Algebra and the concepts of fields, groups, and rings may seem very abstract. But we’ve all been familiar with numbers since first grade. Maybe some of you remember the first time you subtracted a number larger than 3 from 3 and didn’t know what the result would be. The result is a negative number. But if I collect apples, can I end up with half an apple?
Mathematics has names for different number systems and the rules that govern the operations on them. The integers—the positive and negative whole numbers, together with zero—form a so-called group under addition. Every number has an opposite, so adding them gives the neutral element: zero. If I give my friend two apples and he gives them back, he has zero of my apples again.
The concept of zero has a long history. The Bakhshali Manuscript contains an early written symbol for zero: a little dot used as a placeholder. The idea developed over time, but that dot was an important step towards the zero we use today.
The Wisdom of the Dot: Mathematics, Yoga and Life
In Indian tradition, the bindi or tilak is more than an adornment; it is a symbol rich in cultural, spiritual and philosophical meaning.
To a mathematician, a dot is the beginning of geometry—a point without dimensions, yet the foundation of shapes and structures. To a yogi, the centre of the forehead can symbolise awareness, insight and the inward journey of consciousness.
The red bindi has traditionally been associated with womanhood, fertility, auspiciousness and abundance. It can also serve as a poetic metaphor for creation: just as a single fertilised cell begins the remarkable journey of human development, a tiny dot can symbolise the immense potential of life.
Its central placement offers another lesson: the wisdom of balance. Hatha Yoga explores the relationship between opposing forces—effort and relaxation, activity and stillness, sun and moon. The centre becomes a metaphor for harmony, neither blindly favouring one extreme nor rejecting the other.
Modern physiology offers a parallel in homeostasis, the body’s continuous effort to maintain internal stability through adjustment and adaptation. Similarly, a balanced life is not static; it requires constant awareness and intelligent responses to changing circumstances.
Perhaps the bindi or tilak is best understood as a reminder of what we believe in and how we choose to live.
A dot may occupy almost no space, yet represent an entire universe of meaning. The mathematician sees a point, the yogi seeks the centre, and the philosopher discovers a lesson: true balance begins with awareness.
Actually, the integers are even more than a group: they form a so-called ring. That means we can do more than add integers—we can multiply them, too. If I want to sell apples, I might ask: “How much money will I get for my apples?” A similar problem, this time involving asava horses, haya horses, and camels, was already considered in the Bakhshali Manuscript more than a thousand years ago. The problem involves multiplying the number of each kind of animal by its price and adding the values.
In a ring, addition and multiplication are connected by the distributive law: a(b + c) = ab + ac. That might sound complicated, but it means we can work out the same total in different ways. It’s a bit like asking whether we cut the cakes at home and bring it along, or cut it together when we get there.
Cake division brings us closer to what mathematicians call a field. The rational numbers form a field: we can add, subtract, multiply, and divide by any nonzero rational number. All that cake division at school was actually useful: we didn’t have to give our friend a whole cake—we could give them a quarter or a half. How useful!
In German, a field is called a Körper, which also means “body.” That word made me wonder about a connection to yoga. What if we think of physical movement and the mind as two operations? In yoga, we use the body as a tool for the mind. Movement can go in both directions, and the mind can change its thoughts. And what about the distributive law? Well, practising asanas in one session to relieve stress and then restructuring our thoughts has the same effect as practising asanas before every single thought. But now we dive into deeper philosophy.
Many people can’t rest. And if I ever teach yoga in front of mathematicians and guide them into Śavāsana, I’ll simply say: “Let’s turn this ring into a field—or a Körper, as we say in German.”
References
[1] Herstein, I. N. Topics in Algebra. 2nd ed. John Wiley & Sons, 1975.
[2] Hayashi, Takao. The Bakhshālī Manuscript: An Ancient Indian Mathematical Treatise. Groningen Oriental Studies 1. Egbert Forsten, 1995.