Kalawati’s Meadow Mathematics

Fast-forward 800 years. In 1940, 12-year-old Kalawati in Verdant Shimla Hills was grazing her cows in sloping green meadows. She had solved the problem from the Leelawati. This is a book of mathematics written by Bhaskaracharya in 1150 CE. It involves figuring out the depth of the pond from the given length of the lotus plant sticking out of the water. It also uses the distance to the point where it dips underwater when inclined.

As she thought more about the puzzle, she felt increasingly drowsy. She shook herself up and called out to the cow that had wandered too far from the flock. The cow pricked her ears, saw her, and ambled back to the main flock near Kalawati. Meanwhile, Kalawati thought about how Bhaskar had used a lotus plant. It hardly behaved like a stick, swaying and playing in the wind. This was a farfetched approximation for a stick to solve the problem. This notion tickled her so much that she considered writing a letter to Lilawati on a whim. Lilawati was the girl from 12th CE who couldn’t marry but pursued mathematics. She became familiar with approximations not only in numbers but in everyday things. Numbers had become her friends.

Kalawati wrote in her notebook, thought 1+x is a helpful expression that could describe many things, so she made a list to collect her thoughts.

(1+x) is a type of Binomial. It comprises two numbers. It can equal any number depending upon the value of x. The value of x can be smaller, equal, or larger than one. If x is equal and allowed to recurse in the equation, it becomes 2 + x. Then it becomes 3 + x, and so on to infinity. This creates whole numbers. On the other hand, if x is negative, it will create all negative integers. This occurs by sequentially increasing and decreasing in fixed unit lots.

Looking closely, this equation can also be morphed into a straight-line with x as a real number. Although a line itself has only length as a dimension, it is drawn in a plane of two dimensions. It cuts across dimensions defined by the X and Y axes of the cartesian plane. And can point in any direction.

The classical equation for a line is y = mx + c in two dimensions. Here, m is the slope of the line. slope is 0 for a line parallel to the x-axis. It is 1 at 45⁰. It is not defined when parallel to the y-axis. The value c is the y-intercept where the line cuts the y-axis. The slope of the line is the ratio of the perpendicular to the base. This triangle is formed by the y and x intercepts. Thus, tan theta imparts obliquity to the line. Since tan theta has a range between negative infinity to positive infinity, the line can point in any direction

To draw a line parallel to the x-axis, m needs to be zero. For the line to be on the x-axis itself, both m and c need to be zero.

At m=0, line is parallel to x axis cutting y at C and if m= 1 = tan45⁰  and c= 0. The line forms a triangle, originating from zero. With an imagined perpendicular on the x and y axis, it will form a diagonal of a unit square. The line itself will be the hypotenuse, and its hypotenuse is the square root of two. This is a realization of the Pythagorean school and marks the start of incommensurate or irrational numbers.

In any equation, RHS equals LHS. Performing an identical operation on both sides maintains this value. Each value in the equation tells about its domain and final shape. When the domain is a real number, the equation is continuous. Otherwise, restricted domains may be discrete. Although continuous functions are discrete at very small level in some way.

She wrote a summary from a book on a separate page. It was to differentiate functions and equations. She also included some of their core properties.

Understanding Lines and Equations

We began by discussing the range of tanθ in the context of a line’s equation (y=mx+c). The slope (m) of a non-vertical line is equal to tanθ. Its range is (−∞,∞). This means a non-vertical line can point in any direction. Horizontal lines have m=0. Vertical lines have an undefined slope at 90∘. They cannot be represented by the y=mx+c form. Instead, they are described by x=k.

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 Continuity and Differentiability

The concept of continuity was then explored. A function described by y=mx+c (for fixed m and c) is continuous everywhere on its domain. However, the family of lines represented by y=mx+c does not include vertical lines, as the slope m becomes undefined. This led to the fundamental calculus principle: if a function is discontinuous at a point, it cannot be differentiable there. The converse is not true; a function can be continuous but not differentiable (e.g., a sharp corner like f(x)=∣x∣ at x=0).

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Functions vs. Equations

The core distinction between a function and an equation was clarified:

• An equation is a statement asserting the equality (=) of two expressions (e.g., 2x+3=7 or x²+y²=25). It describes a relationship or poses a problem for specific values.
• A function is a special type of relation where each input from its domain is associated with exactly one output. Functions are often expressed as equations (e.g., f(x)=2x+1 or y=2x+1), but not all equations represent functions (e.g., x²+y²=25 does not define y as a function of x because one x can have multiple y values).

This crucial difference is key because:

• Not all equations are functions, but every function can be written as an equation.
• The change in an input variable, leading to a unique output, is precisely what allows functions to be effectively represented as graphs, which typically pass the vertical line test.

This foundational clarity is valuable for anyone encountering mathematical expressions, especially “complex looking equations,” as it helps build correct mental models and apply appropriate analytical tools.

After the straight line come the quadratic function, here  (1 + x)² = 2 results in a quadratic equation: 1+2x+x² = f(x) is a quadratic function. It can be morphed into a two-dimensional parabola and generally represented by (y =ax²+ bx +c).  Raising the power of x doesn’t turn it into 3 D structure, it stays in the plane. Even powers of x will result in narrower tops till it turns into a test tube like structure, odd powers produce a S shaped curve all passing through zero, at very high values of x they hug the x axis between 0 and one and then rockets up and in the negative “x,y” quadrant , it hugs the  negative x and then dives down  the region between -1 and 1 is flattish are below and above the x axis.

(1 + x)ⁿ results in a polynomial of degree n and converts an exponential function to one of addition.

Its expansion follows Newton’s general binomial theorem, which converts multiplication into addition. The expansion is

She also wondered about the path of a projectile and a solid vertical dome from an advanced book by Euler and noted them in her notebook. She realised that the general equation of projectile is vector analysis of velocity V and the force of gravity, so that two motions in vertical and horizontal  incorporate a continuous rise and fall combined with a constant horizontal velocity vector . to produce its curve.

But this looks like inverted the U shape parabola of quadratic , she  thought a little that when she wrote the quadratic as  (-ax² +bx + c), it will produce a parabola equivalent to the projectile path.

When this parabola is rotated around its vertex will produce  a 3 D dome a paraboloid to be exact. Now as every point on the perimeter of a circle can be written as x an y, and the equation of a circle becomes  x² +y^2 = r^2, in a spheroidal dome this translates to x² +y^2 +z^2  = r^2 .

But since a paraboloid dome opening along the y axis is formed by our equation, its equation would be x²+z2=4ay.

 The general form of Binomial is written as :

in which k starts from 0, and the total number of terms is n+1. ⁿCₖ expansion, read as n chose k is n! / (n-k)! k!

Further expansion for (kth term is

n(n-1)(n-2)(n-3)….(n-k-1)/k!

Lilawati already explained why Vishnu, with four hands, holding a separate item in each hand, has 24 names, one for each variation. That is four factorial or 4x³x²x1, a permutation, and Shiva has a maximum of 10 hands. You can imagine and calculate. That is n items all different can be combined in n! ways, but what if some of the items afe same say 10 each pf apples , mangoes and bananas? That requires combination.

 Binomial terms in their sum group items, e. g  –  x²y³. Or x³y², both will have a coefficient of 10, meaning there are 10 occurrences of such patterns in an expansion of (x +y)^5. This is useful in gambling. So, the coefficient of each term… it’s just counting how many ways I can rearrange the bits!” x²y³ , notice that sum of the powers is 5 which is n, now using the

                                We have 5!⁄(2!·3!) = 10, thus 10 is the coefficient for both

Puzzle

 In a team of 11 players, each player shakes hands with all other players once, so how many handshakes were there?

Here, the first player gets to shake ten hands; the second, because he has already shaken hands with the first player, gets to shake hands only nine times, and so on, making it a sum from digit one to 10, which is 55.

The second method visualizes that among 11 players, how many distinct pairs can be formed? And that is  , 11C2 or 110/ 2, which is also 55. In exactly the same way.

In

Meru-Prastar aka Binomial expansion

Although the Binomial theorem has been known since Vedic times, Sir Isaac Newton fully exploited its power by discovering its general form. The theorem also works for fractional and negative powers and can be used for extracting roots. He used an ingenious integration method to extract the value of Pi using binomial expansion.

The sum of all coefficients of Binomial equals 2n, and the value of a particular term is equal to no of possible ways the two terms can be combined into distinct arrangements; thus, for example, xxy can be arranged in 3 ways only as (xxy), (xyx), (yxx), or 3!⁄2! Thus, the coefficient will be 3. In the same way, x^n will have a coefficient of one. In contrast, x^n-1)y) will have n as the coefficient.

If x = 1, the binomial expression is one plus one or two (1+1). If two is multiplied by itself, say 2x², in binomial terms, it becomes (1+1) x (1+1)  or (1+1)^2,  and the result using distributive law of multiplication is 1+1+1+1 = 4 

So, the expansion of this Binomial is the sum of the coefficients in its binomial expansion, which are 1 + 2 + 1 = 4 as both variables are equal to one and thus don’t produce any effect of their own. And if coefficients of all powers of two are put on top of each other, we get the Pascal triangle.

Row 0:                        1
Row 1:                      1   1
Row 2:                    1   2   1
Row 3:                  1   3   3   1
Row 4:                1   4   6   4   1
Row 5:              1   5  10  10   5   1
Row 6:            1   6  15  20  15   6   1
Row 7:          1   7  21  35  35  21   7   1
Row 8:        1   8  28  56  70  56  28   8   1

This is Binomial expansion: When we expand (1+1) ^n, we are essentially looking at the coefficients of the binomial theorem where both terms (x and a) are equal to 1.

In this formula:

•  represents the binomial coefficient, which is “n choose k” (the number of ways to choose k elements from a set of n elements). These are the numbers that form Pascal’s triangle.
• x and a are the two terms in the binomial.
• n is the power to which the binomial is raised. Or multiplied by itself so many times.
• k is the term index, starting from 0.

The sum of the coefficients is equal to the power of 2 raised to an exponent equal to the row number, or in other words; multiplication has been turned into a sum. This magical property becomes even more magical when these are used with unequal numbers. Here is what happens when (1+99) is used instead of (1+1)

Row 0:                        1

Row 1:                   1      99

Row 2:             1      198      9701

The reverse of (99+1) would be (1+ 99). The tilt changes symmetrically. Imagine a scissor opening and closing first from one end, then at the other. 

Row 0:                        1

Row 1:                   99      1

Row 2:          9701      198      1

“Binomial’s rare heartbeat becomes Poisson’s whisper across the infinite.”

Figure: Comparison of Binomial Distribution (n=100, p=0.01) and Poisson Distributions at λ = 1, 3, 5.

 When we talk of Binomial or Poisson distribution, we are talking about the probability of an event happening,  its success or failure. The binomial distribution is based on following premise:-  

Basic Probability Identity

Pₖ + (1 – Pₖ) = 1

This shows that the probability of an event k happening (Pₖ) or not happening (1 – Pₖ) adds up to 1.

 Binomial Term: Probability of k Successes in n Trials

Where:

She found out about how to speak this equation

Component	Spoken As
P(X=k)	“The probability of getting exactly k successes”
nCk	n choose k” or “the number of ways to choose k successes from n
Pkk	“The success probability raised to the power k
(1−Pk)n-k	“The failure probability raised to the power n-k

Put together:

“P of X equals k is equal to n choose k, times the success probability to the power k, times the failure probability to the power n minus k.”

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🔸

Such complexity and it is just a product of 3 numbers, to find the chance of winning exactly k times in n tries:

First, count how many ways such a thing can happen—this is called ‘n choose k’.
Then, multiply by the chance of success happening k times.
Then, again multiply by the chance that failure happens in the rest—that is, n−k times.

Multiply these three together, and you have the answer.

If all the probabilities are counted, it is a binomial distribution unless Pₖ <<(1-Pₖ ); it tends to Poisson’s distribution of rare occurrences. However, when n is very large, it becomes tedious to calculate. The concept of Lambda = np in which n is large, P is small, and their product is constant comes to the rescue when modelling Poisson’s distribution.

Further historical averages may, with stratification of the events may be used to estimate λ. Stratification would be by cause of death or mechanism of injury or disease. This would reduce clustering and make the estimate more accurate.

She compared a binomial with p=0.01 and n=100 using n·p = λ \ λ = 1. It gave the same result more simply.

“This transformation, this ‘flattening’ or ‘diving’ of the binomial into the Poisson distribution that she wrote that letter to Lilavati hinted mysteriously that it seemed to happen under specific conditions,” Kalawati continued to muse. “It wasn’t just any binomial that would morph in this way. It was when one was looking at a great many trials or opportunities – an n so large it stretched towards the horizon or even getting 5 heads sequentially in a 1000 throw of a fair coin! – but where the chance of ‘success’ (p) in any single one of those opportunities was vanishingly small like a single specific raindrop hitting a particular grain of sand on a vast beach during a brief shower. In such cases, calculating the binomial probabilities directly, with all those large factorials, would become unwieldy. Yet, the overall average number of successes one might expect, λ=np, could still be a perfectly sensible, moderate number. It was this λ, this average rate of rare occurrences, that became the heart of the Poisson probability. The Poisson distribution, then, offered a simpler, more elegant way to describe the likelihood of observing 0, 1, 2, or just a few of these rare events without getting lost in the enormity of n or the minuteness of p. It was as if the universe had a shorthand for the mathematics of rarity spread across vastness.”

Plotting a binomial at p = 0.5, as a toss of a fair coin, gives a binomial symmetrical around 50, which is the mean, but when p is taken as very low, here p = 0.01 or 1 %, it transmutes into Poisson type distribution with λ = 1, a distribution of very rare events.

She also realised that calculus, which she was reading in her class, dealt with infinitesimal change and its effect on real-world phenomena. It was still being taught, but she suspected it was connected to uneven binomial in some way.

She finally made a list of number types and mused that if zero and one are known and the operators defined with their commutative, distributive laws, other numbers can be derived from just that or those two numbers, for they are not just numbers; they are full concepts in themselves.

Thinking about various types of numbers, she could not help but wonder how they evolved. She noted in her book-

If we consider just the first number, one, and add one to it, the result is 2. Repeating this process results in counting numbers like 3, 4, and 5, also called natural numbers. Continuous subtraction produces zeros and negative integers.

The usual way of slotting numbers into categories is –

Natural or counting numbers are all positive integers and are named   ( N ), like 1, 2, 3, and 4, and whole numbers, named (W), are all natural numbers ( N ) and zero.

3.         Integers ( Z ) include negative integers and whole numbers.

4.         Rational numbers Symbol (Q): of the form p/q where q is not 0; this means all integral numbers can be expressed as rational numbers, keeping one as the denominator.

5.         Irrational numbers (P) These are non-terminating, non-repeating decimal numbers that cannot be expressed as fractions. These numbers can be algebraic, meaning they are solutions of the root of an algebraic equation:

Transcendental, which are not roots and are non-computable.

6.         All polynomials are algebraic, but not all algebraic numbers are Polynomials. For example, y/(x-1) is not a polynomial but is algebraic as it is discontinuous at x = 1.

7.         Real Numbers (R) make up all rational and irrational numbers

Complex numbers “(C) “,

of the form  a+bi where a and b are real numbers and iota, ( i ) is √(−1). So i+

From the definition of various numbers, it follows. Any rational number can be expressed as

As the sum or difference of two or more numbers, x = (a+b) or (c—d).

As a product or dividend of two or more numbers, any number can be written as  product x = a × b or as a ratio  x = c/d

As an exponent like the nth power of a number like  x = a^n,

As a root of a number like  n√a = x

So, the same number can be expressed in different ways. Thus, the number 8 can be described as addition of its parts, e.g., (4 + 4), as a difference like (10-2), as a product like (4x²), or as an exponent (2^3) or as a root of some number.

(1 + x)ⁿ=1+nx/1!+(n(n-1) x²)/2!+…

Puzzle:

Kalawati writes: –

When we write a number in decimal Notation, it has a decreasing trend from left to right, like 234, but when we represent this number on a number line, you get it by going to the right of zero, in the opposite direction to a number representation. I just had this thought. It puzzled me.

And I thought for hours and then systematically.

The key lies in understanding the place value system that underpins our decimal Notation.

• Decimal Notation (Left to Right): When we write a number like 234, the position of each digit determines its value:
• The leftmost digit, ‘2’, is in the hundreds place, representing 2×100=200.
• The middle digit, ‘3’, is in the tens place, representing 3×10=30.
• The rightmost digit, ‘4’, is in the one’s place, representing 4×1=4.

• The number’s value is the sum of these place values: 200+30+4=234. The digits decrease in the power of ten as you move from left to right (hundreds, tens, ones).
• Number Line (Left to Right): The number line visually represents the magnitude or quantity of numbers in increasing order. As you move to the right from zero:
• You encounter increasingly larger positive values (1, 2, 3).
• Each step to the right represents adding a unit of one.

The “Puzzle” Resolved:

So, writing a number as 234 is “exponential,” with exponents increasing from right to left, and marking a number on a number line, you have to move 234 units to the right. “additive/subtractive captures the core difference in how these two systems represent the numerical value. So it’s a difference between symbolic representation and spatial visualisation.

The observer’s perspective is essential, for if you look at the number line from the opposite side, the perspective changes and the numbers start to grow to the left! Just like you write them.

She smiled at the thought — in numbers, the big part starts at the left; in walking, the right foot leads. So, which direction is truth?

Puzzle:

Hobbit to the Dragon in the North

From Bag End’s door, my worth does start, Each step to left, a greater heart My written form displays with pride, Though smaller steps it does confide.

But should you mark me on a track, Each move to the right, there’s no turning back, My value swells, quite plain to see, A longer stroll for little me.

Yet, if you squint, and walk the way From east to west at close of day, My growing path now seems to wane, Like shadows long on grassy lane.

What am I, simple yet profound, Whose bigness shifts on written ground And walking lines, depending where A Hobbit’s eye begins to stare?

Kalawati Gets a Pleasant Surprise

It was getting late, and cows had stopped eating and were gathering together for a journey back home. Kalawati closed her notebook and went home. It was 5 PM, and some guests were sitting in the garden with her parents. She came and greeted them; Kala,        after putting the cattle in Penn, said your face and come to have some snacks and buttermilk here.  She did as was her daily routine, spruced herself up and joined the parents and the guests.  There was inconsequential chit-chat, and then the guests left. Her mother turned to Kala and asked her, What about marriage? What about it Mama! Kala replied.

 How about marriage to this amiable boy you were chatting to so excitedly, telling him the binomial theorem? He was mostly listening, as you were just gushing. Kala smiled. Yes, he could listen quietly. It is time, Kala. Mama gave me a day, and she went to her room and ruminated on the cud of her future pleasantly. The following day, she said yes, the boy was studying in a college, doing physics honours at Delhi Hindu College.

She said yes, and then things happened quickly: she married in 1942, had her first child, a boy, in 1943, her second son a year later, and her third in 1956. In between, she completed her graduation in mathematics. Still, she never let go of her diary of reminiscences of the meadow and kept thinking even more basic about the heart of mathematics. In 1968, she was detected with advanced choriocarcinoma, and she passed swiftly into her meadow. She had had a full life, no regrets, an acclaimed professor of mathematics, and a great husband who always listened, loved and had talented and loving sons. What else is there? Her heart was at peace.

1974, army headquarters instituted a committee to assess lessons learnt and changes required. Kalawati’s son was assisting in that committee, and he opened her notebooks and saw how beautifully she had derived Poisson’s curve to diving into binomial mathematics; he proposed incorporation into the subject matter but was refused. They said a simple certainty of average is better than imagined extrapolation of complex Poisson’s distribution for military purposes, especially in casualty calculation.

• Lilavati’s Garden of Mathematics
• Lilavati’s Garden of Mathematics — Part II: The Equation of the Heart
• Disquisitions On Life
• Rats Tail Sign
• Physics Explained: From Relativity to Wireless Communication