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For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation).

ⵙ ⵓⵎⴷⵢⴰ, ⵖⵉⵏ ⵉⵏⴳⵎⴰⵎⵏ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵏ ⵓⵙⵉⴹⵏ ⴰⴷ ⴷⵉⵖ ⵙⵙⵡⵓⵔⵉⵏ ⵜⵉⵡⵔⴻⵔⵔⴰⵢ ⵏ ⵓⵔⵏⵏⵓ ⵏ ⵖⵉⵍⴰ, ⵃⴹⵓⵏ ⵜⵉⵡⵔⴻⵔⵔⴰⵢ ⵏⵏⵉⴹⵏ ⵎⴰⵔ ⴰⴷ ⵙⴽⵔⵏ ⵜⵉⴳⴳⵉⵜ ⵏ ⵜⵓⴽⴽⵙⴰ, ⵙ ⵜⴱⵔⵉⴷⵜ ⵏ ⵓⵙⵡⵓⵔⵉ ⵙ ⵙⵉⵏ ⵉⵎⵣⴷⴰⵢⵏ ⴰⵖ ⵢⴰⴽⴽⴰⵏ ⴰⵢⵏⵏⴰ ⴰⵙ ⵉⵜⵜⵔⵏⵓⵏ, ⵉⵡⵀⵏ ⴰⴷ ⵜⵜⵓⵙⴽⵔ ⴳ ⵉⵏⴳⵎⴰⵎⵏ (ⵜⵉⴱⴰⵡⵜ).

Multiplication also combines two numbers into a single number, the product.

ⴷⴰ ⵉⵙⵎⵓⵏ ⵓⴽⴼⵓⴷ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⴳ ⵢⵓⵡⵏ, ⵏⵜⵜⴰ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵙⵢⴰⴼⵓ.

If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say x, is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was.

ⵉⴳ ⴷⴰ ⵏⵙⵡⵉⵏⴳⵉⵎ ⴳ ⵡⵓⵟⵟⵓⵏ ⵉⵙ ⵍⵍⴰⵏ ⴳ ⵜⵡⵏⵖⴰ ⵉⵏⵎⵏ, ⴰⴽⴼⵓⴷ ⴳ ⵡⵓⵟⵟⵓⵏ ⵢⵓⴳⵔⵏ “1”, ⴰⴷ ⵏⵉⵏⵉ “ⴽ” ⵏⵜⵜⴰ ⵏⵏⵉⴽ ⴰⵢⴷ ⵉⴳⴰⵏ ⵜⵉⵏⵣⴷⴰⴷⵜ ⵖⵔ “0” ⵙ ⵜⵎⵓⵏⵉ, ⵉⵜⴳⴳⴰⵏ ⵜⴰⵏⵣⴷⴰⴷⵜ ⵏ ⵡⵓⵟⵟⵓⵏ “1” ⵖⵔ ⵎⴰⴳ ⵉⵍⵍⴰ “ⴽ”.

Any dividend divided by zero is undefined.

ⴰⵢⵏⵏⴰ ⴷ ⴰⴽⴽⵯ ⵢⴰⵖⵓⵍⵏ ⵉⴱⴹⵓ ⵖⴼ ⵓⵎⵢⴰ; ⵀⴰⵜ ⵓⵔ ⵉⵜⵜⵢⴰⵙⵙⵏ.

The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.

ⵜⴰⵎⴰⴳⵓⵏⵜ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵏ ⵓⵙⵙⵉⴹⵏ ⴳ ⵜⵉⵍⴰⵡⵜ ⵜⵉⴽⵍⵜ ⵉⵣⴳⵯⴰⵔⵏ ⵙⴳ ⵖⵓⵔ ⴽⴰⵕⵍ ⴼⵔⵉⴷⵔⵉⵛ ⴳⴰⵡⵙ.

"Positional notation (also known as ""place-value notation"") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the ""ones place"", ""tens place"", ""hundreds place"") and, with a radix point, using those same symbols to represent fractions (e.g., the ""tenths place"", ""hundredths place"")."

ⴷⴰ ⵜⵎⵎⴰⵍ ⵜⴳⵓⵔⵉ “ⴰⵣⵎⵎⴻⵎ ⴰⵙⵓⵔⵙⴰⵏ” (ⵉⵜⵢⴰⵙⵙⵏ ⴰⵡⴷ ⵙ ⵢⵉⵙⵎ “ⴰⵣⵎⵎⴻⵎ ⵏ ⵡⴰⵜⵉⴳ ⴰⴷⵖⴰⵕⴰⵏ”), ⵖⵔ ⴰⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ, ⵏⵖⴷ ⴰⵙⵎⴰⵜⴰⵔ ⵏⵏⵙⵏ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵜⵎⴰⵜⴰⵔⵜ ⵏⵏⴰⵖ ⵏⵏⵉⴽ ⵉ ⵡⵓⵟⵟⵓⵏ ⵓⵔ ⵢⴰⴽⵙⵓⵍⵏ, ⴳ ⵓⴽⵙⴰⵢ (ⵙ ⵓⵎⴷⵢⴰ, ⵢⴰⵏ ⵓⴷⵖⴰⵔ, ⵜⵓⵣⵓⵏⵜ ⵏ ⵜⵎⵔⴰⵡⵉⵏ, ⵜⵓⵣⵓⵏⵉⵏ ⵏ ⵜⵉⵎⵎⴰⴹ), ⴷ ⵜⵏⵇⵇⵉⴹⵜ ⵏ ⵍⵊⵉⴷⵔ, ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵏⵏⴰⵖ ⵏⵏⵉⴽ ⵉ ⵓⵙⵎⴷⵢⴰ ⵏ ⵉⵎⵜⵡⴰⵍⵏ (ⵙ ⵓⵎⴷⵢⴰ, ⵜⵓⵣⵓⵏⵜ ⵏ ⵜⵎⵔⴰⵡⵉⵏ, ⵜⵓⵣⵓⵏⵜ ⵏ ⵜⵉⵎⵎⴰⴹ).

The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci using the Hindu–Arabic numeral system.

ⵉⵜⵜⵓⵣⴳⴰ ⵓⵙⵙⵎⵔⵙ ⵏ 0 ⴰⵎⵎ ⵓⴼⵕⴹⵉⵙ ⴰⴹⴼⴰⵕ, ⵉⵡⴰ ⴰⵙⵙⵎⵔⵙ ⵏ ⵓⵣⵎⵎⴻⵎ ⴰⵙⵓⵔⵙⴰⵏ ⵜⵉⴽⵍⵜ ⵉⵣⵡⴰⵔⵏ ⴳ ⵓⴹⵕⵉⵙ Jain ⵙⴳ ⵍⵀⵉⵏⴷ ⵙ ⵓⵣⵡⵍ Lokavibhâga ⴳ ⵓⵙⴰⴽⵓⴷ 458 ⴳ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵓⵔ ⵜⵜⵓⵎⵓⵜⵜⵉⵢⵏ ⵉⵙⵓⵙⵙⵏ ⴰⴷ ⴰⵔ ⴰⵙⴰⵜⵓ ⵙⵉⵙⵙ 13, ⵜⴰⵡⵙⴰ ⵏ ⵓⵎⴰⴹⴰⵍ ⴰⵄⵕⴰⴱ, ⵜⵢⴰⴽⵛⴰⵎ ⵙ ⵓⵔⵓⴱⴱⴰ ⵙⴳ ⵖⵓⵔ ⴼⵉⴱⵓⵏⴰⵜⵛⵉ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⴰⵏⴳⵔⴰⵡ ⴰⵎⴰⵟⵟⵓⵏ ⴰⵀⵉⵏⴷⵉ-ⴰⵄⵕⴰⴱ.

The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left.

ⴷⴰ ⵜⵜⵓⵙⵙⵉⴹⵏ ⵜⵢⴰⴼⵓⵜ ⵙ ⵜⴱⵔⵉⴷⵜ ⵏ ⵓⵔⵏⵏⵓ ⵉⵣⴷⵉⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵢⵉⵡⵏ ⵉⵙⵡⵓⵔⵉⵏ ⴳ ⵢⵓⵡⵏ ⵓⵙⵓⵔⵙ, ⵙⴳ ⵓⵢⴼⴼⴰⵙ ⵖⵔ ⵓⵣⵍⵎⴰⴹ.

The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero).

ⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵢⴼⴼⴰⵙ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵜⵉⴳ ⵏ ⵓⵙⵓⵔⵙ ⵏ ⴷⵖⵉ, ⴰⵔ ⵜⵜⵓⵔⵏⵓ ⵜⵢⴰⴼⵓⵜ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵣⵍⵎⴰⴹ ⵙ ⵡⴰⵜⵉⴳ ⵏ ⵡⵓⵟⵟⵓⵏ ⵡⵉⵙⵙ ⵙⵉⵏ (ⴰⵣⵍⵎⴰⴹ), ⵉⵜⴳⴳⴰⵏ ⴰⵀⴰ ⵢⴰⵏ (ⵎⴽ ⵓⵔ ⵉⴳⵉ ⴰⵎⵢⴰ).

A multiplication table with ten rows and ten columns lists the results for each pair of digits.

ⵉⵜⵜⴰⵍⵙ ⵓⵙⵎⵢⴰⵍⵍⴰⵢ ⵏ ⵓⴽⴼⵓⴷ ⴳ ⵍⵍⴰⵏ ⵎⵔⴰⵡⵜ ⵏ ⵡⴰⴷⵓⵔⵏ ⴷ ⵎⵔⴰⵡⵜ ⵏ ⵜⵉⵔⵙⴰⵍ, ⵜⴰⵢⴰⴼⵓⵜ ⵏ ⴽⵓ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ.

Similar techniques exist for subtraction and division.

ⵍⵍⴰⵏⵜ ⵜⵉⵇⵏⵉⵢⴰⵜ ⵜⵉⵡⴰⵢⵉⵏ ⵏ ⵜⵓⴽⴽⵙⴰ ⴷ ⵓⴱⵟⵟⵓ.

In mathematical terminology, this characteristic is defined as closure, and the previous list is described as .

ⴳ ⵜⴳⵓⵔⵉⵡⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴷⴰ ⵜⴻⵜⵜⵢⴰⵙⵙⴰⵏ ⵜⵏⴰⵎⴰⵥⵜ ⴰⴷ ⵙ ⵡⵓⵖⵓⵏ, ⴰⵔ ⵜⵜⵓⵙⵏⵓⵎⵎⴰⵍ ⵜⵍⴳⴰⵎⵜ ⵜⴰⵣⵡⵉⵔⵜ ⵙ.

The total in the pence column is 25.

ⵙ ⵓⵎⴰⵜⴰ ⴳ ⵜⵉⵔⵙⵍⵜ ⵏ ⵍⴱⵏⵙ ⵜⴳⴰ 25.

This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column.

ⴷⴰ ⵜⵜⵓⵢⴰⵍⵙ ⵜⵉⴳⴳⵉⵜ ⴰⴷ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵉⵍⵍⴰⵏ ⴳ ⵜⵔⵙⵍⵜ ⵏ ⵛⵉⵍⵏⴰⵜ, ⴷ ⵜⵙⵓⵔⵉⴼⵜ ⵜⴰⵎⵔⵏⵉⵡⵜ ⵎⴰⵔ ⴰⴷ ⴰⵙ ⵉⵜⵜⵓⵔⵏⵓ ⵡⴰⵜⵉⴳ ⵉⵜⵜⵓⵎⵓⵜⵜⵉⵏ ⵖⵔ ⴷⴰⵜ ⵙⴳ ⵜⵉⵔⵙⵍⵜ ⵏ ⵍⴱⵉⵏⵙⴰⵜ.

"One typical booklet that ran to 150 pages tabulated multiples ""from one to ten thousand at the various prices from one farthing to one pound""."

ⵢⵓⵡⵏ ⵓⴷⵍⵉⵙ ⴰⵎⴷⵢⴰ ⵉⴳⵓⵍⴰⵏ 150 ⵏ ⵜⴰⵙⵏⴰ ⵜⴰⵔⴰⵜⵙⴰ ⴷ ⵉⵙⵍⴰⴳⵏ ⵏⵏⵙ, ⵙⴳ ⵢⴰⵏ ⴰⵔ ⵎⵔⴰⵡⵜ ⵏ ⵢⵉⴼⴷⵏ ⵙ ⵡⴰⵜⵉⴳⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ, ⵙⴳ ⵢⴰⵏ ⴼⴰⵔⵜ ⵖⵔ ⵢⴰⵏ ⵊⵉⵏⵉⵀ.

This study is sometimes known as algorism.

ⵜⵢⴰⵙⵙⴰⵏ ⵜⵣⵔⴰⵡⵜ ⴰⴷ ⵙ: “ⴰⵍⴳⵓⵔⵉⵣⵎ”.

Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth.

ⵉⵎⴽⵉⵏⵏⴰ ⵙⵎⵔⵙⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵉⵏⵙⵍⵎⵏ ⴰⵙⵙⵉⴹⵏ ⵎⴰⵔ ⴰⴷ ⵙⵙⵍⵎⴷⵏ ⴰⴼⵔⵔⵓ ⵏ ⵜⴽⵓⵜⵉ ⴷ ⵜⴽⴰⵙⵉⵜ.

Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

ⴰⵙⵎⵓⵏ ( ⵜⵜⴰ ⵉⵜⵜⵓⵏⵄⴰⵜ ⵙ ⵜⵎⴰⵜⴰⵔⵜ ⵏ ⵜⵎⵓⵏⵉ), ⵉⴳⴰ ⵢⴰⵜ ⵜⵉⵎⴳⴳⵉⵜⵉⵏ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⴳ ⴽⴽⵓⵥⵜ, ⴽⵕⴰⴹ ⵜⵎⴳⴳⵉⵜⵉⵏ ⵢⴰⴹⵏ; ⵜⵓⴽⴽⵙⴰ, ⴰⴽⴼⵓⴷ, ⴰⴱⵟⵟⵓ.

In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.

ⴳ ⵍⵊⵉⴱⵔ, ⵉⴳⵔ ⵏⵏⵉⴹⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ, ⵉⵖⵢ ⴰⴷ ⵏⵔⵏⵓ ⵉ ⵉⵎⵖⵏⴰⵡⵏ ⵡⴰⵔⴰⵜⵉⴳ ⵣⵓⵏⴷ ⵉⵎⵏⵉⴷⴰⵏ ⴷ ⵜⴷⵔⴰⵙⵉⵏ ⴷ ⵜⴰⵔⵉⵜⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ, ⴷ ⵜⵔⴰⴱⴱⵓⵜⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ.

"Using the gerundive suffix -nd results in ""addend"", ""thing to be added""."

“ⴰⵙⵎⵔⵙ ⵏ ⵜⵎⴹⴼⵓⵕⵜ “ⵊⵉⵔⵓⵏⴷⵉⴼ”, ⴷⴰⴷ ⵜⴰⴽⴽⴰ “ⴰⴷⴷⵏⴷ”, ⵜⴰⵖⴰⵡⵙⴰ ⵏⵔⴰ ⴰⵙ ⵏⵔⵏⵓ.”

"""Sum"" and ""summand"" derive from the Latin noun summa ""the highest, the top"" and associated verb summare."

“ⵉⴼⴼⵖⴷ “ ⵙⵓⵎ”, ⴷ “ⵙⵓⵎⵎⴰⵏⴷ”, ⵙⴳ ⵢⵉⵙⵎ ⴰⵍⴰⵜⵉⵏⵉ “ⵙⵓⵎⵎⴰ” ⴰⵎⴰⴼⵍⵍⴰ, ⵢⴰⵜⵜⵓⵢⵏ, ⴰⵣⴳⵣⵍ ⵏ ⵓⵎⵢⴰⴳ ⵓⴳⵉⵍ.”

"The later Middle English terms ""adden"" and ""adding"" were popularized by Chaucer."

“ⵜⵉⴳⵓⵔⵉⵡⵉⵏ ⵜⵉⵇⴱⵓⵔⵉⵏ ⵏ ⵜⵏⴳⵍⵉⵣⵜ “ⴰⴷⴷⵏ” ⴷ “ ⴰⴷⴷⵉⵏ” ⵜⵢⴰⴼⵙⴰⵔⵏⵜ ⵙⴳ ⵖⵓⵔ “ⴽⵓⵙⵔ”.”

As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)?

ⵙ ⵓⵎⴷⵢⴰ, ⵉⵙ ⵉⵏⵎ ⵓⵙⵉⵙⵙⵏ ⵏ ⵓⵡⵏⵏⵉ ⴰⴷ; ” ⴰ + ⴱ + ⵙ ”, ⵇⴰⴷ ⴰⵖ ⴷ ⵉⴽ “ ( ⴰ + ⴱ ) + ⵙ “, ⵏⵖⴷ “ ⴰ + ( ⴱ + ⵙ )?

Even some nonhuman animals show a limited ability to add, particularly primates.

ⴰⵡⴷ ⴽⴰⵏ ⵉⵎⵓⴷⴰⵔ ⵓⵔ ⵉⴳⵉⵏ ⴰⴼⴳⴰⵏ, ⵖⵓⵔⵙ ⵜⴰⵣⵎⵔⵜ ⵏ ⵜⵔⵏⵓⵜ , ⵏⵓⵎⴰⵔ ⵜⵉⴷⵙⵍⴰⵏⵉⵏ.

"With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting ""four, five."""

“ⵙ ⵉⵎⵉⴽ ⵏ ⵜⵎⵓⵣⴰⵢⵜ, ⴷⴰ ⵜⵍⵎⴰⴷⵏ ⵉⵛⵉⵔⵔⴰⵏ ⴰⴷ ⵜⵔⵏⵓⵏ ⵙ ⵜⴱⵔⵉⴷⵜ ⵉⵙⵔⴱⵉⵏ ⵙ ⵓⵙⵎⵔⵙ ⵏ ⵓⵙⵎⵓⵏ ⴰⵎⵏⴼⴽ ⵙ ⵓⵙⵙⵉⵟⵏ ⵙⴳ ⵓⵟⵟⵓⵏ ⴰⵅⴰⵜⴰⵔ, ⴳ ⵡⴰⴷⴷⴰⴷ ⴰⴷ; ⴰⵙⵙⵏⵜⵉ ⵙⴳ ⴽⵕⴰⴹ, ⵉⵙⵙⵓⴷⵓ “ⴽⴽⵓⵥ, ⵙⵎⵎⵓⵙ.””

Zero: Since zero is the additive identity, adding zero is trivial.

ⴰⵎⵢⴰ: ⵏⵜⵜⴰ ⴰⵢⴷ ⵉⴳⴰⵏ ⵜⴰⵎⴰⴳⵉⵜ ⵏ ⵜⵎⵔⵏⵓⵜ, ⵙ ⵓⵔⵏⵏⵓ ⵏⵏⵙ ⴽⴰ ⵓⵔ ⵉⵍⵍⵉ.

One aligns two decimal fractions above each other, with the decimal point in the same location.

ⵢⴰⵏ ⵓⵙⵎⵓⵏ ⵏ ⵙⵉⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵉⴷ ⵎⵔⴰⵡ, ⵢⴰⵏ ⴰⴼⵍⵍⴰ ⵏ ⵡⴰⵢⵢⴰⴹ, ⵜⵉⵍⵉ ⵜⵎⴰⵜⴰⵔⵜ ⵜⴰⵎⵔⴰⵡⵜ ⴳ ⵓⴷⵖⴰⵔ ⵏⵏⵙ.

If the addends are the rotation speeds of two shafts, they can be added with a differential.

ⵎⴽ ⴳⴰⵏⵜ ⵜⵎⵔⵏⵓⵜⵉⵏ ⴰⵙⵔⴱⵢ ⵏ ⵡⵓⵜⵓⵢ ⵏ ⵙⵏⴰⵜ ⵜⵉⵔⵙⴰⵍ, ⵀⴰⵜ ⵉⵖⵢ ⴰⴰ ⵉⵜⵜⵓⵔⵏⵓ ⵙ ⵓⵎⵣⵉⵔⴰⵢ.

It made use of a gravity-assisted carry mechanism.

ⴷⴰ ⵜⵙⵙⵎⵔⴰⵙ ⵉⵎⵉⵙ ⵏ ⵢⵉⵙⵢ ⵙ ⵓⵎⵢⵉⵡⴰⵙ ⵏ ⵜⵍⴷⴰⵢⵜ.

To subtract, the operator had to use the Pascal's calculator's complement, which required as many steps as an addition.

ⵉ ⵜⵓⴽⴽⵙⴰ, ⵉⵙⵙⵎⵔⵙ ⵓⵎⵙⵡⵓⵔⵉ ⴰⵙⵎⴷ ⵏ ⵓⵙⵙⵉⵟⵏ “ⴱⴰⵙⴽⴰⵍ, ⵉⵔⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵙⵓⵔⵓⴼⵉⵏ ⵣⵓⵏⴷ “ⵜⴰⵎⵔⵏⵓⵜ”.

Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of full adder circuits which in turn may be combined into more complex logical operations.

XOR ⴷ AND gates ⵡⵀⵏⴻⵏⵜ ⴰⴷ ⵜⵜⵓⵔⵎⴰⵙⵏⵜ ⴳ ⵡⵓⵎⴳⵉⵏ ⴰⵎⴰⵟⵟⵓⵏ ⴰⵢⴷ ⵉⵜⵜⴰⴷⵊⴰⵏ ⵜⵉⴳⴳⵉⵜ ⵏ ⵜⵡⵔⴻⵔⵔⴰⵢⵉⵏ adder ⵎⴰⵕⵕⴰ, ⵏⵏⴰ ⵙ ⵏⵖⵢ ⴰⴷ ⵏⵙⵙⵉⴷⴼ ⴳ ⵜⵉⴳⴳⵉⵜⵉⵏ ⵜⵓⵎⴳⵉⵏⵉⵏ ⵡⴰⵍⴰ ⵉⵔⵡⵉⵏ.

Many implementations are, in fact, hybrids of these last three designs.

ⴽⵉⴳⴰⵏ ⴳ ⵜⵙⵏⵙⵉⵜⵉⵏ ⴰⴷ , ⴳⴰⵏⵜ ⴳ ⵜⵉⵏⴰⵡⵜ ⵜⵓⵛⵓⵔⵜ ⴳⵔ ⵉⵎⴰⵎⴽⵏ ⴰⴷ ⵉⵎⴳⴳⵓⵔⴰ ⵙ ⴽⵕⴰⴹ.

Unanticipated arithmetic overflow is a fairly common cause of program errors.

ⴰⴳⵯⵓⵔ ⵏ ⵓⵙⵙⵉⵟⵏ ⵅⴼ ⵓⵔ ⵏⴳⵉ, ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵙⵔⴰⴳ ⵏ ⵜⵣⴳⴳⴰⵍ ⵏ ⵓⵖⴰⵡⴰⵙ.

Taken literally, the above definition is an application of the recursion theorem on the partially ordered set N2.

ⵎⴽ ⵏⵓⵙⵢ ⴰⵙⵓⵙⵙⵏ ⵏ ⵓⴼⵍⵍⴰ ⵙ ⵎⵉⵎⴽ ⵉⴳⴰ, ⵉⴳⴰ ⴰⵙⵎⵔⵙ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵎⴰⵖⵓⵍⵜ ⵅⴼ ⵜⵔⴰⴱⴱⵓⵜ ⵏ “ ⵏ 2 “, ⵉⵍⵍⴰⵏ ⵙ ⵜⴰⴼⵓⵍⵜ.

If either a or b is zero, treat it as an identity.

ⵎⴽ ⴷ a ⵏⵖⴷ b ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵢⴰ, ⵙⵡⵓⵔⵉ ⵉⴷⵙ ⵉⴷ ⵏⵜⵜⴰⵜ ⴰⵢⴷ ⵉⴳⴰⵏ “ ⴰⵎⵙⴰⵙⴰ “.

Here, the semigroup is formed by the natural numbers and the group is the additive group of integers.

ⴷⴰⴷⵖ ⴰⵣⴳⵏ ⵏ ⵜⵔⴰⴱⴱⵓⵜ ⴷⵉⴽⵙ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏ ⴷ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴳⴰ ⵎⵏⵏⴰⵡⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵉⵜⵜⵓⵔⵏⵓⵏ.

The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity.

ⴰⵙⵏⵎⵔⴰⵔⴰ ⴷ ⵓⵙⵖⵏ ⴷ ⵓⵔⵏⵓ ⵏ ⵜⵉⴷⵜ ⵉⴳⴰ ⴰⴷⵖⵢⴰⵏ, ⴰⵙⵡⵓⵜⵜⵓ ⵏ ⵡⵓⵟⵜⵓⵏ ⵏ ⵜⵉⴷⵜ “0” ⵎⴰⵔ ⴰⴷ ⵉⴳ ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⵙⵔⴰⴳⵏ ⵏ ⵜⵓⵎⴳⵉⵏⵜ ⵜⵓⵣⴷⵉⵔⵜ, ⵉⵡⵀⵏ ⴰⴷ ⵉⵜⵜⵓⵢⴰⵏⵏⴰⵢ ⵉⵙ ⵉⴳⴰ ⵜⴰⵎⴰⴳⵉⵜ ⵜⴰⵎⵔⵏⵉⵡⵜ.

One must prove that this operation is well-defined, dealing with co-Cauchy sequences.

ⵉⵇⵏⴻⵏ ⴰⴼⴳⴰⵏ ⴰⴷ ⵉⴽ ⵜⵉⵍⴰⵡⵜ ⵏ ⵡⵉⵙ ⵜⴳⴰ ⵜⵎⴳⴳⵉⵜ ⵜⵉⵎⵥⵍⵉⵜ, ⴷ ⴷⴰ ⵜⵙⵡⵓⵔⵉ ⴰⴽⴷ ⵜⵎⴹⴼⴰⵕ ⵏ ⴽⵓⵛⵉ.

"The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the ""exclusive or"" function."

ⵜⴰⵔⴰⴱⴱⵓⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, “ⵎⵓⴷⵓⵍ 2”, ⴷⵉⴽⵙ ⵙⵉⵏ ⵉⴼⵔⴹⵉⵚⵏ ⴷⴰⵢ, ⴷ ⵜⵎⴳⴳⵉⵜ ⵜⴰⵎⵔⵏⵓⵜ ⵜⴽⵓⵙⴰ, ⵜⵢⴰⵙⵙⴰⵏ ⴳ ⵓⵎⴳⵉⵏ ⵙ ⵜⵔⴰⴱⴱⵓⵜ “ ⵜⴰⵥⵍⴰⵢⵜ” ⵏⵖ “ ⵜⴰⵖⵍⵉⴼⵜ”.

These give two different generalizations of addition of natural numbers to the transfinite.

ⵜⴰ ⴷⴰ ⵜⴰⴽⴽⴰ ⵙⵉⵏ ⵉⵙⵎⴰⵜⴰⵜⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ, ⵎⴰⵔ ⴰⴷ ⴰⵙ ⵜⵜⵓⵔⵏⵓⵏ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏ ⵖⵔ ⵜⴰⵙⵏⴼⵍⵜ.

There are even more generalizations of multiplication than addition.

ⵍⵍⴰⵏ ⵉⵙⵎⴰⵜⴰⵜⵏ ⵏ ⵓⴽⴼⵓⴷ ⵓⴳⴳⴰⵔ ⵏ ⵡⵉⵏ ⵓⵙⵓⴳⵜ.

In fact, if two nonnegative numbers a and b are of different orders of magnitude, then their sum is approximately equal to their maximum.

ⴳ ⵜⵉⵏⴰⵡⵜ, ⵎⴽ ⵍⵍⴰⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ ⵉⵏⴰⴱⴰⵡⵏ “ⴰ” ⴷ “ⴱ”, ⵖⴰⵔⵙⵏ ⵉⵙⵡⵉⵔⵏ ⵏ ⵓⴽⵙⴰⵢ ⵉⵎⵣⴰⵔⴰⵢⵏ ⵜⴰⵎⵓⵜⵜⵔⵜ ⵏⵏⵙⵏ ⴷⴰ ⴰⵖ ⵜⴰⴽⴽⴰ ⴰⵡⵜⵜⵓ ⵓⵣⵣⵓⵔ

It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero.

ⵉⵍⴰ ⵜⴰⵡⵏⴳⵉⵎⵜ ⵏ ⵜⴰⴳⴰⵔⵜ ⵏ ⵡⵓⵟⵟⵓⵏ ⵢⴰⵏ, ⴷ ⵏⵜⵜⴰ ⵏⵉⵜ ⴰⵜⵉⴳ ⵉⵅⵡⴰⵏ, ⴷ ⵓⵎⵢⴰ.

"Integration is a kind of ""summation"" over a continuum, or more precisely and generally, over a differentiable manifold."

“ⴰⵙⵎⴰⴷ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵏⴰⵡ ⵏ “ ⵓⵣⵣⴳⵣⵍ” ⵙ ⵓⵣⴷⴰⵢ ⵉⵎⴰⵏⴻⵏ, ⵏⵖⴷ ⵙ ⵜⴰⵍⵖⴰ ⵜⵓⵏⵖⵉⴷⵜ ⵙ ⵓⵎⴰⵜⴰ, ⵙ ⵜⵉⴳⴷⵉ ⵉⵖⵉⵏ ⴰⴷ ⵢⵉⵍⵉ ⵜⵓⵍⵍⵓⵖⵜ.”

Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.

ⵙ ⵓⵥⵍⴰⵢ ⵏ ⵉⵙⴰⵜⴰⵍⵏ ⴳ ⵜⵕⵥⵥⴰ ⵜⵎⵔⵏⵉⵡⵜ ⵜⵓⵙⵔⵉⴷⵜ ⵉⵜⵙⵏ ⵉⵍⴳⴰⵎⵏ ⵉⵖⴰⵔⴰⵏ, ⵣⵓⵏⴷ ⵜⵓⵛⵓⵔⵜ ⵏ ⵜⵙⵜⵔⴰⵜⵉⵊⵉⵢⵉⵏ ⴳ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵡⵔⴰⵔⵜ, ⵏⵖⴷ ⴰⵙⵏⴰⵢ ⵏ ⵡⴰⴷⴷⴰⴷⵏ ⴳ ⵜⵎⵉⴽⴰⵏⵉⴽⵉⵜ ⵏ ⵜⵎⴰⴽⵜⴰ.

Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers.

ⵜⴰⴱⴹⵓⵜ ⵜⴳⴰ ⵢⴰⵜ ⵜⵎⴳⴳⵉⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵎⵎ ⴽⴽⵓⵥⵜ, ⵉⴳⴰⵏ ⵜⴰⴱⵔⵉⴷⵜ ⵏ ⵓⵙⵎⵓⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⴼ ⴰⴷ ⵏⵙⵎⵓⵜⵜⴳ ⵓⵟⵟⵓⵏ ⵉⵎⴰⵢⵏⵓⵜⵏ.

Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate (which define multiplication and addition over single-variabled formulas).

ⵜⴰⵏⵏⴰ ⴳ ⵉⵜⵢⴰⵙⴽⴰⵔ ⵓⵙⵉⵙⵙⵏ ⵏ “ ⵓⴽⵍⵉⴷⵏ ⴰⵏⴰⴱⴹⵓ” (ⵙ ⵓⵎⴰⴳⵓⵔ), ⴰⵢⴷ ⵉⴳⴰⵏ ⵉⴳⵔⴰⵏ ⵏ ⵓⴽⵍⵉⴷⵏ ⴷ ⴰⵔ ⵜⵙⵎⵓⵏ ⵜⵉⵅⵔⵚⵉⵏ ⵎⵉ ⴳⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ ⴳ ⵢⵓⵡⵜ ⵓⵔ ⵉⵥⵍⵉⵢⵏ ( ⵉⵙⵜⵜⵉⵏ ⴰⴽⴼⵓⴷ ⴷ ⵜⵎⵔⵏⵓⵜ ⵙ ⵜⵡⵉⵍⴰ ⵏ “ⵢⵓⵡⵏ ⵓⵎⵙⴽⵉⵍ”).

This division sign is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙ ⵜⵎⴰⵜⴰⵔⵜ ⴷⵖ ⵏ ⴱⵟⵟⵓ ⵅⵙ ⵏⵜⵜⴰⵜ, ⵎⴰⵔ ⴰⴷ ⵜⵙⵎⴷⵢⴰ ⵏ ⵜⵎⴳⴳⵉⵜ ⵏ ⵓⴱⵟⵟⵓ ⴷⵖ ⵏⵏⵉⴽ, ⵙ ⵓⵎⴷⵢⴰ ⵜⵉⵙⵓⵔⴰ ⵏ ⵢⵉⵎⵉⵙ ⵏ ⵓⵙⵙⵉⵟⵏ.

Distributing the objects several at a time in each round of sharing to each portion leads to the idea of 'chunking' a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.

ⴷⴰⴷ ⵉⵜⵜⴰⵡⵢ ⵓⴱⵟⵟⵓ ⵏ ⵉⴼⵕⴹⵉⵙⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵉⴽⴽⴰⵍ ⴳ ⵢⵓⵡⵜ ⵜⵉⵣⵉ ⴳ ⴽⵓ ⵜⴰⵙⵓⵜⵍⵜ ⵢⴰⵎⵓⵏ ⵖⴼ ⴽⵓ ⵜⴰⴼⵓⵍⵜ ⵙ ⵜⵡⵏⴳⵉⵎⵜ ⵏ “ ⵜⵓⵟⵟⵓⵜ”, ⵜⴰⵍⵖⴰ ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⵓⵟⵟⵓⵜ, ⵉⵜⵜⴰⴷⵊⴰⵏ ⵢⴰⵏ ⴰⴷ ⵉⴳⴳⴰⵔ ⴰⵙⵍⴰⴳ ⵖⴼ ⵉⵜⵢⴰⴱⴹⴰ ⵙ ⵜⴰⵍⵖⴰ ⵢⵓⵍⵙⵏ ⵙⴳ ⵓⴱⵟⵟⵓ ⵏⵏⴰⵖ ⵏⵏⵉⴽ.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.

ⵉⵣⴹⴰⵕ ⴽⵓ ⵢⴰⵏ ⴰⴷ ⵉⵙⵙⵎⵔⵙ ⵉⵙⵎⵢⴰⵍⵍⴰⵢⵏ ⵏ “ⵍⵓⴳⴰⵔⵉⵜⵎ” ⵎⴰⵔ ⴰⴷ ⵉⴱⴹⵓ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ, ⵙ ⵜⴱⵔⵉⴷⵜ ⵏ ⵜⵓⴽⴽⵙⴰ ⵏ “ⵍⵓⴳⴰⵔⵉⵜⵎ” ⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ, ⴷ ⵢⵉⵏⵉⴳ ⵏ “ⵍⵓⴳⴰⵔⵉⵜⵎ” ⴰⵏⵎⴳⴰⵍ ⵉ ⵜⵢⴰⴼⵓⵜ.

Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer.

ⴷⴰ ⵙⵡⵓⵔⵉⵏⵜ ⵉⵜⵙⵏⵜ ⵜⵓⵜⵍⴰⵢⵉⵏ ⵏ ⵓⵙⵖⵉⵡⵙ ⴰⵎⵎ: “ⵙ”, ⴷ ⵓⴱⵟⵟⵓ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴰⴼⵍⵍⴰ ⴳ ⵡⴰⴷⴷⴰⴷ ⵡⵉⵙⵙ 5, ⴰⴳⴰⵏ ⵜⴰⵎⵔⴰⵔⵓⵜ ⴷ “ⵓⵟⵟⵓⵏ ⴰⵎⴷⴷⴰⴷ”.

Similarly, right division of b by a (written ) is the solution y to the equation .

ⵙ ⵓⵙⵎⴷⵢⴰ, ⵜⴰⴱⵟⵟⵓⵜ ⵜⴰⵎⴷⴷⴰⴷⵜ ⵏ “ⴱ” ⵅⴼ “ⴰ” ( ⵉⵜⵢⴰⵔⴰⵏ), ⴰⴼⵙⵙⴰⵢ “ⵢ” ⴳ ⵜⴳⴷⴰⵣⴰⵍⵜ.

Examples include matrix algebras and quaternion algebras.

ⵙⵎⴰⵏ ⵉⵎⴷⵢⴰⵜⵏ ⵍⵊⵉⴱⵔ ⴰⴷⵔⴰⵙⴰⵏ ⴷ ⵍⵊⵉⴱⵔ ⴱⵓⴽⴽⵓⵥ.

Entry of such an expression into most calculators produces an error message.

ⴰⵙⴽⵛⵎ ⵏ ⵓⵡⵏⵏⵉ ⴰⵎ ⵡⴰ ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵉⵎⴰⵙⵙⵏ ⵏ ⵓⵙⵙⵉⵟⵏ ⴷⴰⵖⴷ ⵢⴰⴽⴽⴰ ⵜⴰⴱⵔⴰⵜ ⵉⵣⴳⵍⵏ.

Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal.

ⵉⵎⴽ ⵉⵙⴽⴰⵔ ⵓⵙⵏⴼⵍ ⵜⴰⴷⵔⵙⵉ ⵏ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵅⴰⵜⴰⵔⵏ, ⴰⵖⴼ ⴰⵖ ⴷ ⵢⴰⴽⴽⴰ ⵢⵉⵍⵙ ⵏ ⵜⵎⴳⴳⵉⵜ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵥⵥⴰⵏ ⵉⵣⴷⵉⵏ ⴰⵔⴷ ⵉⴽⵙⵉⵍⵏ ⵓⵟⵟⵓⵏ ⵙⵙⵉⵏ.

The fact that the GCD can always be expressed in this way is known as Bézout's identity.

ⵜⵢⴰⵙⵙⵏ ⴰⵀⴰ ⵜⵉⴷⵉⵜ ⵏ ⵓⵙⵉⵡⵍ ⵖⴼ “ⴰⵏⴱⴹⵓ ⴰⵅⴰⵜⴰⵔ ⵉⵛⵛⴰⵔⵏ” ⵙ ⵜⴱⵔⵉⴷⵜ ⴰⴷ, ⵜⴰⵎⴰⴳⵉⵜ ⵏ “ⴱⵉⵣⵓⵜ”.

With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer.

ⵙ ⵓⵙⵖⵓⴷⵓ ⴰⴷ, ⵜⴰⵅⴰⵡⴰⵔⵉⵣⵎⵉⵢⵜ ⵓⵔ ⴷⵊⵓⵏ ⵜⵔⵉ ⵜⵉⵙⵓⵔⵉⴼⵉⵏ ⵏⵏⵉⴳ ⵙⵎⵎⵓⵙ ⵏ ⵢⵉⵍⵙⵏ ⵏ ⵡⵓⵟⵟⵓⵏ (ⵜⴰⵙⵉⵍⴰ 10), ⵏ ⵉⵎⵉⴹ ⴰⵎⵥⵥⵢⴰⵏ ⴰⵎⴷⴷⴰⴷ.

The Euclidean algorithm has many theoretical and practical applications.

ⵜⵍⵍⴰ ⴳ ⵜⵅⵡⴰⵔⵉⵣⵎⵜ ⵜⴰⵙⵎⴰⴳⴰⵍⵜ ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⴽⴽⵉⵔⵏ ⵉⵎⴰⴳⵓⵜⵏ ⴷ ⵉⵎⴳⴳⵉⵜⵏ.

The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers.

ⵉⵖⵢ ⴰⴷ ⵜⵜⵓⵙⵎⵔⵙ ⵜⵅⵡⴰⵔⵉⵣⵎⵜ ⵜⴰⵎⵙⴰⴳⴰⵍⵜ ⴳ ⵓⴼⵔⵔⵓ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ “ⴷⵢⵓⴼⴰⵏⵜⵉⵏ”, ⵣⵓⵏ ⵜⵉⴼⵉⵜ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵜⵜⵎⵛⴰⵛⴽⴰⵏ ⴷ ⵉⵎⵙⴰⵙⴰⵜⵏ ⵉⴳⴳⵓⴷⵉⵏ, ⵏⵉⵍ ⵜⵎⴰⴳⵓⵏⵜ ⵏ “ⵜⵓⴳⵔⵜ ⵜⴰⵙⵉⵏⵉⵢⵜ”, ⵉ ⵜⵓⵙⴽⴰ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵣⴷⴰⵢⵏ, ⴷ ⴰⴷ ⵏⴰⴼ ⴰⴷⴰⵙ ⴰⵎⵥⵍⴰⵏ ⵉⵏⵖⴷⵏ ⵉ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ.

The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b), although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD.

ⴰⵏⴱⴹⵓ ⴰⵎⵙⵙⵓⵔ ⴰⵅⴰⵜⴰⵔ ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵢⴰⵔⴰ ⵙ “ⴰ,ⵎ,ⵅ”(ⴰ, ⴱ), ⵏⵖⴷ ⵙ ⵓⵎⴷⵢⴰ ⵏⵏⵉⴹⵏ (ⴰ, ⴱ), ⵎⵇⵇⴰⵔ ⵓⵔ ⵡⴰⵍⴰ ⵉⵙⵙⵓⴷⴷⵉ ⵓⵎⴻⴳⴳⴰⵔⵓ ⴰⴷ, ⵎⴰⴽⴰ ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⵉ ⵉⵔⵎⵎⵓⵙⵏ ⵣⵓⵏⴷ: “ ⵜⴰⴼⵉⵍⴰⵡⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ”, ⵏⵏⴰ ⵉⵟⴼⵏ ⵙ ⴰⵏⴱⴹⵓ ⴰⵎⵙⵙⵓⵔ ⴰⵅⴰⵜⴰⵔ.

For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7.

ⵙ ⵓⵎⴷⵢⴰ, ⵉⵎⵉⴹ 6 ⵓⵍⴰ 35; ⵓⵔ ⴳⵉⵏ ⵉⵎⵉⴹ ⴰⵎⵣⵡⴰⵔⵓ, ⴰⵛⴽⵓ ⵙⵙⵉⵏ ⵉⵜⵙⵏ ⵖⵓⵔⵙⵏ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⵜⵏ ⵉⵙⵎⴰⵏⴻⵏ: 6 = 2 × 3 ⴷ 35 = 5 × 7.

Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility.

ⴷⴰ ⵉⵜⵜⵓⵢⴰⵖⴰⵍ ⵉⴷ ⴰⴼⴰⵔⵙ ⵏ ⵉⵎⴹⴰⵏ ⵉⵅⴰⵜⴰⵔⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⴰⵢⴷ ⵉⵙⴽⴰⵔⵏ ⵜⴰⵎⴳⴳⵉⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⵉⵛⵇⵇⴰⵏ ⴽⵉⴳⴰⵏ, ⴰⵔ ⵉⵜⵜⵉⵍⵉ ⵓⴼⵔⴰ ⵏ ⴽⵉⴳⴰⵏ ⵏ ⴱⵕⵓⵜⵓⴽⵓⵍⴰⵜ ⵏ ⵓⵙⵏⵜⵍ ⵉⵜⵜⵓⵙⵡⵓⵔⵉⵏ ⴳ ⵓⴱⴰⵔⴰⵣ ⴰⵅⴰⵜⴰⵔ ⵏⵏⴰ ⵓⵔ ⵉⵍⵍⵉⵏ ⵏⵉⵍ ⵓⵙⵙⵎⵔⵙ.

The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer).

ⵜⴰⵔⴰⴱⴱⵓⵜ ⵏ ⵜⵏⴼⴰⵍⵉⵜⵉⵏ ⵜⵉⵡⵏⵖⴰⵏⵉⵏ ⵉⵙⵎⴷⵏ “ⴰ” ⴷ “ⴱ”, ⴳ ⵜⵉⵏⴰⵡⵜ ⴳⴰⵏⵜ ⵜⵉⵔⵓⴱⴱⴰ ⵜⵉⵙⵍⴰⴳⵉⵏ ⵏ “ⴳ” (“ⵎ,ⴳ”, ⴰⵖ ⵢⴰⴽⴽⴰⵏ “ⵎ” ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ).

In other words, multiples of the smaller number rk−1 are subtracted from the larger number rk−2 until the remainder rk is smaller than rk−1.

ⵙ ⵓⵏⴰⵎⴽ ⵏⵏⵉⴹⵏ, ⴷⴰ ⵜⵜⵢⴰⴽⴰⵙⵏⵜ ⵜⵙⵍⴰⴳⵉⵏ ⵏ ⵢⵉⵎⴹ ⴰⵎⵥⵥⴰⵏ “ⵔⴽ-1” ⵙⴳ ⵢⵉⵎⵉⴹ ⴰⵅⴰⵜⴰⵔ “ⵔⴽ-2”, ⴰⵔ ⴷ ⴰⵖⴷ ⵉⵇⵇⵉⵎ “ⵔⴽ” ⵉⵎⵥⵥⵉⵢ ⵖⴼ “ⵔⴽ-1”.

Therefore, c divides the initial remainder r0, since r0 = a − q0b = mc − q0nc = (m − q0n)c.

ⵖⴰⵢⴰⵏ ⴰⵖⴼ ⵉⴱⵟⵟⵓ “ⵙ” ⴰⵎⴰⴳⵓⵔ ⴰⵎⵣⵡⴰⵔⵓ “ⵔ0”, ⴰⵛⴽⵓ “ⵔ0” = ⴰ - ⵇ0ⴱ = ⵎⵙ - ⵇ0ⵏⵙ = (ⵎ - ⵇ0ⵏ)ⵙ.

We first attempt to tile the rectangle using b-by-b square tiles; however, this leaves an r0-by-b residual rectangle untiled, where r0 < b. We then attempt to tile the residual rectangle with r0-by-r0 square tiles.

ⴷⴰ ⵏⵜⵜⴰⵔⵎ ⴰⴷ ⵏⴽⴽ ⵜⴰⵙⴳⴰ ⵉ ⵡⵓⵏⵣⵉⵖ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵓⵎⴽⴽⵓⵥ, “ⴱ-ⴱⵢ-ⴱ” ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴰⵢⴰ ⴷⴰⵖⴷ ⵉⵜⵜⴰⴷⵊⴰ ⵓⵏⵣⵉⵖ “ⵔ0-ⴱⵢ-ⴱ” ⴰⵔⴷ ⵉⴼⵓⴽⴽ, ⵉⴳⴰⵏ “ⵔ0” ⵉⵎⵥⵥⵉⵢ ⵅⴼ “ⴱ”. ⴷⴰⵕⵜ ⵓⵢⴰ ⴷⴰ ⴰⵙ ⵏⵜⴳⴳⴰ ⵉ ⵡⵓⵏⵣⵉⵖ ⴷ ⵢⵓⴳⵔⵏ ⵜⵉⴱⵍⴰⴹⵉⵏ ⵏ ⵓⵎⴽⴽⵓⵥ “ⵔ0-ⴱⵢ-ⵔ0”.

The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique.

ⵜⵓⵡⵉⴷ ⵜⵎⴰⴳⵓⵏⵜ ⵖⴼ ⵉⵜⵜⵡⴰⴳⴰ ⵏ ⵓⴱⵟⵟⵓ ⴰⴽⵍⵉⴷⵏ, ⵣⵓⵏⴷ ⵜⴰⵢⴼⵓⵜ ⴷ ⵓⵎⴰⴳⵓⵔ ⴰⴷ ⵍⵍⴰⵏ ⴰⵀⴰ ⴳⴰⵏ ⵉⵎⵥⵍⴰⵢ.

At the end of the loop iteration, the variable b holds the remainder rk, whereas the variable a holds its predecessor, rk−1.

ⴳ ⵜⵢⵉⵔⴰ ⵏ ⵢⵉⵍⵙ ⵏ ⵜⵅⵔⵙⵜ ⴷⴰ ⵉⵃⵟⵟⵓ ⵓⵙⵏⴼⵍ “ⴱ” ⵙ ⴰⵎⴰⴳⴰⵔⵓ “ⵔⴽ”, ⵉⵎⴽⵉⵏⵏⴰ ⵉⵀⵟⵟⵓ ⵓⵎⵙⵏⴼⵍ “ⴰ” ⵙ ⵓⵎⵣⵡⴰⵔⵓ, “ⵔⴽ-1”.

The mathematician and historian B. L. van der Waerden suggests that Book VII derives from a textbook on number theory written by mathematicians in the school of Pythagoras.

ⴰⵎⵓⵙⵏⴰⵡ ⵏ ⵜⵓⵙⵏⴰⴽⵜ “ⴱ.ⵍ.ⴼⴰⵏ .ⴷⵉⵔ. ⵡⵉⵔⴷⵏ, ⵉⵜⵜⵉⵏⵉ ⵉⴷ ⴰⴷⵍⵉⵙ ⵡⵉⵙⵙ ⵙⴰ, ⵉⵜⵜⵓⵙⵓⴼⵖⴷ ⵙⴳ ⵓⴷⵍⵉⵙ ⴰⵏⵎⵍⴰⵏ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵎⴹⴰⵏ ⵏⵏⴰ ⴰⵔⴰⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴳ ⵜⵉⵏⵎⵍ ⵏ ⴼⵉⵜⴰⵖⵓⵔⵙ.

Centuries later, Euclid's algorithm was discovered independently both in India and in China, primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.

ⵜⵉⵙⵓⵜⵉⵡⵉⵏ ⴰⵢⴰ ⵜⵢⴰⴼⴰ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵓⵇⵍⵉⴷ” ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⵏⵙⵉⵎⴰⵏⵜ ⴳ ⵍⵀⵉⵏⴷ ⴷ ⵚⵚⵉⵏ, ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⵙⵉⵍⴰⵏⵜ ⵎⴰⵔ ⴰⴷ ⵜⴳ ⴰⴼⵙⵙⴰⵢ ⵉ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵏ ⴷⵢⵓⴼⴰⵏⵜⵉⵏ ⴷ ⵉⴳⵎⴰⵏ ⴳ ⵜⴰⵡⵙⵙⵏⵉⵜⵔⴰⵏ ⴷ ⵓⵙⴽⴰⵔ ⵏ ⵉⵙⵜⴰⵍⵏ ⵉⵎⵏⵖⴰⴷⵏ.

The Euclidean algorithm was first described numerically and popularized in Europe in the second edition of Bachet's Problèmes plaisants et délectables (Pleasant and enjoyable problems, 1624).

ⵜⵜⵓⵙⵏⵓⵎⵍ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵓⵇⵍⵉⴷⵏ” ⴳ ⵜⵉⴽⵍⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵙ ⵉⵎⵉⴹ, ⵜⴰⵖ ⴰⴽⴽⵯ ⵓⵔⵓⴱⴱⴰ ⴳ ⵡⵓⴼⵓⵖ ⵡⵉⵙⵙ ⵙⵉⵏ ⵏ “ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⴱⴰⴽⵉⵜⵙ ⵉⴼⵊⵊⵉⵊⵏ ⵉⴹⴼⵉⵜⵏ”( ⵉⵎⵢⴽⵔⵉⵙⵏ ⵉⵎⵎⵔⵏ, ⴼⵊⵊⵉⵊⵏ 1624).

In the 19th century, the Euclidean algorithm led to the development of new number systems, such as Gaussian integers and Eisenstein integers.

ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 19 ⵜⵓⵡⵢ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵓⴽⵍⵉⴷⵏ” ⴰⵙⴱⵓⵖⵍⵓ ⵉⵎⴰⴳⴰⵏ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴰⵢⵏⵓⵜⵏ, ⵣⵓⵏⴷ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵏ ⴳⵓⵙⵢⴰⵏ, ⴷ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵏ ⵉⵣⵏⵛⵜⴰⵢⵏ.

Peter Gustav Lejeune Dirichlet seems to have been the first to describe the Euclidean algorithm as the basis for much of number theory.

ⴱⵉⵜⵔ ⴳⵓⵙⵜⴰⴼ ⵍⵉⴳⵓⵏ ⴷⵉⵔⵉⵜⵛⵍⵉⵜ, ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵙⵏⵓⵎⵍⵏ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵓⴽⵍⵉⴷⵏ” ⴰⵙⵉⵍⴰⵏ ⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵎⵏⵓⴳⴰⵍⵉⵏ ⵏ ⵉⵎⴹⴰⵏ.

For example, Dedekind was the first to prove Fermat's two-square theorem using the unique factorization of Gaussian integers.

ⵙ ⵓⵎⴷⵢⴰ, ⴷⵉⴷⴽⵉⵏ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵏⵥⵉⵏ ⵜⴰⵎⴰⴳⵓⵏⵜ “ⴼⵉⵔⵎⴰ” ⵎⵎ ⵙⵉⵏ ⵉⵎⴽⴽⵓⵥⵏ, ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵓⴼⵙⵙⴰⵢ ⵉⵥⵍⵉⵏ “ⴳⵓⵙⵢⴰⵏ” ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ.

Other applications of Euclid's algorithm were developed in the 19th century.

ⵜⵉⵙⵏⵙⵉⵜⵉⵏ ⵢⴰⴷⵏ ⵏ “ⵉⴽⵍⵉⴷⵙ ⴰⵍⴳⵓⵔⵉⵜⵎ”, ⵜⵜⵓⵙⴱⵓⵖⵍⵍⴰⵏⵜ ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 19.

Several novel integer relation algorithms have been developed, such as the algorithm of Helaman Ferguson and R.W. Forcade (1979) and the LLL algorithm.

ⵜⵜⵓⵙⴱⵓⵖⵍⵍⴰⵏⵜ ⴽⵉⴳⴰⵏ ⵏ “ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵜⴰⵎⵣⴷⴰⵢⵜ ⵜⴰⵎⴷⴷⴰⴷⵜ ⵉⴳⵏ ⵜⴰⵎⴰⵢⵏⵓⵜ” ⵣⵓⵏⴷ “ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵀⵉⵍⴰⵎⴰⵏ ⴼⵓⵔⴳⵓⵙⵓⵏ ⴷ ⵔ.ⵡ.ⴼⵓⵔⴽⴰⴷ (1979) ⴷ ⵍⵍⵍ ⴰⵍⴳⵓⵔⵉⵜⵎ.”

The players take turns removing m multiples of the smaller pile from the larger.

ⵓⵎⵣⵏ ⵉⵎⵉⵔⴰⵔⵏ ⵜⴰⵡⴰⵍⴰ ⵖⴼ ⵓⵙⴼⵜⴰⵢ ⵏ ⵜⴰⴽⵓⵔⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ ⴳ ⵜⵅⴰⵜⴰⵔⵜ.

By allowing u to vary over all possible integers, an infinite family of solutions can be generated from a single solution (x1, y1).

ⴰⵙⵎⵓⵔⵓⴼ ⵏ “ⵓ” ⵙ ⵓⵏⵓⵃⵢⵓ ⴳⵔ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⵉⴳ ⵡⵉⵏ ⴰⴷ ⵉⵍⵉⵏ” ⵏⵖⵢ ⴰⴷ ⵏⵙⴽⵔ ⵜⴰⵡⵊⴰ ⵓⵔ ⵉⵜⴼⴹⴹⵓⵏ ⵏ ⵉⴼⵓⴽⴽⵓⵜⵏ ⵏ ⵉⴼⵙⵙⴰⵢⵏ ⴳ ⵢⵓⵡⵏ ⵓⴼⵙⵙⴰⵢ (ⴽ1, ⵢ1).

In this field, the results of any mathematical operation (addition, subtraction, multiplication, or division) is reduced modulo 13; that is, multiples of 13 are added or subtracted until the result is brought within the range 0–12.

ⴳ ⵢⵉⴳⵔ ⴰⴷ, ⴷⴰ ⵉⵙⵙⴷⵔⵉⵙ ⵜⵉⵢⴰⴼⵓⵜⵉⵏ ⵏ ⴽⴰ ⵉⴳⴰⵜ ⵜⵉⴳⴳⵉ ⵏ ⵓⵙⵙⵉⵟⵏ ( ⵜⴰⵔⵏⵓⵜ, ⵜⵓⴽⴽⵙⴰ, ⴰⴽⴼⵓⴷ, ⵜⵓⵟⵟⵓⵜ) ⵙ ⵜⴱⵔⵉⴷⵜ 13, ⵉⴳⴰⵏ ⵉⵙ ⴷⴰ ⵉⵜⵜⵔⵏⵓ ⵏⵖⴷ ⵜⵓⴽⴽⵙⴰ ⵏ ⵉⵙⴼⵜⴰⵢⵏ ⵏ ⵉⵎⵉⴹ 13, ⴰⵔⴷ ⵉⵙⵓⵊⴷ ⵜⴰⵢⴰⴼⵓⵜ ⴳ ⵓⴰⵙⴳⵔⴰⵔ 0 - 12.

Now assume that the result holds for all values of N up to M − 1.

ⵎⵔⴷ ⴷⵖⵉ ⵉⴷ ⵜⴰⵢⴰⴼⵓⵜ ⵜⵜⵓⵙⵎⵔⴰⵙ ⵖⴼ ⵜⵉⵏⴷⵉⵜⵉⵏ ⴰⴽⴽⵯ ⵏ “ⵏ” ⴰⵔ “ⵎ - 1”.

For illustration, the probability of a quotient of 1, 2, 3, or 4 is roughly 41.5%, 17.0%, 9.3%, and 5.9%, respectively.

ⵉ ⵓⵙⵉⵙⵙⴼⵉⵡ, ⵉⵖⵢ ⴰⴷ ⵏⴰⴼ ⵜⵉⴱⴹⵓⵜ ⵏ 1, 2, 3, ⵏⵖⴷ 4 ⵉⵏⵎⴰⵍⴰⵏⵏ ⵉ 41.5%, 17.0%, 9.3%, ⴷ 5.9%, ⵙ ⵓⵎⴹⴼⴰⵕ.

One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor.

ⵢⴰⵏ ⴳ ⵡⴰⵎⵎⴰⴽⵏ ⵓⵔ ⵉⵎⵕⵡⵉⵏ ⴳ ⵜⵉⴼⵉⵜ ⵏ GCD ⵉ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏⴻⵏ ⵉⵖⴰⵔⴰⵏⴻⵏ a ⴷ b, ⵉⴳⴰ ⴰⵙⵙⵉⵟⵏ ⵏ ⴰⵎⴳⴰⵍ ⴰⵎⵛⵛⵓⵔ ⴳⵔⴰⵜⵙⵏ, ⴷ GCD ⵉⴳⴰ ⵓⵎⴳⴰⵍ ⴰⵎⵛⵛⵓⵔ ⴰⵅⴰⵜⴰⵔ.

As noted above, the GCD equals the product of the prime factors shared by the two numbers a and b. Present methods for prime factorization are also inefficient; many modern cryptography systems even rely on that inefficiency.

ⵉⵎⴽ ⵉⵜⵢⴰⵏⵏⴰ ⴰⴼⵍⵍⴰ, ⴷⴰⵖ ⵢⴰⴽⴽⴰ GCD ⵜⴰⵢⴰⴼⵓⵜ ⵏ ⵓⵙⴼⴼⵓⴽⵜⵉ ⵏ ⵉⵎⴳⴳⵉⵜⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵉⵎⵙⵙⴰⵔⵏ ⴳⵔ ⵙⵉⵏ ⵉⵎⴹⴰⵏ a ⴷ b. ⵜⵉⴱⵔⵉⴷⵉⵏ ⵏ ⴷⵖⵉ ⵉ ⵉⵎⴳⴳⵉⵜⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵓⵔ ⴳⵉⵏ ⴰⵡⴷ ⵉⵎⵕⵡⵉⵜⵏ, ⴰⵡⴷ ⴽⵉⴳⴰⵏ ⵏ ⵉⴳⵔⵔⴰⵢⵏ ⵏ ⵓⴼⵙⵙⵓ ⵜⴰⵜⵔⴰⵔⵜ, ⵜⴱⴷⴷⴰ ⵖⴼ ⴳⴰⵔ ⵜⴰⵣⵎⵔⵜ.

Lehmer's GCD algorithm uses the same general principle as the binary algorithm to speed up GCD computations in arbitrary bases.

ⴷⴰ ⵙⵙⵎⵔⴰⵙⵏⵜ ⴰⵍⴳⵓⵔⵉⵜⵎ Lehmer's GCD, ⴰⵎⵏⵣⴰⵢ ⴰⵏⵎⴰⵜⵜⴰⵢ ⵏⵏⵉⴽ, ⵣⵓⵏⴷ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵜⴰⵎⵙⵏⴰⵜ ⵏ ⵓⵙⵔⴱⵢ ⵏ ⵉⵙⵙⵉⵟⵏ ⵏ GCD ⴳ ⵉⵍⴳⴰⵎⵏ ⵉⵎⵅⵔⴱⵇⵏ.

The Euclidean algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can also be defined.

ⵏⵖⵢ ⴰⴷ ⵏⵙⵙⵎⵔⵙ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵓⴽⵍⵉⴷⵏ ⵏ ⵓⴼⵙⵙⴰⵢ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵜ ⴷⵢⵓⴼⴰⵏⵜⵉⵏ ⴰⵡⵏⵖⴰⵏ ⴷ ⵉⵙⵇⵙⴰⵢ ⵏ ⵉⵎⴰⴳⵓⵔⵏ ⵉⵚⵉⵏⵉⵢⵏ ⵏ ⵉⴷ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ, ⵉⵖⵢ ⴰⵡⴷ ⵓⵔⵎⵎⵓⵙ ⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵣⴷⴰⵢ ⵙⴳ ⵉⴷ ⵎⵎ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ.

Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.

ⴽⴰ ⵉⴳⴰⵜ ⵢⵉⴳⵔ ⵏ ⵓⴽⵍⵉⴷⵏ; ⵉⴳⴰ ⵉⴳⵔ ⵏ ⵓⴼⴰⵔⵙ ⵉⵥⵍⵉⵏ (UFD), ⵡⴰⵅⵅⴰ ⵓⵔ ⵉⴳⵉ ⵡⵓⵖⵓⵍ ⴰⵎⴷⴷⴰⴷ.

A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal.

ⵉⴳⴰ ⵢⵉⴳⵔ ⵏ ⵓⴽⵍⵉⴷⵏ; ⵉⴳⴰ ⴰⵀⴰ ⵉⴳⵔ ⴰⵎⴰⵜⵜⴰⵢ ⴰⴷⵙⵍⴰⵏ (PID), ⴷ ⵏⵜⵜⴰ ⴷ ⵉⴳⵔ ⴰⵙⵎⴷⴰⵏ ⵉⵍⴰⵏ ⴰⵎⴷⵢⴰ ⵢⴰⵜⵜⵓⵢⵏ.

Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵉⵎⴹⴰⵏ ⴷ ⵉⵎⵛⵛⵓⵔⵏ ⴰⵡⴷ ⴳ ⵉⵎⵜⵡⴰⵍⵏ ⵓⵔ ⵡⴰⵍⴰ ⵉⵍⵍⵉⵏ, ⴳ ⴰⵎⵓⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵓⴷⴷⵉⵙⵏ ⴷ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵔⵡⵉⵏ ⴷ ⵉⵎⴹⴰⵏ ⵉⵛⵛⴰⵔⵏ.

The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.

ⵜⴰⴳⵓⵔⵉ ⵜⵜⵓⵙⵎⵔⴰⵙ ⴳ ⵓⵙⴰⵍⴰ ⵎⴰⵔ ⴰⴷ ⵜⵙⵏⵓⵃⵢⵓ ⴰⵏⴰⵡ ⴰⴷ ⵏ ⵓⵎⵜⵡⴰⵍ ⵖⴼ ⵓⵎⵜⵡⴰⵍ ⴰⵚⴹⵉⵙⵎⵔⴰⵡ, ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵜⴰⵡⵙⵙⵏⵉⵜⵔⴰⵏ.

This was explained in the 17th century textbook The Ground of Arts.

ⵉⵜⵜⵓⵙⴼⵔⴰ ⵓⵢⴰ ⴳ ⵓⴷⵍⵉⵙ ⴰⵏⵎⵍⴰⵏ “ⴰⴽⴰⵍ ⵏ ⵜⵥⵓⵕⵉⵡⵉⵏ” ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 17.

The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction.

ⵜⴰⵢⴰⴼⵓⵜ ⵏ ⵓⵎⵜⵡⴰⵍ ⴷ ⵓⵎⵢⴰⵖ ⵏⵏⵙ ⵉⴳⴰⵜ 1, ⵅⴼ ⴰⵢⴰ ⴰⵙ ⵉⴳⴰ ⵓⵎⵢⴰⵖ ⴰⵎⴰⵖⵓⵍ ⵏ ⵓⴽⴼⵓⴷ ⵖⴼ ⵓⵎⵜⵡⴰⵍ.

The remainder becomes the numerator of the fractional part.

ⴷⴰ ⵉⵜⴳⴳⴰ ⵓⵎⴰⴳⵓⵔ ⴰⵏⵣⵍ ⵏ ⵜⴰⴼⵓⵍⵜ ⵜⴰⵎⵜⵡⴰⵍⵜ.

Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is .

ⵙⴳ ⵎⴰⵢⴷ ⵉⴳⴰ 5×17 (= 85) ⵢⵓⴳⵔ 4× 18 (= 72), ⵜⴰⵢⴰⴼⵓⵜ ⵏ ⵓⵙⵎⵣⴰⵣⴰⵍ ⵜⴳⴰ.

Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth.

ⵙⴳ ⵎⴰⵢⴷ ⵡⵉⵙⵙ ⴽⵕⴰⴹ ⵏ ⵡⵉⵙⵙ ⴽⴽⵓⵥ ⴰⵢⴷ ⵉⴳⴰⵏ ⵢⴰⵏ ⵖⴼ ⵙⵉⵏ ⴷ ⵎⵔⴰⵡ, ⴷⴰ ⵉⵜⴳⴳⴰ ⵙⵉⵏ ⵏ ⵉⴷ ⵡⵉⵙⵙ ⴽⵕⴰⴹ ⴳ ⵡⵉⵙⵙ ⴽⴽⵓⵥ ⵙⵉⵏ ⵖⴼ ⵙⵉⵏ ⴷ ⵎⵔⴰⵡ.

Sometimes an infinite repeating decimal is required to reach the same precision.

ⵉⵜⵙⵏⵜ ⵜⵉⴽⴽⴰⵍ ⴷⴰ ⵉⵜⵜⵉⵔⵉ ⵡⴰⴷⴷⴰⴷ ⵉⵎⵉⴹ ⴰⵎⵔⴰⵡⴰⵏ ⵡⴰⵔⵜⵎⵉ ⵢⵓⵍⵙⵏ, ⵎⴰⵔ ⴰⴷ ⵏⴰⵡⴹ ⵜⴰⵎⵏⵖⵓⵜ ⴰⵎⵎ ⵜⴰⵏⵏⴰ.

The Egyptians used Egyptian fractions  BC.

ⵙⵎⵔⵙⵏ ⵉⵎⵉⵚⵕⵉⵢⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵉⵚⵕⵉⵢⵏ ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ.

Their methods gave the same answer as modern methods.

ⵜⴽⴰ ⵜⵖⴰⵔⴰⵙⵜ ⵏⵙⵏ ⵜⴰⵎⵔⴰⵔⵓⵜ ⵣⵓⵏⴷ ⵜⵉⵏ ⵜⵖⴰⵔⴰⵙⵜ ⵜⴰⵜⵔⴰⵔⵜ.

A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta, Brahmagupta, and Bhaskara.

ⵉⴳⴰ ⵓⵡⵏⵏⵉ ⴰⵜⵔⴰⵔ ⵖⴼ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵜⵜⵢⴰⵙⵏ ⵙ ⵢⵉⵙⵎ ⵏ “ⴱⵉⵏⴰⵔⴰⵙⵉ” ⵉⴳⴰⵏ ⵙⴳ ⵍⵀⵉⵏⴷ ⵉⵜⵢⴰⵙⵙⵏ ⵙ ⵜⵡⵓⵔⵉⵡⵉⵏ “ⴰⵔⵢⴰⴱⴰⵜⴰ”, “ⴱⵔⴰⵎⴰⴳⵓⵜⴰ”, ⴷ “ⴱⴰⵙⴽⴰⵔⴰ”

"In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers ""wrap around"" when reaching a certain value, called the modulus."

ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⴰⵙⵙⵉⵟⵏ ⴰⵏⴰⵡⴰⵏ, ⵉⴳⴰ ⴰⵏⴳⵔⴰⵡ ⵏ ⵓⵙⵙⵉⵟⵏ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⴰⴳ ⴷⴰ ⵜⵎⵓⵏ ⵉⵎⴹⴰⵏ ⵉⴳⵏ ⵓⵡⴹⵏ ⵢⴰⵏ ⵡⴰⵜⵉⴳ; ⵉⵙⵎ ⵏⵏⵙ ⴰⵎⵢⵉⵡⵏ.

A very practical application is to calculate checksums within serial number identifiers.

ⵜⵉⵙⵏⵙⵉⵜ ⵏ ⵉⵎⵙⴽⵔ ⵜⴳⴰ ⴰⵙⵙⵉⵟⵏ ⴰⵎⵎⵓⵜⵔⵏ ⴰⵔⵓⵛⵛⵉⵍ ⴰⴳⵏⵙⵓ ⵏ ⵡⵓⵏⵎⵉⵍⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵙⵏⵙⵍ.