A few years ago this equation spread in Vines and on the internet:

9 10 = 21

It is not true in the standard base 10 decimal system. But what if we modify the equation a bit with other number bases? Suppose the numbers on the left are in base *x* and the number on the right is in base *y*:

(9 10) (base *x*) = 21 (base *y*)

For what values of *x* and *y* is this equation true? This is actually a fun little problem. Watch the video for a solution.

Or keep reading.

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**Answer To The Viral Meme 9 10 = 21 Solved**

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Let’s expand each side.

(9 10) (base *x*) = 21 (base *y*)

9(1) [1(*x*) 0(1)] = 2(*y*) 1

Now we simplify and solve for *y*:

2*y* = 8 *x*

*y* = 4 *x*/2

Since we have number bases, we want *x* and *y* to be positive integers. The term *x*/2 requires that *x* be a positive even number.

Also since 9 is in base *x*, we have *x* ≥ 10, as the digit 9 would not be used for a base 9 or smaller.

Thus we have the pairs of solutions:

*x* = 10, so *y* = 9

*x* = 12, so *y* = 10

*x* = 14, so *y* = 12

…

*x*, *y* = 4 *x*/2

So at first 9 10 = 21 seems like a simple false equation. But if we think about number bases, there are an infinite* number of solutions–pretty neat!

(*countably infinite to be precise)

**Sources for meme**

https://www.quora.com/Is-9- -10-21

https://knowyourmeme.com/memes/9-10-21

**The base 10 number system**

The development and spread of decimal numerals is a fascinating history. I want to share a few interesting parts from Wikipedia:

Fact 1: the base 10 system was developed by Aryabhata in India, and Brahmagupta introduced the symbol for 0.

https://en.wikipedia.org/wiki/Numeral_system

(Quoting Wikipedia)

The most commonly used system of numerals is the Hindu–Arabic numeral system. Two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The numeral system and the zero concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial and military activities with India. The Arabs adopted and modified it. Even today, the Arabs call the numerals which they use “Raqam Al-Hind” or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the western world due to their trade links with them. The Western world modified them and called them the Arabic numerals, as they learned them from the Arabs. Hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is still used in India and neighbouring Nepal.

**Fact 2**: Fibonacci shared the method “how the Indians multiply” in 1202, but it took Europe hundreds of years to adopt the method. For all the poeple that think base 10 is natural since we have 10 fingers, I wonder: why did it take so long for Europe to adopt a “natural” system? I don’t think it’s so natural–the decimal system s a revolutionary idea, and we should give proper credit to the Indian mathematicians who developed it.

(I am intrigued at the similarities with a more recent episode. The method how the Japanese multiply is a fun way–not as revolutionary–to visualize multiplication and to learn group theory. I feel a bit like Fibonacci as others are very slow to accept the method’s value!)

https://en.wikipedia.org/wiki/Liber_Abaci#Modus_Indorum

(Quoting Wikipedia)

In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum (the method of the Indians), today known as Hindu–Arabic numeral system or base-10 positional notation. It also introduced digits that greatly resembled the modern Arabic numerals.

(Quoting translated *Liber Abaci* on Wikipedia): “There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method.

…

(Quoting Wikipedia)

In other words, in his book he advocated the use of the digits 0–9, and of place value. Until this time Europe used Roman Numerals, making modern mathematics almost impossible. The book thus made an important contribution to the spread of decimal numerals. The spread of the Hindu-Arabic system, however, as Ore writes, was “long-drawn-out”, taking many more centuries to spread widely, and did not become complete until the later part of the 16th century, accelerating dramatically only in the 1500s with the advent of printing.