The same holds for the hops along the negative axis.

**Multiplication along the negative axis**

(-2 x 3) = (-2) + (-2) + (-2) = (-3 x 2) = (-3) + (-3) = -6

Visually, it tells you how fast and far you have moved away from 0 along either axes. This amounts to scaling. The term ‘x 2’ implies twice the scale, ‘x 3’ is thrice the scale and so on. i.e. if you multiply some entity with any number, the effect is that of proportionate scaling of that entity.

Also, observe from Fig. 5 and Fig. 6 that, when you multiply a positive number by another positive number, you go along the positive direction away from 0 and when you multiply a negative number by a positive number, you go along the negative direction that many times far away from 0. **This is an empirical proof that (plus into plus) will always remain plus and (minus x plus) will always stay minus.**

It is fairly easy to multiply so long as we multiply an entity (positive or negative) by another positive entity. This positive entity tells you how many times you ought to add the given entity to get the final product. So, we can say :

**Multiplication of a number ‘a’ by ‘b’ is equivalent to adding ‘a’ repetitively ‘b’ number of times OR hopping in strides of ‘a’, ‘b’ number of times, where ‘b’ is a positive number.**

So far, we’ve been able to visualize the arithmetic operations and translate them into movements this way or that way on the number line. The complication is about to begin.

What if both the numbers are negative? What is -2 x -3 = ? How do we add -2 some negative number of times?

It is impossible to calculate this unless we convert it into a form where one of the numbers in the equation is a positive number. The way to do this is by expressing a negative number in terms of its relative position with respect to any other number on the scale. As an example, -3 = (0-3) = (1-4) = (2-5) = … For convenience, we shall rewrite (-2 x -3) as :

-2 x -3 = -2 x (1 – 4)

By distributive property,

-2 x (1 – 4) = (-2 x 1) – (-2 x 4)

Now, we have transformed one multiplication operation into two, however with each having a positive number to multiply with.

-2 x -3 = -2 – (-8) = 6

By this very approach, we can also prove, -1 x -1 = +1 :

-1 x (1 – 2) = (-1 x 1) – (-1 x 2) = -1 – (-2) = +1

**Again, an empirical proof that minus into minus is always a plus!**

**Division**

Division is the reverse process of multiplication. With multiplication, it is about finding the destination that we should reach, given the methods. With division, it is about finding the method to reach the given destination with a partial knowledge of the method. This involves repetitive subtraction process.

**Division along the positive axis**

In the equation 2 x y = 8, finding y is the goal of division. Here, we know the destination we want to reach i.e. 8. We know what we’re equipped with i.e. 2. Now how many times should we add 2 to itself to reach 8? That gives us ‘y’.

The way to find ‘y’ is by starting at 8 and hopping towards 0 in steps of 2. The number of times you need to do this to reach 0 from 8 gives you ‘y’.

8 – 2 – 2 – 2 – 2 = 0

Here, the number 2 has been subtracted 4 times. So, y = 4. Another way to write the equation 2 x 4 = 8 without changing its meaning is, 4 x 2 = 8. Here, we hop from 8 in steps of 4. It takes 2 such hops to reach 0.

**Division along the negative axis**

Given -2 x y = -6, find y.

By the same approach as in the previous case,

-6- (-2) – (-2) – (-2) = 0

We get y = 3.

Let’s step up the challenge : Given, 4 x y = -20, find y.

-20 – 4 – 4 – …? Can we ever reach 0 by this method? No.

Here, we need to approach with a different interpretation of the question : Which number when subtracted 4 times from -20 gives us 0?

-20 – (-5) – (-5) – (-5) – (-5) = 0

We get y = -5.

Now for the real challenge : Given -4 x y = 20, find y.

20 – (-4) – (-4) – (-4) …?

This is kind of an impossible case. It’s as good as traversing in the negative direction to reach a destination on the positive side. Looks like we have hit the limitation at last! However, we need to admit that **not everything about mathematics is conceivable**. Sometimes, we need to work our way around to find a solution if one exists. That’s what we’re going to do now. We flip our destination point and place ourselves on the negative side. How do we do it? By multiplying by -1.

-1 x (-4 x y) = -1 x 20

By associative property, we have

(-1 x -4) x y = -1 x 20

4 x y = -20

Now this equation is as good as the previous one that we solved. Hence, y = -5.

The table in Fig. 9 shows how the multiplication and division operations work for different combinations of positive and negative integers :

]]>*Why do we need Negative numbers?*

*Why is -1 less than 0? How can there be something smaller than nothing?*

*Is there anything rational about Rational numbers?*

*What is behind the random sequence of unending decimal part in Irrational numbers?*

For answers, read on…

We get imbued with the idea of numbers very early in life – so early that we consume it the way it is presented to us. We first learn to count 0, 1, 2… As we grow, more categories get added on – integers, rational numbers, irrational numbers, complex numbers etc. We even learn about different systems of numeration apart from the decimal system (base 10), such as binary system (base 2), octal system (base 8) and hexadecimal system (base 16) from the computing lingo. This is how we come to traverse the Number Line.

Because the idea of numbers predated writing, they happened to get their own symbols called numerals. In the hands of different civilizations who individually worked on the numeric concepts, numbers found their diverse representation schemes. The artifacts from those times give us a fair idea of the symbols used and the different numeration systems that were in vogue among civilizations. As an example, the inscribed clay tablets found from the Babylonian era indicate the existence of sexagesimal system of numeration (base 60). This is still being used by us for calibration of time, as in hours-minutes-seconds. The dozen measure that groups numbers in units of 12 stands for the duodecimal system.

Many such systems of numeration came into being, but the one that became ubiquitous is the decimal system with base 10. Though not used in the same sense as decimal system, the number 10 has a special meaning in all the numeral systems. It stands for the respective base.

“10” => 10 in Decimal system (base 10)

“10” => 2 in Binary system (base 2)

“10” => 8 in Octal system (base 8)

“10” => 16 in Hexadecimal system (base 16)

Before we delve into the world of numbers under the decimal system and the categories thereof, let us attempt to get to a somewhat subjective understanding of measures.

Things around us exist in measures of nought (denoted by ‘0’) to infinity (denoted by ‘∞’). Nought is the lack of something and infinity is the abundance of something without limit. What connects the two is the infinite series of infinitesimals – the minutest immeasurable granules – acting as a filler between the two extremities of nought and infinity.

It’s important to understand that the scale of number line itself is relative to the viewer’s degree of vision. As an example, if you take countable objects visible to the naked eye, the demarcation between any two consecutive numbers is very clear.

0 apple = no apple

1 apple = there IS an apple

2 apples = there is ONE MORE apple

Now let us get to the microscopic scale of measurement. Can you distinguish between 0 and the smallest indivisible measure next to 0? If you are to say it is 0.000000000…1, after how many zeroes should you place that digit 1 there? You cannot finitely define this. Can you? That opens up the world of infinitesimality as opposed to infinity that we have been so much familiarized with. Just as in the case of infinity, there is no limit to infinitesimality as well. While infinity stands for never-ending expansion, infinitesimality means incessant withdrawal to something deep within. In this dichotomy lies the philosophy that we can neither contain nor discretize what is perceived as a continuum. This is the very essence of the continuum of space and time – entities with unknown beginning or end.

Back to the Number line. Let us traverse it in the order that makes sense.

**Natural numbers**

The set of natural numbers is denoted by N as, **N = {1, 2, 3, 4, …}**. They are mainly used for counting.

**Whole numbers**

The addition of zero to the set of natural numbers gives the set of whole numbers denoted by W as, **W = {0, 1, 2, 3, 4, …}**

**Integers**

The addition of negative numbers to the set of whole numbers gives integers. They are denoted by I or Z, as, **I or Z = {…, -3, -2, -1, 0, 1, 2, 3, …}**

The** negative numbers** can be interpreted in many ways depending on the contexts that made its existence essential. It can mean **undoing an action, or, the absence or lack of something** : If you have 5 objects put together, removing 2 of them (-2) implies the absence of 2 objects.

In the context of pairs of opposites, **negative numbers complement positive numbers enabling us to quantify the opposing forces in nature**, such as, that of electrons and protons, opposite poles of north and south in a magnet, opposing forces of pull and push, centrifugal-centripetal forces etc. If you represent the force of push to be something positive, say +10 Newtons, the same force applied in the opposite direction by means of pulling action is -10 Newtons.

The idea verily extends to the relative measures of things : We denote electrons by -ve charge and protons by +ve charge. In a neutral atom, the number of electrons is equal to the number of protons. The charge of a neutral atom therefore equals 0. If we remove a few electrons from an atom, the atom gains positive charge in proportion to the number of electrons lost. Similarly, if we add electrons to a neutral atom, the atom becomes negatively charged in proportion to the number of electrons added. In this context, the charge of the atom whether +ve or -ve, is said to be relative to the charge 0 when the atom is in neutral state.

An understanding gained thus of the negative numbers helps us explain the position of these numbers in the number line shown in Fig.4. In reality, is there a number less than zero? Can anything be smaller than nothing? No. Then, **what puts negative numbers ahead of zero or positive numbers in the number line? The answer lies in their relatively complementing nature with respect to positive numbers**.

A philosophical rub to this whole idea of number line with positive and negative numbers on either side of zero is that, when you are zero, what you see within i.e. -1, -2, -3 … is a mere reflection of what you see without i.e. 1, 2, 3, … Considering that a number line is infinite in length, your zero can be a moving one, nailing it right at the point that you are at any given moment of time.

So far, so good. But we’re not there yet. The numbers covered thus far are all distinct numbers – the transition from one number to the other is abrupt. The space between any two consecutive numbers has not been taken into account. The next two categories of numbers are inclusive of this space taking you to the world of fractions and decimals.

**Rational numbers**

A rational number is a number that can be expressed as a “ratio”. So, there’s nothing “rational” about this number if you had ever got misled by the use of this term. They’re just fractions. Perhaps, “fractional” number would have been a better name. It is denoted by Q, as, **Q = { p/q | p, q E I and q is not 0}**

Needless to say, any natural number, whole number or integer can be expressed as a rational number by virtue of the property of 1. The number 1 divides all numbers without changing its value or property. e.g. 3/1 = 3, -2/2 = -2, 0/1 = 0

Another way to look at rational numbers is, **every rational number can be expressed as a terminating decimal or a periodic (recurring) non-terminating decimal.**

*Terminating decimals* : 1/8 = 0.125 , 97/8 = 12.125

*Non-terminating periodic decimals* : 1/7 = 0.142857142857142857… (the pattern 142857 repeats itself in the decimal part) , 100/11 = 9.0909090909… (the pattern 09 repeats itself in the decimal part)

**Irrational numbers**

Any number that is not a rational number, something that cannot be expressed as a “ratio”, is an irrational number. Again, there is nothing “irrational” about irrational numbers.

Famously falling in this category are the roots of numbers that are not perfect squres of any number e.g.√2, √3 etc. Also, falling in this category are the surds – the n-th roots of numbers that cannot be expressed as the nth power of any other number e.g. 3rd root of 7 denoted by 3√ 7 etc.

Keep in mind though that the irrational numbers are not limited to roots or surds as they’re otherwise made to look. For example, pi = 3.14159265358979323846… that is an irrational number cannot be expressed as the n-th root of any other number. Such a number is called *transcendental* number. The best definition for irrational number therefore is that, it is any number that is not a rational number.

Another way to look at irrational numbers is,** an irrational number is a non-periodic non-terminating decimal number**. e.g. √2 = 1.4142136… is a non-periodic number that grows infinitely with no sequence of digits repeating itself systematically. This property implies that the irrational numbers interpenetrate the space left behind by the rational numbers.

**Convergence aspect of non-terminating decimals **

Though it appears that a non-terminating decimal number is an unending number, it is actually a recursively converging number. Its beauty is that, while it extends the decimal part digit by digit from one end, it is actually limiting the scope of the decimal number each time the digit is added. Here’s how :

Let’s take x = 2. We can say, 2 ≤ x < 3. We add a decimal part, say, 0.5 to this. Now, x = 2.5. By adding this decimal part, we increased the value of x above 2. At the same time, we limited the scope of x from 3 to 2.6 (next higher number at that decimal position). By limiting scope, we mean, now on, no matter how many digits you add to the decimal part of x, the value of x will never go beyond 2.6. i.e. 2.5 ≤ x < 2.6.

Now, let’s add a digit 8 to the decimal part i.e. 0.08 to x. x = 2.58. The scope of x is now reduced from 2.6 to 2.59. i.e. 2.58 ≤ x < 2.59.

Next, we add 0.003. x = 2.583. The scope of x is limited to 2.584. i.e. 2.583 ≤ x < 2.584.

Effectively, by adding decimal digits, the range of x has been narrowed down from 2 ≤ x < 3 to 2.583 ≤ x < 2.584 – a clear convergence process. Fig. 5 shown below illustrates this process.

Step 1 : We start with stretching the space between the numbers 2 and 3 to a larger scale. Divide this space into 10 parts (because we follow base 10 method). We now mark 2.5 as the mid-point on this new scale.

Step 2 : Now, stretch the space between 5 and 6 on the second scale to a larger scale. Once again divide this space into 10 parts. We now mark 2.58 on this newer scale. This process repeats as we keep adding each digit to the decimal part.

Though we cannot attach much physical significance to the repeating (rational) or non-repeating (irrational) nature of non-terminating decimals, **the non-terminating property of decimal numbers in itself is indicative of the deep dive of these numbers into the world of infinitesimality. **

The detour that we took early on was to set the premise for these non-terminating decimals. **It is the world of an unending division process of number space into smaller and smaller units. ** An impossible task, whatsoever. We can’t express something that is continuous in terms of something that is discrete. Can we?

** Real numbers
**Real numbers is just a name given to the union of the set of rational numbers Q and the set of irrational numbers Q`. It’s denoted as R = Q U Q`.

Real numbers are therefore the totality of all the numbers discussed so far – from infinitesimals to infinite numbers, from negative to positive numbers.

]]>Why do we express angles in terms of fractions or multiples of π i.e. 30°, 90°, 180°, 360°… as π/6, π/2, π, 2π … ?

*Why does a circle have 360°? Why is 360° = 2π?
*

*What is behind the periodicity of trigonometric functions?
*

For answers, read on…

An angle is a measure of rotation of a ray OA about its initial point.

The concept of angle and its units of measurement are best understood with a little knowledge of the contexts in which angle took its very birth and flourished thereon :

**<Phase I : The birth of Angle via Line geometry>
**

**Angle is a concept that evolved out of Line geometry i.e. geometry involving the meeting of straight lines.** This makes it an integral attribute of polygons, where lines join to make a closed circuit. The ancient people who studied triangles in particular, observed a relation between the angles and the ratios of the sides of a right-angled triangle :

For the two similar triangles shown in Fig 2, it is seen that, for a given angle Θ,

X/Z = X`/Z`

Y/Z = Y`/Z`

X/Y = X`/Y`

The ratios being functions of Θ were subsequently tabulated for varying values of Θ and came to be known as sin(Θ), cos(Θ) and tan(Θ) respectively. This field of study of the relationship between the angles of triangles and the ratios of its sides came to be known as Trigonometry and the functions – Trigonometric functions. It helped mankind a great deal in the measurement of distances and heights of remote inaccessible objects scaling up to astronomical lengths.

**<Phase II : The growth of Angle via Circular geometry>
**

**Angle is a natural attribute of a Circle. ** When the rotation of a ray OA about its initial point O is complete, the locus of the path is a circle. At any instant of revolution along the circular path, the projections of A onto the x and y axes give us a rectangle with the diagonal OA`bisecting the rectangle into two right-angled triangles. Thus the unfolding of angle Θ along a circle can be mapped to right-angled triangles varying along with Θ, linking trigonometric functions with circular geometry.

The beauty of linking an angle with circle is that, as A moves along the circumference revolutions after revolutions, Θ keeps growing with infinity being its limit. This treatment extends the span of trigonometric functions to infinity. Because a circular movement is repetitive in nature, the graph of trigonometric functions should also repeat for every revolution of Θ. It naturally follows that trigonometric functions have been used to model the periodic phenomena in nature, such as the motion of waves, simple harmonic motion, sound vibrations etc.

Two popular units of measurement of angles have emerged over time. They are (1) Degrees and (2) Radians.

**(1) Degrees**

The degree is a very ancient unit of measurement, so much so that its origin is not exactly established yet. Multiple theories have been posited by those who sought to solve the mystery of 360 degrees associated with circles. These are the two that stand out :

*– It is believed that the number 360 is an approximation of the number of days in a year. It is said that the ancient astronomers, who were keen observers of the sky, used the delta shift in the daily positions of the stars and constellations as a reference to mark a degree.*

*– Another theory suggests that the basis of degrees, minutes and seconds is the sexagesimal system. Sexagesimal is a numeral system with sixty as a base. The number 60 being a highly composite number i.e. it has twelve factors {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, was perhaps adopted by the Sumerians and the Babylonians for the convenience of measurements.*

Both the theories can be equally argued in favor of. The fact that a year has 365 ¼ days and a circle has 360 degrees is quite a coincidence. However, if we were to presume that there was a numeral system in vogue, at the time degree was fixed to be the unit of angular measurement, then this theory of sky observation does not appeal. The fact that a degree is further divided into 60 minutes and a minute is sub-divided into 60 seconds is an evidence of the permeance of sexagesimal system into the angular measurement.

That evokes a question :

**If degree is a unit based on sexagesimal system, then why doesn’t a circle have 60 degrees instead of 360 degrees?**

This perhaps has to do with the chronology of geometry. The concept of angles emerged out of the study of lines and triangles first. Further, an equilateral triangle makes the same angle at its three corners. That must have made it convenient to use it as the basic unit of angle measurement, by marking each angle to be of 60 parts based on the sexagesimal system.

Over time, when the concept of angles extended to circles, it was convenient to use equilateral triangle with side length equal to the radius of the circle as shown in Fig. 5, to measure the angle enclosed by circles.

Placing six such triangles together gave a circle with 360 degrees angle.

**(2) Radians**

The radian marks its birth around the 18th century on the timeline. A radian is defined as the angle subtended at the center of a circle by an arc equal to the radius of the circle.

**A visual comparison of Fig. 7 with Fig. 5 says much in one glance. ** Fig. 7 carves out a sector of arc length equal to the radius enclosed by two radii. Fig. 5 carves out an equilateral triangle with radii as its sides. (Yes, the equilateral triangle is further divided into 60 parts, whereas the sector is not. That explains the finer granularity of degree measure.)

The radian measure is based on a unique property observed of circles : For a circle, the ratio of the circumference to the diameter (= twice the radius) is a constant, regardless of its size. This constant called ‘pi’ (written as Π) is an irrational number that approximates to a value of 3.14159.

When we calibrate angle in terms of radians, we actually fragment the circumference into arcs of length equal to the radius. Fig. 8 shows six complete arcs with one fractional part making it about 6.28318 radian measure i.e. 2Π radians makes one revolution.

An angle specified in radians is therefore a direct measure of how many times the radius occurs on the circumference of a circle for that angle. For instance, three radians provide an arc coverage equal to three radii length. This means, if the circle rotates by a measure of three radii length, it implies an angle coverage of three radians. **This maps the math of radians to the physics of rotatory movement also viewed as periodic movement. **

Radian is a dimensionless entity – a ratio of the arc length covered by the angle to the radius of the circle. Giving it the name of radian makes it convenient for use.

**Degree is a product of Line geometry :** Degrees work well with angular measurements in triangles and polygons. They offer a finer granularity compared to radian measure as well. One radian = 57.2957795 degrees. In fields like navigation, land surveying and astronomy, where minute angles are involved, degrees become the obvious choice of angle measurement.

**Radian is a product of Circular geometry :** Radian measure is well coupled with the radius of the circle, making it strictly a property of circular geometry. A circle means periodicity. That makes radian a convenient unit of measurement for any periodic phenomena plotted against angular frequency. Clearly, the trigonometric functions seem to make better sense when expressed in radians.

जैसे भीगा हुआ एह्सास ओस के गिरने से

खिल उठता है दिल फूलों के खिलने से

उकसा जाती हूँ दरिया के समँदर से मिलने से

क्यों … ?

क्यों गुदगुदी सी होती है नसीम के छूने से

गाता है मेरा दिल पँछियों के चह्चहाने से

सरकती हूँ मैं सूखे पत्तों के सरसराने से

वह हल्का सा एह्सास तितली के फ़रफ़राने से

क्यों … ?

क्यों दहलाती है मुझे बादलों की गरज

चौँकाते हैं मुझे बर्क़ के शोले

वह घेर लेनेवाला सर्द बारिश का क़तरा

मुन्तज़ीर करते हुए तूफ़ान के थमने का

क्यों … ?

क्यों मुरझाती हूँ मैं चमन के उजडने से

मायूस सी होती हूँ सूरज के ढ़लने से

मचलता है बदन गरमी की बढ़ने से

गल जाने का एह्सास बर्फ़ के पिघलने से

क्यों … ?

सोचती हूँ मैं मन ही मन, आखिर

क्या रिश्ता है मेरा क़ुदरत के वारीदात से

क्यों माइल है यूँ मेरी सरिश्त…

क्यों महरूम हूँ मैं इस राज़-ए-निहाँ से?

क्यों … ?

**Glossary**

ओस = dew

उकसा जाना = propelled

दरिया = river

गुदगुदी = tickling

नसीम = gentle breeze

चह्चहाना = twitter

सरकना = slide

फ़रफ़राना = flutter

दहलाना = terrify

बर्क़ = lightning

सर्द = cold

क़तरा = droplet

मुन्तज़ीर = awaiting

वारीदात = events, happenings (plural of वारदात)

माइल =inclined, biased

सरिश्त = temperament

महरूम = deprived