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Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

ⵎⴰⵛⴽⵓ ⵜⴳⴰ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⴳⴳⴰⴹⵜ ⵏ ⵓⵙⵉⵍⴰ ⵏ 1, (ⴽ1) ⵜⴳⴰ ⴰⵀⴰ 1, ⵓⴹⵓ ⵏⵏⵙ ⵓⵔ ⵉⵍⵍⵉ (ⵏⵏⴰ ⵎⵉ ⵔⴰⴷ ⵙⵎⵎⴰⵏ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵙⵉⵍⴰ 1, ⵉⴳ ⵉⵍⵍⴰ).

Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties.

ⵙ ⵓⵏⴰⵡⴰⵢ, ⴷⴰ ⵡⴰⵍⴰ ⵜⵜⵓⵔⵛⴰⵎⵏ ⵉⵙⵓⵜⵜⵉⵢⵏ ⴳ ⵉⵎⵏⵉⴷⵉⵍⵏ ⴰⵡⵢⵉⵡⵏ ( ⵉⵎⵏⵉⴷⵉⵍⵏ ⵉⴷ ⵎ ⵢⵓⵡⵏ ⵓⴽⵙⴰⵢ), ⴰⵛⴽⵓ ⴷⴰ ⵡⴰⵍⴰ ⵖⵓⵔⵙ ⵜⵜⵉⵍⵉⵏ ⵉⵎⵥⵍⴰⵢ ⵉⵜⵜⵓⵔⴰⵏ ⴽⵉⴳⴰⵏ.

It is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

ⵏⵜⵜⴰ ⴰⵡⴷ ⴰⵢⴷ ⵉⴳⴰⵏ ⵓⵟⵟⵓⵏ ⴰⵎⵣⵡⴰⵔⵓ ⴷ ⵡⵉⵙⵙ ⵙⵉⵏ ⴳ ⵜⵙⵏⵙⵍⵜ ⵏ “ⴼⵉⴱⵓⵏⴰⵜⵛⵉ (0 ⵉⵙⵎⴷⵢⴰ ⴰⵎⵢⴰ), ⵏⵜⵜⴰ ⴷ ⵓⵟⵟⵓⵏ ⴰⵎⵣⵡⴰⵔⵓ ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⵎⴹⴼⵕⵉⵏ ⵜⵓⵙⵏⴰⴽⵉⵏ ⵢⴰⴷⵏ.

Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵉⵖⵢ ⵍⵊⵉⴱⵔ ⴰⵡⵏⴳⵉⵎ ⴰⴷ ⵉⵙⴽⵙⵡ ⵖⵔ ⵉⴳⵔ ⵙ ⵢⵓⵡⵏ ⵓⴼⵕⴹⵉⵚ, ⴷ ⵏⵜⵜⴰ ⵓⵔ ⵉⴳⵉ ⴰⴼⵕⴹⵉⵚ ⵉⵥⵍⵉⵏ ⵡⵍⴰ ⵉⴳⴰ ⵜⴰⵔⴰⴱⴱⵓⵜ ⴰⴽⴽⵯ.

A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.

ⵍⴽⵓⴷ ⴱⵓ-ⵙⵉⵏ ⵉⴳⴰ ⵜⴰⴳⴼⴼⵓⵔⵜ 1 ⴷ 0 ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵉⵏⴳⵎⴰⵎ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵎⴰⵔ ⴰⴷ ⵙⵎⴷⵢⴰⵏ ⵙ ⴽⴰ ⵉⴳⴰⵜ ⵡⴰⵏⴰⵡ ⵏ ⵓⵏⵎⵎⴰⵍ.

+1 is the electric charge of positrons and protons.

+1 ⵜⴳⴰ ⵜⴰⴽⵜⵔⵜ ⵜⴰⵎⵥⵥⴰⵕⵓⵕⵜ ⵏ ⵍⴱⵓⵥⵕⵓⵏⴰⵜ ⴷ ⵍⴱⵓⵟⵕⵓⵏⴰⵜ.

The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number.

ⵉⵙⵍⴽⵏ ⵓⴼⴰⵢⵍⴰⵙⵓⴼ ⵏⵢⵓⴼⵉⵜⴰⵖⵓⵔⵉ “ ⵏⵉⵎⵓⴽⴰⵅⴰⵔⵓⵙ” ⴳ ⴳⵔⴰⵙⴰ ⵉⵙ ⵓⵔ ⵉⴳⵉ ⵢⴰⵏ ⵓⵟⵟⵓⵏ, ⵎⴰⴽⴰ ⴰⵙⴰⴳⵯⵎ ⵏ ⵉⵎⵉⴹ.

We Are Number One is a 2014 song from the children's TV show LazyTown, which gained popularity as a meme.

“ⵏⴽⵯⵏⵉ ⵓⵟⵟⵓⵏ ⵢⴰⵏ” ⵉⵙⵎ ⵏ ⵢⴰⵜ ⵜⵣⵍⵉⵜ ⴳ 2014, ⵙⴳ ⵓⵖⴰⵡⴰⵙ ⴰⵜⵍⴼⴰⵣ ⵏ ⵉⵎⵥⵥⴰⵏ “ⵍⴰⵣⵉ-ⵜⴰⵡⵏ” ⵉⵍⴰⵏ ⵢⴰⵜ ⵜⵎⴷⵏⴰⵏⵜ ⴰⵛⴽⵓ.

In association football (soccer) the number 1 is often given to the goalkeeper.

ⴳ ⵜⴰⴽⵓⵔⵜ ⵏ ⵓⴹⴰⵕ ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵓⴼⴽⴰ ⵡⵓⵟⵟⵓⵏ 1 ⵉ ⵉⵎⵃⴹⵉ ⵏ ⵜⵙⵉⵙⵡⵉⵜ.

1 is the lowest number permitted for use by players of the National Hockey League (NHL); the league prohibited the use of 00 and 0 in the late 1990s (the highest number permitted being 98).

1 ⴷ ⵓⵟⵟⵓⵏ ⴰⵎⴰⵣⴷⴰⵔ ⵙ ⵥⴹⴰⵕⵏ ⴰⴷ ⵜⵜ ⵙⵙⵎⵔⵙⵏ ⵉⵎⵉⵔⴰⵔⵏ ⵏ ⵜⴱⵔⵣⴰ ⵜⴰⵏⴰⵎⵓⵔⵜ ⵏ “ⵍⵀⵓⴽⵉ” (ⵜ.ⵜ.ⵍ), ⵜⵜⵓⵙⴳⴷⵍ ⵜⴱⵔⵣⴰ ⴰⵙⵙⵎⵔⵙ ⵏ 00 ⴷ 0 ⴳ ⵜⵢⵉⵔⵉⵡⵉⵏ ⵏ 1990 ( ⵓⵟⵟⵓⵏ ⴰⵎⴰⴼⵍⵍⴰ ⵉⴳ ⵡⵉⵏ ⴰⴷ ⵉⵜⵜⵓⵙⵎⵔⵙ ⵉⴳⴰ ⵜ 98).

Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem.

ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵙⵏⵙⵍⵜ ⵜⴰⵎⵔⵡⵉⵜ ⵏ ⵡⵓⵟⵟⵓⵏ ⴷⵉⴽⵙ ⵜⵉⵍⵙⵉⵜⵉⵏ ⵜⵉⵎⴹⴼⵕⵉⵏ ⵜⵉⵖⵣⵣⴰⴼⵉⵏ ⵙ ⵜⴰⵍⵖⴰ ⵏ ⵓⵙⴽⴽⵛⴷ ⵓⵔ ⵜⵏⵏⵉ ⴰⴷ ⴰⵔⵡⴰⵢ, ⵙ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵣⵄⴹⵓⴹ ⵡⴰⵔⵜⵎⵉ.

"Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to ""square the circle""."

“ ⵡⵉⵙⵙ ⵙⵉⵏ, ⴰⵛⴽⵓ ⵓⵔ ⵜⵍⵍⵉ ⵜⵣⵎⵔⵜ ⵎⴰⵔ ⵉⵜⵜⵓⵙⴽⵔ ⵡⵓⵟⵟⵓⵏ ⴰⵎⴰⴼⵍⵍⴰ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵜⴰⴱⵉⵚⵓⵍⵜ ⴷ ⵜⵏⵓⵎⵉ, ⵓⵔ ⵏⵣⴹⴰⵕ ⴰⴷ ⵏⵔⴰⵔ ⵜⴰⵣⴳⵓⵏⵜ ⴰⴷ ⵜⴳ ⵜⴰⵎⴽⴽⵓⵥⵜ”.

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD).

ⵉⵙⵙⵎⵔⵙ ⵓⵎⵙⵏⵉⵜⵔⴰⵏ ⴰⵀⵉⵏⴷⵉ “ⴰⵔⵢⴰⴱⴰⵟⴰ” ⴰⵜⵉⴳ ⵏ 3.1416 ⴳ ⵓⴷⵍⵉⵙ ⵏⵏⵙ “ⴰⵔⵢⴰⴱⵉⵟⵉⵢⴰ” (499 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ).

The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×228 sides, which stood as the world record for about 180 years.

ⵉⴼⴰⵔⵙ ⵓⵎⵓⵙⵏⵉⵜⵔⴰⵏ ⴰⴼⴰⵔⵉⵙⵉ “ ⵊⴰⵎⵛⵉⴷ ⴰⵍⴽⴰⵛⵉ” 9 ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵚⴹⵉⵚ-ⵎⵔⴰⵡ, ⵉⵜⴳⴳⴰ ⵣⵓⵏⴷ 16 ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵔⴰⵡ ⴳ ⵓⵣⵙⴳⴳⵯⴰⵙ ⵏ 1424 ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵓⵎⵖⵣⴷⵙ ⴳ ⵍⵍⴰⵏ 3×228 ⵏ ⵓⵙⴳⴰ, ⵏⵏⴰ ⵉⵙⵎⴷⵢⴰⵏ ⴷ ⵓⵟⵟⵓⵏ ⴰⵎⵙⵖⴰⵍ ⴰⵎⴰⴹⵍⴰⵏ ⴽⴰⵏ 180 ⵏ ⵓⵙⴳⴳⵯⴰⵙ.

These avoid reliance on infinite series.

ⵜⴰⴷⵖ ⵜⵜⴰⵔⵉ ⵉ ⵜⴰⵙⵖⵣⵏⵜ ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵡⴰⵔⵜⵎⵉ.

As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

ⵉⵎⴽⵉⵏⵏⴰ ⵜ ⵉⵙⵏⵎ ⵙⴰⵍⴰⵎⵉⵏ ⴷ ⴱⵔⴰⵏⵜ, ⴷ ⵜⵜⵓⵢⴰⵙⵏ ⴰⵡⴷ ⵙ ⵢⵉⵙⵎ ⵏ “ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⴱⵔⴰⵏⵜ- ⵙⴰⵍⴰⵎⵉⵏ”.

This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.

ⵓⵔ ⵉⴷ ⴰⵎ “ⴰⵍⴳⵓⵔⵉⵜⵎ ⵡⴰⵔⵜⵎⵉ” ⵏⵖⴷ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵜⵉⵎⵢⴰⵍⵙⵉⵏ” ⵏⵏⴰ ⵉⵃⵟⵟⵓⵏ ⵓⵟⵟⵓⵏ ⵉⵎⴰⵣⴰⵏ ⴽⵓⵍ, ⴰⵔⵜⵏ ⵜⵙⵙⵎⵔⴰⵙ ⴰⵔⴷ ⵜⵙⵙⵓⴼⵖ ⵜⴰⵢⴰⴼⵓⵜ ⵜⴰⵔⵜⵎⵉ.

Such memorization aids are called mnemonics.

ⵜⵜⵓⵢⴰⵙⵙⵏ ⵉⵙⴳⴳⵓⵔⵏ ⴰⴷ ⵏ ⵓⵃⵟⵟⵓ ⵙ ⵜⴰⵥⵓⵕⵉ ⵏ ⵓⵙⵎⵎⴽⵜⵉ.

The digits are large wooden characters attached to the dome-like ceiling.

ⵓⵟⵟⵓⵏ ⴳⴰⵏ ⵉⵙⴽⴽⵉⵍⵏ ⵉⵅⴰⵜⴰⵔⵏ ⵏ ⵓⴽⵛⵛⵓⴹ ⵎⵎⵣⴷⴰⵢⵏ ⵙ ⵓⵙⵔⴰⴳ ⵢⴰⵖⵏ ⴳ ⵜⴽⵔⴱⵓⵙⵜ.

"A numerical digit is a single symbol used alone (such as ""2"") or in combinations (such as ""25""), to represent numbers in a positional numeral system."

ⵓⵟⵟⵓⵏ ⴰⵎⵉⴹⴰⵏ ⵉⴳⴰ ⵜⴰⵎⴰⵜⴰⵔⵜ ⵉ ⵡⴰⴹⵓⵏ ⵏⵏⵙ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵉ ⵡⴰⴹⵓⵏⵙ (ⴰⵎⵎ “2”), ⵏⵖⴷ ⴳ ⵜⵔⵓⴱⴱⴰ ( ⵣⵓⵏⴷ “25”), ⵉ ⵓⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵓⵙⵏⵇⴹ ⴰⵙⵓⵔⵙⴰⵏ.

A positional number system has one unique digit for each integer from zero up to, but not including, the radix of the number system.

ⵉⵍⵍⴰ ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵙⵓⵔⵙⴰⵏ; ⵢⵓⵡⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵥⵍⵉⵏ ⴳ ⴽⵓ ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⴰⵡⴷ ⴳ ⵓⵎⵢⴰ, ⵎⴰⴽⴰ ⵓⵔ ⵉⵙⵎⴰⵏ ⴰⵙⵉⵍⴰ ⴰⵏⴰⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ.

The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.

ⴽⴽⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵥⵖⵓⵕⴰⵏⴻⵏ ⴰⴽⵙⵓⵍⵏ ⴽⵉⴳⴰⵏ ⴷ ⵡⵓⵟⵟⵓⵏ ⵉⵜⵔⴰⵔⵏ, ⴰⵔ ⴷ ⵏⵏ ⴰⴽⴽⵯ ⵏⴰⵡⴹ ⵓⵟⵟⵓⵏ ⵏ ⵡⵓⵏⵓⵖ ⵉⵜⵜⵓⵙⵎⵔⵙⵏ ⴰⴷ ⵙⵎⴷⵢⴰⵏ ⵓⵟⵟⵓⵏ.

The Mayas used a shell symbol to represent zero.

ⵉⵙⵙⵎⵔⵙ ⵎⴰⵢⴰⵣ “ⵜⵉⵇⵛⵔⵜ” ⴰⴷ ⵙⵎⴷⵢⴰⵏ ⵉ ⵓⵎⵢⴰ.

The Thai numeral system is identical to the Hindu–Arabic numeral system except for the symbols used to represent digits.

ⵉⵎⵙⴰⵙⴰ ⵓⵏⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ ⵏ ⵟⴰⵢⵍⴰⵏⴷ ⴰⴽⴷ ⵓⵏⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ ⵏ ⵀⵉⵏⴷⵓ- ⴰⵄⵕⴰⴱ ⵅⵙ ⵜⵉⵎⴰⵜⴰⵔⵉⵏ ⵉⵜⵜⵓⵙⵎⵔⵙⵏⵜ ⴰⴷ ⵙⵎⴷⵢⴰⵏⵜ ⵓⵟⵟⵓⵏⵏ.

They are both base 3 systems.

ⴽⵓⵍⵍⵓⵜⵏ ⴳⴰⵏ ⴱⴷⴷⴰⵏ ⵖⴼ 3 ⵉⵎⴰⴳⴰⵡⵏ.

Several authors in the last 300 years have noted a facility of positional notation that amounts to a modified decimal representation.

ⴰⵏⵏⴰⵢⵏ ⵡⴰⵀⵍⵉ ⵏ ⵉⵎⴳⴰⵢⵏ ⴳ 300 ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⵉⵣⵔⵉⵏ ⵏ ⵡⵉⵙ ⵉⵍⵍⴰ ⵡⴰⵎⵎⴰⴽ ⵏ ⵓⵣⵎⵎⴻⵎ ⴰⵙⵓⵔⵙⴰⵏ ⵜⴳⵓⵍⴰⵏ ⴰⵙⵡⵉⵔ ⵏ ⵓⵙⵎⴷⵢⴰ ⴰⵎⵔⴰⵡ ⵉⵜⵢⴰⵍⵙⵏ.

For example, 1111 (one thousand, one hundred and eleven) is a repunit.

ⵙ ⵓⵎⴷⵢⴰ, 1111 ( ⵉⴼⴹ ⴷ ⵜⵎⵉⴹⵉ ⴷ ⵢⴰⵏ ⴷ ⵎⵔⴰⵡ) ⵉⴳⴰⵏ ⵢⴰⵜ ⵜⴰⵍⵍⴰⵙⵜ.

Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers.

ⴳ ⵜⵙⴳⴰ ⵏ ⵓⵙⵙⵉⵟⵏ ⵏ ⵎⵔⴰⵡ ⵉⴹⵓⴷⴰⵏ, ⴽⴰⵏ ⵜⴷⵍⵙⵉⵡⵉⵏ ⴰⵙⵙⵉⵟⵏ ⵏ ⵜⵖⵏⴼⵉⴼⵉⵏ ⴷ ⵓⵙⵜⵓⵎ ⵉⵍⵍⴰⵏ ⴳⵔ ⵉⴹⵓⴷⴰⵏ ⵏ ⵉⴼⴰⵙⵙⵏ ⴷ ⵡⵉⵏ ⵉⴹⴰⵕⵏ ⴷ ⵉⴹⵓⴷⴰⵏ.

Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods.

ⵙⵙⵎⵔⵙⵏⵜ ⵜⴷⵍⵙⵉⵡⵉⵏ ⵏ ⵓⵣⵎⵣ-ⴰⴳⴳⵓⵏ ⴳ ⴰⵎⵓⵏⵜ ⵜⵔⵓⴱⴱⴰ ⵜⵉⵎⵉⵔⵉⴽⴰⵏⵉⵏ ⵜⵉⵇⴱⵓⵕⵉⵏ, ⵉⵎⴹⴰⵏ ⵉ ⵜⵇⴱⴱⴰⴹⵜ ⴷ ⵜⵡⵓⵔⵉⵡⵉⵏ ⵜⵉⵏⵎⵖⵓⵔⵉⵏ ⴷ ⵜⵖⴰⵡⵙⵉⵡⵉⵏ ⵏ ⵎⵏⵣⵉⵡⵜ.

Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked.

ⵙⴳ ⵡⴰⵜⵜⴰⵢⵏ ⵏ 3500 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ ⵜⵜⵓⵙⵏⴼⵍⵜ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵏ ⵡⴰⵍⵓⴹ ⵉⵎⵉⴽ ⵙ ⵉⵎⵉⴽ ⵖⵔ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵙⵙⵉⴳⴳⵣⵏ ⵙ ⵓⵖⴰⵏⵉⴱ ⴰⵡⵔⴻⵔⵔⴰⵢ ⵉⵍⴰⵏ ⵜⵉⵖⵎⵔⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ ⴳ ⵜⴼⵉⵍⵉⵜⵉⵏ ⵏ ⵡⴰⵍⵓⴹ ( ⵉⴽⴽⴰⵏ ⴳⴰⵏⵜ ⵜⵉⵏ ⵜⵜⵓⴽⵏⴰⵜ) ⵏⵏⴰ ⵢⴰⴷ ⵉⵏⵡⴰⵏ.

These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs.

ⴰⵖⵏⵜ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⴽⵎⴰⵎ ⴰⴷ, ⵜⵉⵎⵉⵜⴰⵔ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵡⵔⴻⵔⵔⴰⵢ ⵏⵏⴰ ⵉⵍⵍⴰⵏ ⴳ ⵓⴷⵖⴰⵔ ⵏⵏⵙ ⵜⵃⴹⵓ ⴰⵜⵉⴳ ⵏ ⵓⵣⵎⵎⴻⵎ ⵏ ⵜⵎⵉⵜⴰⵔ ⵏ ⵜⵓⵔⵏⵓⵜ ⵉ ⵜⵎⵉⵜⴰⵔ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵇⵍⴰⵍⵍⴰⵢ.

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons.

ⴽⴽⴰⵏⵜ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵚⴹⵉⵚ-ⵎⵔⴰⵡ ⴳⴰⵏ ⴰⵏⴳⵔⴰⵡ ⵏ ⵍⵊⵉⴷⵔ ⵉⵛⵛⴰⵔⵏ ⵉⵃⵟⵟⵓⵏ ⵜⴰⵍⴳⴰⵎⵜ 10 ⵜⴰⵡⴰⵍⵉⵜ ⴷ ⵜⴰⵍⴳⴰⵎⵜ 6 ⴳ ⵜⴳⴼⴼⵓⵔⵜ ⴷ ⵜⴳⵓⵙⵉⵏ ⵜⵉⴽⵎⴰⵎⵉⵏ ⵉⴱⴷⴷⴰⵏ ⴷ ⵛⵉⴼⵔⵓⵏ.

Unique numbers of troops and measures of rice appear as unique combinations of these tallies.

ⴷⴰⴷ ⵜⴼⴼⵖⵏ ⵉⵎⴹⴰⵏ ⵉⵥⵍⵉⵏ ⵏ ⵜⴰⴳⴳⴰⵍⵉⵏ ⴷ ⵉⵎⵙⵖⴰⵍⵏ ⵏ ⵉⴷⴳⵍ ⴰⵎ ⵜⵔⵓⴱⴱⴰ ⵉⵙⵜⵉⵏ ⴳ ⵉⵎⴹⴰⵏ ⴰⴷ.

Conventional tallies are quite difficult to multiply and divide.

ⵉⵛⵇⵇⴰ ⴽⵉⴳⴰⵏ ⵓⵙⴼⵓⴽⵜⵉ ⴷ ⴱⵟⵟⵓ ⵏ ⵉⵎⴹⴰⵏ ⵉⵣⴰⵢⴽⵓⵜⵏ.

Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.

ⵙⵙⵏⵜⵉⵏ ⵡⵓⴷⴰⵢⵏ ⴰⵙⵙⵎⵔⵙ ⵏ ⵓⵏⴳⵔⴰⵡ ⴰⵎⵎ ⵡⴰ (ⵓⵟⵟⵓⵏ ⵏ ⵍⵄⵉⴱⵔⵉⵢⴰ), ⴰⵛⴽⵓ ⴽⴽⴰⵏⵜ ⵉⵎⴷⵢⴰⵜⵏ ⵉⵇⴱⵓⵔⵏ ⵉⵜⵢⴰⵙⵙⵏ ⴰⴷⵔⵉⵎ ⴰⵣⴰⵖⵓⵔ ⴰⵜⵜⴰⵢⵏ ⵏ 100 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ.

The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.

ⵜⵙⵙⵎⵔⵙ ⵍⵎⴰⵢⴰ ⴳ ⴰⵎⵔⵉⴽⴰ ⵏ ⵡⴰⵎⵎⴰⵙ; ⴰⵏⴳⵔⴰⵡ ⵉⵛⵛⴰⵔⵏ ⴳⵔ ⵜⵍⴳⴰⵎⵜ 18 ⴷ ⵜⵍⴳⴰⵎⵜ 20, ⵉⵖⵢ ⵉⵙ ⴰⵙⵏⵜⵉⴷ ⵓⴷⵊⴰⵏ ⵓⵍⵎⴽ, ⴳ ⵜⵍⵍⴰ ⵜⴱⵖⵓⵔⵜ ⵢⴰⵜⵜⵓⵢⵏ ⵣⵓⵏⴷ ⴰⵣⵎⵎⴻⵎ ⴰⵙⵓⵔⵙⴰⵏ ⴷ ⵓⵎⵢⴰ.

Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

ⵜⵜⵡⴰⵎⵙⴰⵢ ⵜⵓⵙⵙⴰⵏ ⵏ ⵉⵙⵏⵜⵍⵏ ⵏ ⵓⴳⵍⵍⴰⵢ ⴷ ⵉⴽⵯⵍⴰⵏ ⴳ ⵖⵓⵔ ⵡⴰⵣⵣⴰⵖⵏ ⵏ ⵙⴱⴰⵏⵢⴰ ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 16, ⵓⵔ ⵜⵏⵊⵉⵎ ⵡⴰⵅⵅⴰ ⵍⵍⴰⵏ ⵉⵏⴳⵎⴰⵎ ⵓⵏⵣⵉⵍⵏ ⵏ ⵓⵣⵎⵎⴻⵎ ⵢⴰⵖⵏ ⴳ “ⴽⵉⴱⴱⵓ”, ⵜⵙⵓⵍ ⴷⴰ ⵜⵜⵓⵙⵙⵎⵔⴰⵙ ⴳ ⵡⴰⵏⵙⴰ ⵏ “ⵍⴰⵏⴷⵉⵣ”.

Zero was first used in India in the 7th century CE by Brahmagupta.

ⵉⵜⵜⵓⵙⵎⵔⵙ ⵓⵎⵢⴰ ⵜⵉⴽⵍⵜ ⵉⵣⵡⴰⵔⵏ ⴳ ⵍⵀⵉⵏⴷ ⵙⴳ ⵖⵓⵔ “ⴱⵔⴰⵀⵎⴰⴳⵓⵜⴰ”, ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 7 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⴰⵎⵙⵉⵃ.

Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th  century.

ⵙⴱⵉⵔⵡⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⴰⵄⵔⴰⴱⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴰⵏⴳⵔⴰⵡ ⵎⴰⵔ ⴰⴷ ⵢⴰⵎⵥ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵔⴰⵡⵏ, ⴷ ⵢⴰⵔⵓ ⵎⵓⵃⵎⵎⴰⴷ ⴱⵏ ⵎⵓⵙⴰ ⵍⵅⴰⵡⴰⵔⵉⵣⵎⵉ ⵢⴰⵜ ⵜⵡⵓⵔⵉ ⵉⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⵖⴼ ⵓⵢⵏⵏⴰⵖ, ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 9.

The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz.

ⵉⵜⵢⴰⴼⵙⴰⵔ ⵓⵏⴳⵔⴰⵡ ⴰⵎⵙⵉⵏ (ⵜⴰⵍⴳⴰⵎⵜ 2), ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 17 ⵙⴳ ⵖⵓⵔ ⴳⵓⵜⴼⵔⵉⴷ ⵍⵉⴱⵏⵉⵣ.

The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation.

ⵉⵎⵙⴽⵉⵍⵏ ⵙ ⵏⵏ ⵉⵇⵏ ⴰⴷ ⴰⵙ ⵜⵢⴰⴼⵙⴰⵢ ⵜⴳⴷⴰⵣⴰⵍⵜ, ⵜⵜⵓⵢⴰⵙⵙⵏ ⵙ ⵡⴰⵔⵉⵙⵎ, ⴷ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵜⵉⵍⵉⵎⵙⵍⵉⵜⵉⵏ ⵏⵏⴰ ⵉⵙⴽⴰⵔⵏ ⴰⵙⵉⴽⵙⵍ ⴳ ⵉⴼⵙⵙⴰⵢⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ.

A conditional equation is only true for particular values of the variables.

ⵜⴰⴳⴷⴰⵣⴰⵍⵜⵏ ⵜⴼⴰⴷⴰ ⵜⴳⴰ ⴷⴰⵢ ⵜⴰⵎⴷⴷⴰⴷⵜ ⵎⴰⵔ ⴰⴷ ⵜⴳ ⵜⴰⵣⵍⵖⴰ ⵉ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵏ ⵉⵙⵏⴼⴰⵍⵏ.

Very often the right-hand side of an equation is assumed to be zero.

ⴷⴰ ⵡⴰⵍⴰ ⵜⵜⴳⴳⴰ ⵜⵙⴳⴰ ⵜⴰⵢⴼⴼⴰⵙⵜ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴰⵎⵢⴰ.

An equation is analogous to a scale into which weights are placed.

ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵡⴰⵢⵜ ⵏ ⵓⵎⵙⵖⴰⵍ ⴳ ⵜⵜⵓⵢⴰⴳⴰⵏ ⵉⵙⵜⴰⵍⵏ.

This is the starting idea of algebraic geometry, an important area of mathematics.

ⵜⴰⴷⵖ ⴰⵢⴷ ⵉⴳⴰⵏ ⵜⴰⵡⵏⴳⵉⵎⵜ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵏ ⵓⵜⵡⴰⵍ ⴰⵍⵊⵉⴱⵔⵉ, ⴷ ⵜⴳⴰ ⵉⴳⵔ ⵉⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⴳ ⵜⵓⵙⵏⴰⴽⵜ.

To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.

ⵎⴰⵔ ⴰⴷ ⵏⴼⵙⵉ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵉⵍⵍⴰⵏ ⴳ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵡⵊⴰ, ⴷⴰ ⵉⵙⵙⵎⵔⴰⵙ ⵢⴰⵏ ⵜⴰⵜⵉⵇⵏⵉⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⵏⵖⴷ ⴰⵜⵡⴰⵍ ⴷ ⵉⴳⵎⴰⵏ ⴳ ⵍⵊⵉⴱⵔ ⴰⵡⵏⵖⴰⵏ, ⵏⵖⴷ ⴰⴼⵙⵙⴰⵢ ⴰⵡⵙⵏⴰⴽ.

These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

ⵜⵉⴳⴷⴰⵣⴰⵍⵉⵏ ⴰⴷ ⵛⵇⵇⴰⵏⵜ ⵙ ⵓⵎⴰⵜⴰ, ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵉⵏⵉⴳ ⵓⴼⴳⴰⵏ ⴰⴷ ⵢⴰⴼ ⴰⴼⵙⵙⴰⵢ ⵏⵖⴷ ⵡⴰⵔ ⴰⴼⵙⵙⴰⵢ, ⵎⴽ ⵉⵍⵍⴰ, ⴷⴰ ⵉⵙⵙⵉⵟⵉⵏ ⵎⵛⵜⴰ ⵏ ⵉⴼⵙⵙⴰⵢⵏ.

In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight.

ⴳ ⵡⵓⵏⵓⵖ ⵏ ⵓⵙⵙⵉⴽⵣ, “ⴽ”, “ⵢ” ⴷ “ⵣ” ⴳⴰⵏ ⴰⴽⴽⵯ ⴰⵏⵛⵜ ⵉⵎⵣⴰⵔⴰⵢⵏ ( ⴳ ⵡⴰⴷⴷⴰⴷ ⴰⴷ ⵓⵟⵟⵓⵏ ⵏ ⵜⵉⴷⵜ), ⵙ ⵎⴷⵢⴰⵏ ⵙ ⵉⵙⵜⴰⵍⵏ ⵉⵡⵔⴻⵔⵔⴰⵢⵏ, ⴷ ⵉ ⴽⵓ “ⴽ”, “ⵢ”, ⴷ “ⵣ” ⴰⵙⵜⴰⵍ ⵉⵙⵜⵉⵏ.

Hence, the equation with R unspecified is the general equation for the circle.

ⵙⴳ ⵓⵢⴰ, ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⴷ “ⵔ” ⵓⵔ ⵉⵥⵍⵉⵢⵏ, ⵜⴳⴰ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⵏⵎⴰⵜⴰⵢⵜ ⵏ ⵓⵡⵔⴻⵔⵔⵢ.

The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation.

ⴷⴰ ⴰⵙ ⵜⵜⵉⵏⵉⵏ ⵜⵉⴳⴳⵉⵜ ⵏ ⵢⵉⴼ ⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵏⵖⴷ ⴰⵙⵉⵡⵍ ⵖⴼ ⵉⵍⵉⵎⵙⵍⵉ ⴳ ⵜⵙⵖⵍⵜ, ⵎⴽ ⵜⴳⴰ ⵜⵙⵖⵍⵜ ⵜⵉⵏ ⵓⴼⵙⵙⴰⵢ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ.

Multiplying or dividing both sides of an equation by a non-zero quantity.

ⴰⵙⴼⵓⴽⵜⵉ ⵏⵖⴷ ⵜⵓⵟⵟⵓⵜ ⵙⵉⵏ ⵉⵎⵏⴰⴹⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵖⴼ ⵜⵙⵎⴽⵜⴰ ⵓⵔ ⵉⴳⵉⵏ ⴰⵎⵢⴰ.

An algebraic equation is univariate if it involves only one variable.

ⴷⴰ ⵜⴳⴳⴰ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴰⵍⵊⵉⴱⵔ ⵎⵎ ⵢⵓⵡⵏ ⵓⵙⵏⴼⵍ ⵉⴳ ⴷⵉⴽⵙ ⵢⴰⵏ ⵓⵙⵏⴼⵍ ⴷⴰⵢ.

In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics.

ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴳⴰ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⴳⵔⵔⴰⵢⵏ ⵉⵡⵏⵖⴰⵏⵏ ⴰⵙⵉⵍⴰⵏ, ⴷ ⵉⵎⵉⴽ ⴰⴷⵙⵍⴰⵏ ⴳ ⵍⵊⵉⴱⵔ ⴰⵡⵏⵖⴰⵏ, ⵏⵜⵜⴰ ⴷ ⴰⵙⴳⵓⵎ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⴽⵉⴳⴰⵏ ⵉⴼⵔⴷⴰⵙ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵜⴰⵜⵔⴰⵔⵜ.

This formalism allows one to determine the positions and the properties of the focuses of a conic.

ⴷⴰ ⵉⵜⵜⴰⴷⵊⴰ ⵓⴽⵓⵔⵎⵉⵙ ⴰⴷ ⴰⴼⴳⴰⵏ ⴰⴷ ⵉⵙⵜⵉ ⵉⵎⵏⴰⴷⵏ ⴷ ⵜⵥⵍⴰⵢⵉⵏ ⵏ ⵉⵖⵉⵙⴰⵏ ⵉⵛⵄⴱⴰ.

This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

ⵜⴰⵏⵏⴰⵢⵜ ⴰⴷ, ⵏⵏⴰ ⵉⵥⵍⵉ ⴷⵉⴽⴰⵔⵜ ⴷⴰ ⵜⵀⵢⵢⴰ ⴰⵔ ⵜⴳⴰⴷⴷⴰ ⴰⵏⴰⵡ ⵏ ⵜⵏⵣⴳⵉⵜ ⵏⵏⴰ ⵙⵏⵓⵎⵍⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵉⵢⵓⵏⴰⵏⵉⵢⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵜⴰⵇⴱⵓⵔⵜ.

An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⴷⵢⵓⴼⴰⵏⵜⵉⵏ ⵜⴰⴳⴳⴰⴷⵜ, ⵜⴳⴰ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵏⵏⴰ ⴳ ⵉⵖⵢ ⴰⴷ ⵢⵉⵍⵉ ⵡⵉⵏⵏⴰ ⵉⵇⵇⴰⵔⵏ ⵙ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵎⵎ ⵙⵉⵏ ⵡⴰⵔⵉⵙⵎⴰⵡⵏ.

Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

ⵜⴱⴷⴷⴰ ⵜⵏⵣⴳⵉⵜ ⵜⴰⵜⵔⴰⵔⵜ ⵏ ⵍⵊⵉⴱⵔ ⵖⴼ ⵜⵉⵇⵏⵉⵜⵉⵏ ⵡⴰⵍⴰ ⵉⵏⵣⵣⵖⵏ ⵍⵊⵉⴱⵔ ⴰⵡⵏⴳⵉⵎ, ⵏⵓⵎⴰⵔ ⵍⵊⵉⴱⵔ ⵉⵜⵜⵎⵏⴼⴽⴰⵏ ⴰⴽⴷ ⵜⵓⵜⵍⴰⵢⵜ ⴷ ⵜⵎⵓⴽⵔⵉⵙⵉⵏ ⵏ ⵜⵏⵣⴳⵉⵜ.

A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.

ⴷⴰ ⵜⵜⵓⵎⵓ ⵜⵏⵇⵇⵉⴹⵜ ⵜⴰⵙⵡⵉⵔⴰⵏⵜ ⴳ ⵉⵣⵔⵉⵔⵉⴳ ⵏ ⵍⵊⵉⴱⵔ ⵉⴳ ⵟⴼⵏ ⵉⵣⴷⴰⵢⵏ ⵏⵏⵙ ⵖⵔ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵉⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵡⵜⵜⴰ.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation.

ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⵉⵎⵣⴰⵔⴰⵢⵏ, ⴷⴰ ⵜⵜⵓⵖⵔⴰⵏⵜ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵜⵉⵎⵓⴼⴰⵢⵉⵏ ⴳ ⵡⴰⵀⵍⵉ ⵏ ⵜⵖⵎⵔⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ, ⵡⴰⵀⵍⵉ ⴷⵉⴽⵙ ⵉⴳⴰ ⵡⵉⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵏⵏⵙ ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⵙⵖⵏⴰⵏ ⵉⵜⴳⴳⴰⵏ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained.

ⵜⵉⴳⴷⴰⵣⴰⵍⵉⵏ ⵜⵉⵣⵔⴰⵔⴰⴳⵉⵏ ⵉⵎⵣⴰⵔⴰⵢ, ⵏⵏⴰ ⵉⵍⴰⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵉⵖⵢ ⴰⴷ ⵜⵜⵓⵔⵏⵓ ⵜⵜⵓⵙⴼⵓⴽⵜⵓ ⵙ ⵡⴰⵜⵉⴳ ⵉⴱⴷⴷⴰⵏ, ⵜⵥⵍⵉ ⴷ ⵜⵜⵓⵔⵎⴰⵙ ⴽⵉⴳⴰⵏ, ⴷ ⴷⴰ ⵜⵜⴰⴼⴰ ⵉⴼⵙⵙⴰⵢⵏ ⵉⵍⴰⵏ ⵜⵍⴰⵖⴰ ⵜⵓⵏⵖⵉⴷⵜ ⴳⵉⵏ ⵉⵎⴰⵇⵇⴰⵏ.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.

ⵉⵖⵢ ⴰⴷ ⵏⵙⵎⵔⵙ ⵉⴳⵎⴰⵎⵏ ⵏ “PDE” ⵎⴰⵔ ⴰⴷ ⵏⵙⵏⵓⵎⵎⵍ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⵏⴰⵡⵏ ⵏ ⵜⵓⵎⴰⵏⵉⵏ ⵣⵓⵏⴷ ⵉⵎⵙⵍⵉ ⴷ ⵜⵉⵔⵖⵉ ⵏⵖⴷ ⵉⵍⵓⴽⵜⵔⵓⵙⵜⴰⵜⵉⴽ, ⵉⵍⵓⴽⵜⵔⵓⴷⵉⵏⴰⵎⵉⴽ, ⵏⵖⴷ ⵜⴰⵏⵖⵍⴰ ⵏ ⵉⴱⵍⵓⵍⵉⵡⵏ, ⵏⵖⴷ ⴰⵔⵜⵓⵜⵎ, ⵏⵖⴷ ⵜⴰⵎⵉⴽⴰⵏⵉⴽⵜ ⵏ ⵉⴷⴰⵎⵏ.

A solution is an assignment of values to the unknown variables that makes the equality in the equation true.

ⴰⴼⵙⵙⴰⵢ ⵉⴳⴰⵜ ⴰⴷ ⵏⴰⴷⵊ ⵉ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵜⵉⵎⵙⵏⴼⴰⵍⵉⵏ ⵓⵔ ⵉⵜⵢⴰⵙⵙⵏ ⵏⵏⴰ ⵉⵜⵜⴰⴷⵊⴰⵏ ⴰⵏⴳⵉⴷⴷⵉ ⴳ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⴷⴷⴰⴷⵜ.

The set of all solutions of an equation is its solution set.

ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⴼⵙⵙⴰⵢⵏ ⴰⴽⴽⵯ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴰⵢⴷ ⵉⴳⴰⵏ ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵉⵥⵍⵉⵏ ⵥⴰⵕⵙ.

Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given interval.

ⵙⴳ ⵓⵙⴰⵜⴰⵍ ⵉⵖⵢ ⴰⴷ ⵉⵜⵜⵓⴳ ⵓⴼⵙⵙⴰⵢ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴰⴱⵔⵉⴷ ⵖⵔ ⵉⴼⵙⵙⴰⵢⵏ ⵏⵏⵉⴹⵏ (ⵅⵙ ⴰⴷ ⴷⴰⵢ ⵜⴰⴼⴷ ⵢⴰⵏ ⵓⴼⵙⵙⴰⵢ), ⵏⵖⴷ ⵉⴼⵙⵙⴰⵢⵏ ⵎⴰⵕⵕⴰ, ⵏⵖⴷ ⴰⴼⵙⵙⴰⵢⵏ ⵉⵎⵙⴰⵙⴰⵏ ⴷ ⵜⵎⵉⵜⴰⵔ , ⵣⵓⵏⴷ ⵜⵉⵍⵉⵜ ⵙⴳ ⴽⴰⵏ ⵓⵣⵎⵣ.

In this case, the solutions cannot be listed.

ⴳ ⵡⴰⴷⴷⴰⴷ ⴰⴷ, ⵓⵔ ⵏⵣⴹⴰⵔ ⴰⴷ ⵏⴳ ⴰⵍⵍⴰⵙ ⵏ ⵉⴼⵙⵙⴰⵢⵏ.

The variety in types of equations is large, and so are the corresponding methods.

ⵜⴰⵏⴰⵡⴰⵢⵜ ⴳ ⵡⴰⵏⴰⵡⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵉⵎⵇⵇⵓⵔ, ⵓⵍⴰ ⴰⵡⴷ ⵜⵉⴱⵔⵉⴷⵉⵏ ⵏ ⵓⵎⵢⴰⵡⴰⴹ.

This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.

ⵉⵖⵢ ⴰⴷ ⵉⴳ ⵓⵢⴰ ⴰⵙⵔⴰⴳ ⵏ ⵜⴷⵔⵙⵉ ⵏ ⵜⵎⵓⵙⵏⴰ ⵜⵓⵙⵏⴰⴽⵜ, ⵄⴰⴷ ⴰⴼⵙⵙⴰⵢ ⵏ ⵉⵜⵙⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴹⴰⵕⵜ ⵜⵙⵓⵜⵉⵏ ⵏ ⵜⵣⵎⵎⴰⵔ.

Polynomials appear in many areas of mathematics and science.

ⴷⴰⴷ ⵜⴱⴰⵢⴰⵏⵜ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵢⵉⴳⵔⴰⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⵎⴰⵙⵙⴰⵏⵉⵏ.

Many authors use these two words interchangeably.

ⴷⴰ ⵉⵙⵙⵎⵔⴰⵙ ⴽⵉⴳⴰⵏ ⵏ ⵉⵎⴳⴰⵢⵏ ⵙⵏⴰⵜ ⵜⴳⵓⵔⵉⵡⵉⵏ ⴰⴷ ⵙ ⵓⵎⵔⴰⵔⴰ.

Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.

P ⴰⵢⴷ ⵉⴳⴰⵏ ⵉⵙⵎ ⵓⵏⵚⵉⴱ ⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵡⵜⵜⴰ ⵓⵔ ⵉⴷ P(x), ⵎⴰⴽⴰ ⴰⵙⵎⵔⵙ ⵏ ⵓⵏⵜⴰⵍ ⵓⵖⵔⵉⴼ P(x), ⵉⴷⴷⴰⴷ ⵙⴳ ⵓⵣⵎⵣ ⴳ ⵓⵔ ⵉⵚⴼⵉ ⵓⵙⵏⵓⵃⵢⵓ ⵏ ⴽⵉⴳⴰⵏ ⵉⵡⵜⵜⴰ ⴷ ⵜⵎⵙⴽⴰⵔⵜ ⵣⴰⵕⵙ ⵉⵙⵍⵖⵏ.

However, one may use it over any domain where addition and multiplication are defined (that is, any ring).

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵉⵖⵢ ⵓⴼⴳⴰⵏ ⴰⴷ ⵜⵜ ⵉⵙⵙⵎⵔⵙ ⴳ ⴽⴰ ⵉⴳⴰⵜ ⵉⴳⵔ ⴳ ⵉⵜⵢⴰⵙⵙⴰⵏ ⵓⵙⵎⵓⵏ ⴷ ⵓⵙⴼⵓⴽⵜⵉ ⵙ (ⴽⴰ ⵉⴳⴰⵜ, ⴷ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵅⵔⵙⵜ).

Polynomials of small degree have been given specific names.

ⵜⵉⴷ ⵎⵉ ⴳⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ ⵉⵍⴰⵏ ⵜⴰⵙⴽⴼⵍⵜ ⵎⵥⵥⵉⵢⵏ, ⴰⵢⴷ ⵉⴽⴰⵏ ⵉⵙⵎⴰⵡⵏ ⵉⵎⵥⵍⴰⵢⵏ.

The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial.

ⴱⵓ-ⵉⵡⵜⵜⴰ 0, ⵙ ⵏⵖⵢ ⴰⴷ ⵏⵉⵏⵉ ⵓⵔ ⴰⴽⴽⵯ ⵉⵍⵉ ⵉⵡⵜⵜⴰ ⴷⴰ ⵉⵜⵙⵎⵎⴰ ⴰⵎⵢⴰ ⵎⴳⴳⵓⴷⵢ ⵉⵡⵜⵜⴰ.

Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

ⵎⴰⵛⴽⵓ ⵍⴰⵏⵜ ⵜⵙⴽⴼⴰⵍ ⵜⴰⵔ-ⴰⵎⵢⴰ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ, ⵜⴳⴰ ⵜⴰⵙⴽⴼⵍⵜ ⵜⴰⵅⴰⵜⴰⵔⵜ ⴳ ⵢⵓⵡⵜ ⵜⴳⵓⵔⵉ, ⴷⴰ ⵜⵜⵉⵍⵉ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⵜⵙⴽⴼⵍⵜ ⵜⵉⵙⵙ ⵙⵏⴰⵜ.

Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial.

ⵏⵖⵢ ⴰⴷ ⵏⴳ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⵙⴳ ⵎⵏⵛⴽ ⵏ ⵜⴳⵓⵔⵉⵡⵉⵏ ⵉⵍⴰ ⵉⴳⴳⵉⵜⵏ ⵓⵔ ⵉⴳⵉⵏ ⴰⵎⵢⴰ, ⵉⵎⴽ ⵉⵜⵜⵢⴰⵏⵏⴰ ⵉ ⴱⵓ-ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⵙ ⵢⵓⵡⵜ ⵜⴳⵓⵔⵉ ⵉⵙⵎ ⴱⵓ-ⵢⵓⵡⵏ ⵓⵡⵜⵜⵓ, ⴰⵔ ⵉⵜⵢⴰⵏⵏⴰ ⵉ ⴱⵓ-ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⵙ ⵙⵏⴰⵜ ⵜⴳⵓⵔⵉⵡⵉⵏ ⴱⵓ-ⵙⵉⵏ ⵉⵡⵜⵜⴰ, ⴰⵔ ⵉⵜⵢⴰⵏⵏⴰ ⵉ ⴱⵓ ⵎⵏⵏⴰⵡ ⵏ ⵉⵡⵜⵜⴰ; ⴱⵓ-ⴽⵕⴰⴹ ⵉⵡⵜⵜⴰ.

When it is used to define a function, the domain is not so restricted.

ⵉⴳ ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⵎⴰⵔ ⴰⴷ ⵉⵙⵜⵉ ⵜⴰⵎⵔⵙⵜ, ⵓⵔ ⴷⴰ ⵉⵜⵜⵓⴽⵔⴰⴼ ⵢⵉⴳⵔ.

A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.

ⴱⵓ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⵢⵓⵡⵏ ⵓⵡⵜⵜⵓ ⵓⵔ ⵉⵜⵢⴰⵙⵙⴰⵏ ⴷⴰ ⴰⵙ ⵏⵜⵜⵉⵏⵉ ⴱⵓ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⵉⵍⴰⵏ ⵢⵓⵡⵏ ⵓⵙⵏⴼⵍ, ⴱⵓ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⴽⵉⴳⴰⵏ ⵓⵔ ⵉⵥⵍⵉⵢⵏ ⴷⴰ ⴰⵙ ⵏⵜⵜⵉⵏⵉ ⴰⵎⴳⴳⵓⴷⵢ ⵏ ⵉⵡⵜⵜⴰ, ⴰⵎⴳⴳⵓⵡⴷⵢ ⵏ ⵉⵙⵏⴼⴰⵍⵏ.

In the case of the field of complex numbers, the irreducible factors are linear.

ⴳ ⵡⴰⴷⴷⴰⴷ ⵏ ⵢⵉⴳⵔ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ, ⴷⴰ ⵜⴳⴳⴰⵏ ⵉⵎⵙⴽⴰⵔⵏ ⵉⵜⵢⴰⴽⴽⴰⵙⵏ; ⵉⵣⵔⵉⵔⵉⴳ.

If the degree is higher than one, the graph does not have any asymptote.

ⵉⴳ ⵜⴽⴽⴰ ⵜⵙⴽⴼⵍⵜ ⵏⵏⵉⴳ ⵢⴰⵏ, ⵓⵔ ⴷⴰ ⵉⵜⵜⵉⵍⵉ ⵉ ⵡⵓⵏⵓⵖ ⵏ ⵓⵙⵎⵎⴰⵍ ⴰⵡⴷ ⵢⴰⵏ ⵉⵣⵔⵉⵔⵉⴷ ⵉⵏⵎⴰⵍⴰⵏ.

In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable.

ⴳ ⵍⵊⵉⴱⵔ ⴰⵎⵣⵡⴰⵔⵓ, ⴷⴰ ⵜⵜⵓⵙⵖⵔⴰⵏⵜ ⵜⴱⵔⵉⴷⵉⵏ ⵣⵓⵏⴷ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴽⴽⵓⵥⵜ ⵉ ⵓⴼⵙⵙⴰⵢ ⵏ ⵎⴰⵕⵕⴰ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵉⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵡⵜⵜⴰ ⵉⵍⵍⴰⵏ ⴳ ⵜⵙⴽⴼⵍⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵓⵍⴰ ⵜⵉⵙⵙ ⵙⵏⴰⵜ ⴳ ⵢⵓⵡⵏ ⵓⵙⵏⴼⵍ.

However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ ⵏⵖⵢ ⴰⴷ ⵏⵙⵙⵎⵔⵙ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵓⵔⵣⵣⵓ ⵅⴼ ⵓⵥⵓⵕ, ⵎⴰⵔ ⴰⴷ ⵏⴰⴼ ⴰⵏⵎⴰⵍⴰ ⵓⵟⵟⵓⵏ ⵉ ⵉⵥⵖⵕⴰⵏ ⵏ ⵜⴰⵍⵖⴰ ⵉⵍⴰⵏ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵙⴽⴼⵍⵜ.

Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation).

ⵙⴳ ⵓⵙⴰⵜⵉ ⵡⵉⵙⵙ 16 ⵜⵜⵓⵢⴰⵙⵙⵏⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵓⵙⵎⵙⴽⵙⵍ ( ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵣⵖⵕⴰⵏ ⵉⴳⵏⵜⵔⵏ ⴷ ⵉⵣⵖⵕⴰⵏ ⵉⵎⴽⴽⵓⵥⵏ), ⵎⴰⴽⴰ ⵡⴰⵏⵏⴰ ⵡⴰⵍⴰ ⵉⵛⵇⵇⴰⵏ ⵉⴳⴰⵜ ⵜⵉⴳⴷⴰⵣⴰⵍⵉⵏ ⵏ ⵜⵙⴽⴼⵍⵜ ⵜⵉⵙⵙ ⴽⵕⴰⴹⵜ ⴷ ⵜⵉⵙⵙ ⴽⴽⵓⵥⵜ ( ⵥⵕ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⴳⵏⵜⵔⵜ ⴷ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⴽⴽⵓⵥⵜ).

In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it.

ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1830, ⵉⵙⵡⵔ ⵉⴼⴰⵔⵉⵙⵜ ⴳⴰⵍⵓ ⵏ ⵡⵉⵙⴷ ⴽⵉⴳⴰⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵏ ⵜⵙⴽⴼⵍⵜ ⵜⴰⵎⴰⴼⵍⵍⴰⵜ ⵙⴳ ⴽⴽⵓⵥ, ⵓⵔ ⵏⵣⴹⴰⵕ ⴰⵜ ⵏⴼⵙⵉ ⵙ ⵓⵥⵓⵕ, ⵉⵙⵙⴼⵔⵓⴷ ⵉⴷ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵉⵖⵢ ⵓⴼⴳⴰⵏ ⴰⴷ ⵢⵉⵏⵉ ⵉⵙ ⵜⵍⴰ ⴰⴼⵙⵙⴰⵢ ⵙ ⵜⴱⵔⵉⴷⵜ ⵜⵣⵖⵕⴰⵏⵜ, ⵎⴽ ⵉⵖⵢ, ⵉⴼⵙⵉ ⵜⵜ.

Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵜⵢⴰⴼⵙⴰⵔⵏⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵉⵍⴰⵏ ⴰⴼⵙⵙⴰⵢ ⵙ ⵜⵙⴽⴼⵍⵜ ⵜⵉⵙⵙ 5 ⴷ 6 (ⵥⵕ ⵜⴰⵎⵔⵙⵜ ⵜⴰⵎⵙⵎⵎⵓⵙⵜ ⴷ ⵜⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⵚⴹⵉⴹⵜ).

The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).

ⴷⴰ ⵜⵜⴰⴷⵊⴰⵏⵜ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵜⵓⵏⵚⵉⴱⵉⵏ ⴰⴷ ⵜⵜⵢⴰⴼⵙⴰⵢⵏⵜ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵉⴷ ⵎⵎ ⵉⵡⵜⵜⴰ ⵙ ⵓⵎⵏⵀⴰⵍ ( ⴳ ⵓⵎⵙⵙⵓⴷⵙ), ⵙ ⵜⵙⴽⴼⵍⵜ ⵉⴽⴽⴰⵏ ⵏⵏⵉⴳ 1,000 ( ⵥⵕ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵢⵉⴼ ⵏ ⵉⵣⵖⵕⴰⵏ).

For a set of polynomial equations in several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions.

ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵉⴷ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵉⴷ ⵡⴰⵔ-ⵉⵙⵎ, ⵍⵍⴰⵏⵜ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵉⵜⵜⵉⵏⵉⵏ ⵉⵙ ⵖⴰⵔⵙⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵎⵥⵍⴰⵢ ⵉⵛⵇⵇⴰⵏ, ⴷ ⵎⴽ ⵉⴳⴰ ⵡⵓⵟⵟⵓⵏ ⴰⴷ ⴰⵎⵥⵍⴰⵢ ⵉ ⵓⵙⵙⵉⵟⵏ ⵏ ⵉⴼⵙⵙⴰⵢⵏ.

A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation.

ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⴷⴰ ⵜⴳⴳⴰ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ ⵏⵏⴰ ⵎⵉ ⵢⴰⴽⴽⴰ ⵓⴼⴳⴰⵏ ⵜⴰⵖⴷⴼⵜ ⵙ ⵉⴼⵙⵙⴰⵢⵏ ⵉⴳⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵙ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴷⵢⵓⴼⴰⵏⵜⵉⵏ.

The coefficients may be taken as real numbers, for real-valued functions.

ⵏⵖⵢ ⴰⴷ ⵏⴳ ⵉⵎⴳⴳⵓⵜⵏ ⴷ ⵓⵟⵟⵓⵏ ⵉⵖⴰⵔⴰⵏ ⵏ ⵜⵉⴳⴳⵉⵜⵉⵏ ⵉⵍⴰⵏ ⴰⵜⵉⴳ ⴰⵖⴰⵔⴰⵏ.

This equivalence explains why linear combinations are called polynomials.

ⴷⴰ ⵉⵙⵙⴼⵔⵓ ⵓⵙⵙⴽⵙⵍ ⴰⴷ ⴰⵎⵏⵜⵉⵍ ⵏ ⵢⵉⵙⵎ ⵏ ⵜⵔⵓⴱⴱⴰ ⵜⵉⵣⵔⵉⵔⵉⴳⵉⵏ ⵉⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵡⵜⵜⴰ.

"In the case of coefficients in a ring, ""non-constant"" must be replaced by ""non-constant or non-unit"" (both definitions agree in the case of coefficients in a field)."

ⴳ ⵡⴰⴷⴷⴰⴷ ⵏ ⵉⵎⴳⴳⵓⵜⵏ ⴳ ⵜⵅⵔⵙⵜ ⵉⵇⵏⴻⵏ ⴰⴷ ⵏⵙⵏⴼⵍ ⵓⵏⵏⴰ ⵓⵔ ⵉⵣⵣⴳⴰⵏ ⵖⵔ ⵓⵏⵏⴰ ⵓⵔ ⵉⵡⵉⵔⵏ, ⵏⵖⴷ ⵓⵏⵏⴰ ⵓⵔ ⵉⴳⵣⵉⵎⵏ (ⵉⵙⵓⵙⵙⵏ ⵙⵙⵉⵏ ⵜⵜⵎⵙⴰⵙⴰⵏ ⴳ ⵡⴰⴷⴷⴰⴷ ⵏⵡⴰⴳⴳⵉⵜⵏ ⴳ ⵢⵉⴳⵔ).

When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials).

ⵉⴳ ⴷⴰ ⵜⵜⵓⵖⵓⵍⵏ ⵡⴰⴳⴳⵉⵜⵏ ⵖⵔ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵏⵖⴷ ⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ ⵏⵖⴷ ⵉⴳⵔ ⵉⵥⵍⵉⵏ, ⵜⵍⵍⴰ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵎⴰⵔ ⴰⴷ ⵜⴳ ⵉⵔⵎ ⵏ ⴳⴰⵔ ⵜⴰⵣⵎⵔⵜ ⵏ ⵓⵣⴳⵣⵍ ⴷ ⵓⵙⵙⵉⵟⵏ ⴷ ⵢⵉⴳⴳⵉⵜⵏ ⵏ ⵜⵉⵏⵏⴰ ⵎⵉ ⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ ⵓⵔ ⵉⴳⵉⵏ ⵜⵉⵏ ⵓⵣⴳⵣⵍ (ⵥⵕ ⴰⵙⴼⵙⵉ ⵎⵉ ⴳⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ ⵙ ⵉⵎⵙⴽⵉⵔⵏ).

The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues.

ⵉⵍⵍⴰ ⴳ ⵉⵎⴳⴳⵓⴷⵉⵏ ⵏ ⵉⵡⵜⵜⴰ ⵙ ⵜⵥⵍⵉ ⵜⴷⵔⴰⵙⵜ, ⵏⵖⴷ ⴰⵎⵙⵡⵓⵔⵉ ⴰⵣⵔⵉⵔⵉⴳ ⵖⴼ ⵉⵏⵖⵎⵉⵙⵏ ⵏ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵏ ⵜⵏⴽⴽⵉⵏⵜ ⵏ ⵓⵎⵙⵡⵓⵔⵉ.

However, the elegant and practical notation we use today only developed beginning in the 15th century.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ ⴰⵙⴷⵓⴽⵎ ⴰⵏⵡⵓⵔⵉ ⵉⵥⵉⵍⵏ ⵏⵙⵙⵎⵔⴰⵙ ⴰⵙⵙⴰ ⵉⵜⵜⵓⵙⴱⵓⵖⵍⵍⴰ ⴳ ⵜⵉⵣⵡⵓⵔⵉ ⵏ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 15.

"This ""completes the square"", converting the left side into a perfect square."

ⵖⴰⵢⴰ “ⵉⵜⵙⵎⴰⴷ ⴰⵎⴽⴽⵓⵥ”, ⴷ ⴰⵔ ⵉⵜⵔⴰⵔ ⵜⴰⵙⴳⴰ ⵜⴰⵣⵍⵎⴰⴹⵜ ⴰⴷ ⵜⴳ ⴰⵎⴽⴽⵓⵥ.

Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.

ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⴷⵉⴽⴰⵔ ⴷⴰ ⵜⵜⵉⵏⵉ ⵍⵍⴰⵏⵜ ⵖⵓⵔ ⴽⴽⵓⵥⵜ ⵏ ⵜⵡⵔⴻⵔⵔⴰⵢ ⵜⵓⴳⴷⵓⵜ (ⴰⵎⴰⵍⵓ ⵓⴳⴷⵓ), ⴳⴰⵏ ⵉⵣⴳⵏ ⵏ ⵡⴰⴳⵎ ⵏⵏⵙ ⵉⵊⵊ ⵏ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⴽⴽⵓⵥⵜ.

Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.

ⵙⵙⵎⵔⵙⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵉⴱⴰⴱⵉⵍⵉⵢⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ; ⵖⵓⵔ ⵡⴰⵜⵜⴰⵢⵏ ⵏ 400 ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⴷ ⵓⵎⵓⵙⵏⴰⵡⵏ ⵉⵚⵉⵏⵉⵢⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ, ⴰⵜⵜⴰⵢⵏ ⵏ 200 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵜⵉⴱⵔⵉⴷⵉⵏ ⵏ ⵊⵢⵓⵎⵉⵜⵉⴽ ⵉ ⵓⵙⴼⵙⵢ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵜⵉⵎⴽⴽⵓⵣⵉⵏ ⵉⵍⴰⵏ ⵉⵥⵖⵕⴰⵏ ⵓⵎⵏⵉⴳⵏ.

Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC.

ⵉⵙⵙⴽⵔ ⵓⴽⵍⵉⴷ ⴰⵎⵓⵙⵏⴰⵡ ⴰⵢⵓⵏⴰⵏⵉⵢ ⵏ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴰⴱⵔⵉⴷⵜ ⵜⴰⵊⵢⵓⵎⵉⵜⵔⵉⴽⵜ ⴰⵜⵜⴰⵢⵏ ⵏ 300 ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ.

Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.

ⴷⴷⴰⵏ ⵍⵅⴰⵡⴰⵔⵉⵣⵎ ⵖⵔ ⵓⴳⴳⴰⵔ ⵏ ⵓⵢⴰ ⴳ ⵓⵙⵙⵏⴽⴷ ⵏ ⵓⴼⵙⵙⴰⵢ ⴰⴽⴽⵯ ⵉⵙⵎⴰⵏ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⴽⴽⵓⵥⵜ ⵜⴰⵎⴰⵜⵜⴰⵢⵜ, ⵉⵜⵜⵉⵔⵉⵏ ⵢⴰⵜ ⵜⵎⵔⴰⵔⵓⵜ ⵏⵖⴷ ⵙⵏⴰⵜ ⵜⵎⵔⴰⵔⵓⵜⵉⵏ ⵉ ⴽⵓ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⴽⴽⵓⵥⵜ, ⴷ ⵜⵉⴽⴽⵉ ⵏ ⵡⴰⵏⵥⵉⵡⵏ ⴰⵊⵢⵓⵎⵉⵜⵔⵉⴽⵏ ⴳ ⵜⵉⴳⴳⵉ ⴰⴷ.

Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.

ⴰⴱⵓ ⴽⴰⵎⵉⵍ ⵛⵓⵊⴰⵄ ⵉⴱⵏ ⴰⵙⵍⴰⵎ (ⵎⵉⵚⵕ, ⴰⵙⴰⵜⵓ ⵡⵉⵙⵙ 10); ⵙ ⵓⵥⵍⴰⵢ, ⴷ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵏⵏⴰⵏ ⴰⵀ ⵉ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵙⵖⴰⵏ ( ⴳ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵏⵖⴷ ⴰⴳⵏⵜⵔ, ⵏⵖⴷ ⵡⵉⵙⵙ ⴽⴽⵓⵥ) ⵣⵓⵏⴷ ⵉⴼⵙⵙⴰⵢⵏ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵜⵉⵎⴽⴽⵓⵥⵉⵏ ⵏⵖⴷ ⵉⵎⵙⵖⴰⵍ ⴳ ⵜⴳⴷⴰⵣⴰⵍⵜ.

His solution was largely based on Al-Khwarizmi's work.

ⴰⴼⵙⵙⴰⵢ ⵏⵏⵙ ⴷⴰ ⵉⵜⵜⵓⵖⵓⵍ ⴽⵉⴳⴰⵏ ⵖⴼ ⵜⵡⵓⵔⵉ ⵏ ⵍⵅⴰⵡⴰⵔⵉⵣⵎⵉ.