We invited Geoffrey Shen to lead us through an exercise, starting from the seemingly simplest questions, triggering and expanding our in-depth thinking on mathematical problems, and cultivating core mathematical literacy.
In fact, among all numbers, 0 is one of the top five most important, if not the most important number.
What are the five most important numbers?
These are the five numbers that appear once each in Euler’s famous identity:
1, i, π, e, 0
What is 1?
1 is the real unit.
One begets two, two begets three, and three begets all things…
What is i?
i is the imaginary unit.
What is π?
π is 3.14…
Right?
The percentage of students who answer that π is 3.14 is usually more than 50%.
If someone answers that π is 3.14, there are only two possibilities: either the person is not familiar with mathematics, or their math teacher has taught them incorrectly.
So, what is π?
π is the ratio of a circle’s circumference to its diameter. The value of π is approximately 3.14.
What is e?
e is the first letter of the term “exponential” and also the first letter of the great mathematician Euler. This is quite a coincidence.
e is a mathematical constant often used as the base for exponential and logarithmic functions.
The common definition of e is the limit of (1 + 1/n)^n as n approaches infinity. The concept of limits falls into the realm of advanced mathematics.
Having introduced the four main characters on the left side of Euler’s identity, let’s turn our attention to the last character on the right side.
What is 0?
0. Looking at this symbol, it resembles a hollow circle.
Yes, hollow…representing nothingness.
According to archaeological research, the invention of the number 0 dates back several hundred years later than other numbers.
Why?
Because when you look around, you don’t see “0”.
Our eyes can only see tangible things, not intangible ones, even though the latter is far more important.
One fruit is visible, two fruits are visible, three fruits are visible. But where do you see 0 fruits?
How was this invisible “0” discovered?
According to archaeological research, this symbol 0 first appeared on the wall of a temple in India.
Why India?
India’s religious beliefs strongly advocate nothingness. They believe that the universe was born from nothingness, and nothingness is the ultimate goal of humanity.
People who advocate nothingness are more likely to discover nothingness and define it.
Nothingness is not optional; it is essential.
Ancient mathematicians, while doing arithmetic with stones on the beach, noticed that when they placed a stone, then removed one, a circular mark was left in the sand – this may have been the origin of the symbol 0.
The discovery of 0 significantly simplified the expression of numbers, making numerical representation much more straightforward.
Without 0, expressing 305 would be ambiguous. Writing 35 or 3 5 (with a space in between) would create confusion or misunderstanding. To avoid ambiguity, one would have to write 3 hundred 5, making it unnecessarily long.
Moreover, 0 has a special property: a product of a series of factors equals 0 if at least one factor is 0. No other number has a similar property.
In fact, in solving equations or inequalities, we often rely on this special property of 0.
For example, when solving the equation (x – 3)(x + 1) = 12, the first step is to set the expression on the right side (usually the right side) equal to 0.
The last step also relies on the special property of 0.
For example, in solving inequalities with high-degree polynomials, the commonly used method or technique essentially relies on the special property of 0.
The single-side zeroing method is often used in some sequence problems involving inequality proofs.
Don’t forget, in the proof that -3 times -4 equals 12, 0 played an indispensable and crucial role.
0 also has a unique characteristic: it cannot be a divisor in division. Because no number multiplied by 0 equals 1.
However, as the divisor becomes smaller and smaller, the quotient becomes larger and larger. Dividing 1 by 0 introduces the concept of infinity, another breakthrough in mathematics.
Don’t underestimate 0, don’t ignore anything.
From nothing to something, and then from something to nothing, isn’t this the essence of human life?
One flower, one world; one leaf, one enlightenment.
A tiny 0 holds so much wisdom, let alone the myriad of other things in the world.
Our knowledge, compared to all the knowledge in the world, is like a 0. Our years on Earth, compared to the age of the Earth and the universe, are also like a 0. So, what is there to be proud of?
Only by maintaining humility, maintaining an open-minded attitude, like the 0 in division, can we break through ourselves and move towards infinity.
