Tamazight-NLP/NLLB-Seed_Tamasheq-Tifinagh-Script huggingface dataset:
English: By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well.
Tamasheq: ⵙⴳ ⵢⵉⵙⵢ ⵏ ⵉⵡⵜⵜⴰ, ⵏⵖⵢ ⴰⴷ ⵏⴳ ⴰⵙⵜⴰⵍ ⵉ ⵉⵜⵙⵏ ⵉⵙⵏⴼⵍⵓⵍⵏ ⵓⵎⴳⵉⵏ, ⴳ ⵉⵍⵍⴰ ⵉⵎⵉⴹ ⵡⴰⵔⵜⵎⵉ ⵏ ⵉⴼⴰⵜⵔⵏ ⵙⴳ ⵜⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.
English: Its history, notably documented by Philip J. Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century.
Tamasheq: ⴷⴰ ⵢⴰⴽⴽⴰ ⵓⵎⵣⵔⵓⵢ ⵏⵏⵙ ⵏⵏⴰ ⵢⴰⵔⵓ ⴼⵉⵍⵉⴱ ⵊ. ⴷⵉⴼⵉⵙ, ⴳ ⵓⵎⴳⵔⴰⴷ ⵢⵓⵡⵉⵏ ⵜⴰⵙⵎⵖⵔⵜ ⵏ ⵛⵓⴼⵉⵏⵉ ⴱⵔⵉⵣ ⴰⵙⴳⴳⵯⴰⵙ ⵏ1963, ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⴱⵓⵖⵍⵓⵜⵏ ⵉⴷⵙⵍⴰⵏ ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⵙⴳ ⵓⵙⴰⵜⵓ 18.
English: Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.
Tamasheq: ⵅⴼ ⴰⴷ ⵉⵜⵢⴰⴼ ⵓⵏⵎⵎⴰⵍ ⵉⵥⵍⵉⵏ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ, ⵢⵓⴼ ⴰⴷ ⴷⴰⵔⴽ ⵜⵉⵍⵉ ⵜⴱⵔⵉⴷⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵏ ⵢⵉⴼ ⵏ ⵜⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.
English: However, the gamma function does not appear to satisfy any simple differential equation.
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ ⴷⴰ ⵜⵎⵙⴰⵙⴰ ⴷ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⵢⴰⴼⵜ ⵜⵓⵏⵣⵉⵍⵜ.
English: The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function.
Tamasheq: ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⴱⵓⵔⵀ-ⵎⵓⵍⵉⵔⵓ ⵜⵓⴱⵖⵉⵔⵜ, ⴰⵛⴽⵓ ⵉⵡⵀⵏ ⴰⴷ ⵉⵜⵜⵓⵡⵔ ⵡⵓⴹⵓ ⴰⵍⵓⴳⴰⵔⵉⵜⵎⵉ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵎⴰⵔ ⴰⴷ ⵏⵙⵜⵉ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.
English: As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. National Bureau of Standards.
Tamasheq: ⴷⴷⴰ ⴳ ⴳⴰⵏ ⵉⵏⴳⵎⴰⵎⵏ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵉⵍⵉⴽⵜⵔⵓⵏⵉ ⵡⵉⵏ ⵓⵙⵏⴼⵍ ⵏ ⵉⵎⵢⴰⵍⵍⴰⵢⵏ ⴳ 1950, ⵜⵜⵢⴰⴼⵙⴰⵔⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⵏⵎⴰⵍⵍⴰⵢⵏ ⵉⵙⵎⴰⵏ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ ⵉⵛⵇⵇⴰⵏ ⵎⴰⵔ ⴰⴷ ⵜⵜⵓⴳ ⵜⵓⵜⵜⵔⴰ, ⴳ ⵢⴰⵎⵓ ⵢⴰⵏ ⵓⵙⵏⵎⴰⵍⵍⴰⵢ ⴰⵎⵖⴷⴰⵏ ⵏ 12 ⵏ ⵜⵓⵣⵓⵏⵜ ⵜⴰⵎⵔⴰⵡⵜ, ⵙⴳ ⵖⵓⵔ ⵓⵙⵉⵔⴰ ⴰⵏⴰⵡⴰⵢ ⴰⵏⴰⵎⵓⵔ ⴰⵎⵉⵔⵉⴽⴰⵏⵉ.
English: In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula.
Tamasheq: ⴳ ⵜⵓⵙⵏⴰ, ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴳⴰ ⵜⴰⴱⵔⵉⴷⵜ ⵉⵙⴰⵏⴼⵏ ⵏ ⵓⵙⵉⵡⵍ ⵖⴼ ⵉⵏⵖⵎⵉⵙⵏ ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴰⵜⴰⵔⵜ, ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ ⵜⴰⵍⵖⴰ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⴰⵍⵖⴰ ⵏ ⵛⵉⵎⵉ.
English: In mathematics, a formula generally refers to an identity which equates one mathematical expression to another, with the most important ones being mathematical theorems.
Tamasheq: ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⵎⵎⴰⵍ ⵜⴰⵍⵖⴰ ⵙ ⵓⵎⴰⵜⴰ ⵜⴰⵎⴰⴳⵉⵜ ⵉⴳⴰⵏ ⴰⵙⵉⵡⵍ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵢⴰⴹⵏ, ⵙⵜⴰⵡⵀⵎⵎⴰⵏⵜ ⴷⵉⴽⵙ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.
English: This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex.
Tamasheq: ⵜⴰⵎⵢⴰⵇⵇⴰⵏⵜ ⴰⴷ ⵎⵇⵇⴰⵔ ⵓⵔ ⵡⴰⵍⴰ ⵜⵙⵜⴰⵡⵀⵎⵎⴰ ⴳ ⵜⴰⵍⵖⴰ ⵜⴰⵎⵏⵣⴰⵍⵜ, ⴷⴰ ⵜⵜⵉⵏⵉ ⵉⵙ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵖⵉⵏ ⴰⴷ ⵜⵎⵛⴰⵛⴽⴰⵏ ⵙ ⴽⵉⴳⴰⵏ ⵏ ⵓⵙⵔⴱⵉ ⴰⴽⴷ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⵉⵅⴰⵜⴰⵔⵉⵏ ⴷ ⵜⵉⵏⵏⴰ ⵡⴰⵍⴰ ⵉⵛⵇⵇⴰⵏ.
English: For example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom.
Tamasheq: ⵙ ⵓⵎⴷⵢⴰ, H2O ⵜⴳⴰ ⵜⴰⵍⵖⴰ ⵜⴰⴽⵉⵎⵉⵢⵜ ⵏ ⵡⴰⵎⴰⵏ ⵏⵏⴰ ⵉⵙⵜⵜⵉⵏ ⵉⴷ ⴽⵓ ⵉⵎⵉⴽ ⴷⵉⴽⵙ ⵙⵏⴰⵜ ⵜⴱⵍⴽⵉⵎⵉⵏ ⵏ ⵀⵉⴷⵔⵓⵊⵉⵏ (H) ⴷ ⵜⴱⵍⴽⵉⵎⵜ ⵏ ⵓⴽⵙⵉⵊⵉⵏ (O).
English: In empirical formulas, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element.
Tamasheq: ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⵉⵔⵎⵉⵜ, ⴷⴰ ⵜⵜⵓⵙⵏⵜⴰⵢⵏⵜ ⵜⵙⵖⴰⵍⵉⵏ ⴰⴷ ⵙ ⵓⴷⵕⴹⵉⵚ ⴰⴷⵙⵍⴰⵏ ⴷ ⴰⵔ ⵜⵙⵜⵜⵉⵢ ⵎⵏⵏⴰⵡⵜ ⵏ ⵜⴱⵍⴽⵉⵎⵉⵏ ⵏ ⵉⴼⵕⴹⵉⵚⵏ ⵢⴰⴹⵏ ⴳ ⵜⴰⵏⴰⵡⵜ, ⵉⴳⴰⵏ ⵉⵙⵖⴰⵍⵏ ⵏ ⵓⴼⵕⴹⵉⵚ ⴰⴷⵙⵍⴰⵏ.
English: Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers.
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵓⵔ ⵏⵣⴹⴰⵕ ⴷ ⵏⵓⵔⵓ ⵉⵜⵙⵏ ⵡⴰⵏⴰⵡⵏ ⵏ ⵜⵓⴷⴷⵉⵙⵉⵏ ⵜⴰⵢⵢⵓⵏⵉⵏ ⴰⴷ ⴳⵉⵏⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⵉⵔⵎⵉⵜ ⴳ ⵍⵍⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⴷⴰⵢ.
English: There are several types of these formulas, including molecular formulas and condensed formulas.
Tamasheq: ⵍⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⵏⴰⵡⵏ ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⴰⴷ ⴳ ⵜⴰⵎⵓ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⵉⴳⵣⵣⵓⵎⵉⵏ ⴷ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⴰⵏⵥⵥⵉⵡⵉⵏ.
English: Functions were originally the idealization of how a varying quantity depends on another quantity.
Tamasheq: ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⴳ ⵓⵥⵓⵕ ⵏⵏⵙⵏⵜ ⴳⴰⵏⵜ ⵜⴰⵙⴼⵉⵍⴰⵡⵜ ⵏ ⵎⴰⵎⵏⴽ ⵙⵏⵣⵖⵣⴰⵏ ⵜⴰⵎⴰⴽⵜⴰ ⵜⴰⵎⵙⵏⴼⵍⵜ ⵖⴼ ⵜⵎⴰⴽⵜⴰ ⵏⵏⵉⴹⵏ.
English: This definition of "graph" refers to a set of pairs of objects.
Tamasheq: ⴷⴰ ⵉⵎⵎⴰⵍ ⵓⵙⵉⵙⵙⵏ ⵏ “ⵓⵏⵓⵖ ⴰⵙⵏⵎⵎⴰⵍ” ⴰⴷ, ⵖⵔ ⵜⵔⴰⴱⴱⵓⵜ ⵏ ⵉⵏⴰⵔⴰⴳⵏ ⵉⵎⵖⵏⴰⵡⵏ.
English: When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
Tamasheq: ⴰⴷⴷⴰⵢ ⵉⴳ ⵢⵉⴳⵔ ⴷ ⵢⵉⴳⵔ ⵉⵛⵛⴰⵔⵏ ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵉⵖⵢ ⴰⴷ ⵉⴳ ⵓⵏⴰⵔⴰⴳ ⴳ ⵉⵏⴰⵔⴰⴳⵏ ⴰⴷ ⵜⵉⵎⵉⵜⴰⵔ ⵜⵉⴷⵉⴽⴰⵔⵜⵉⵢⵢⵉⵏ ⵏ ⵜⵏⵇⵇⵉⴹⵜ ⴳ ⵓⵖⴰⵡⴰⵙ.
English: Occasionally, it may be identified with the function, but this hides the usual interpretation of a function as a process.
Tamasheq: ⴳ ⵜⵉⵣⵉ ⵖⵔ ⵜⴰⵢⴹ, ⵉⵖ ⴰⴷ ⵜⵜⵓⵙⵜⵢ ⴰⴽⴷ ⵜⵡⵓⵔⵉ, ⵎⴰⴽⴰ ⴰⵢⴰ ⴷⴰ ⵉⴷⴷⴰⵍ ⴰⴼⵙⵙⴰⵢ ⵢⴰⴷ ⵉⵍⵍⴰⵏ ⵉ ⵜⵡⵓⵔⵉ ⴰⵎⵎ ⵜⵉⵣⵉⴳⵣⵜ.
English: A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of real or complex numbers.
Tamasheq: ⵉⵖ ⴰⴷ ⵢⵉⵍⵉ ⴳ ⵜⴽⴰⵕⴹⴰ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵔⴰⴱⴱⵓⵜ ⴰⴼⵓⵖⴰⵍ ⴰⵎⴰⵜⴰⵔ ⵏⵏⵙ, ⵎⴰⴽⴰ ⴳ ⵉⵜⵙⵏ ⵉⵙⴰⵜⵉⵍⵏ, ⵡⴰⵍⴰ ⴳ ⵉⴷⵍⵉⵙⵏ ⵉⵇⴱⵓⵔⵏ, ⴷⴰ ⵉⵜⴳⴳⴰ ⵢⵉⴳⵔ ⵜⴰⵎⴰⵜⴰⵔⵜ ⵏ ⵜⵡⵓⵔⵉ, ⵏⵜⵜⴰ ⵏⵏⵉⴽ ⵙ ⵓⵥⵍⴰⵢ; ⵜⴰⵔⴰⴱⴱⵓⵜ ⵏ ⵉⵎⴹⴰⵏ ⵜⵉⴷⵜ ⵏⵖⴷ ⵓⴷⴷⵉⵙⵏ.
English: Another common example is the error function.
Tamasheq: ⴰⵎⴷⵢⴰ ⵢⴰⴷⵏ ⵉⵜⵢⴰⴼⵙⴰⵔⵏ; ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵉⵣⴳⵍⵏ.
English: Power series can be used to define functions on the domain in which they converge.
Tamasheq: ⵉⵖⵢ ⴰⴷ ⵏⵙⵡⵓⵔⵉ ⵙ ⵜⴳⴼⴼⵓⵔⵜ ⵏ ⵜⵣⴹⴰⵕⵜ ⵎⴰⵔ ⴰⴷ ⵏⵙⵜⵉ ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⴳ ⵢⵉⴳⵔ ⴳ ⵎⵎⵍⵎⴰⵍⴰⵏⵜ.
English: Then, the power series can be used to enlarge the domain of the function.
Tamasheq: ⴹⴰⵕⵜ ⵓⵢⵏⵏⴰⵖ ⵏⵖⵢ ⴰⴷ ⵏⵙⵡⵓⵔⵉ ⵙ ⵜⴳⴼⴼⵓⵔⵜ ⵏ ⵜⵣⴹⴰⵕⵜ ⵎⴰⵔ ⴰⴷ ⵏⵙⵅⵉⵜⵔ ⵉⴳⵔ ⵏ ⵜⵡⵓⵔⵉ.
English: Parts of this may create a plot that represents (parts of) the function.
Tamasheq: ⵉⵖⵢ ⴰⴷ ⴰⵡⵉⵏ ⵉⴳⵣⵣⵓⵎⵏ ⵙⴳ ⵓⵢⴰ ⵖⵔ ⵓⵙⴽⴰⵔ ⵏ ⵓⵚⵟⵟⴰ ⵉⵙⵎⴷⵢⴰⵏ (ⵜⴰⴳⵣⵣⵓⵎⵜ ⵙⴳ) ⵙ ⵜⵏⴰⵎⴽⴰⵏⵜ.
English: This is the canonical factorization of .
Tamasheq: ⵡⴰⴷ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⴼⴰⵔⵙ ⴰⴷⵙⵍⴰⵏ ⵏ.
English: At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth.
Tamasheq: ⴳ ⵜⵉⵣⵉ ⵏⵏⴰⵖ, ⵜⴰⵏⵏⴰⵢⵜ ⵖⴼ ⵜⵡⵓⵔⵉⵡⵉⵏ ⵉⵍⴰⵏ ⴰⵜⵉⴳ ⵏ ⵜⵉⴷⵜ ⵏ ⵓⵎⵙⵏⴼⵍ ⵏ ⵜⵉⴷⵜ, ⴷ ⵉⵇⵏⴻⵏ ⴰⴷ ⴰⴽⴽⵯ ⴳⵉⵏⵜ ⵜⵏⴰⵎⴽⴰⵜⵉⵏ ⵜⴰⴳⴼⴼⵓⵔⵜ.
English: Functions are now used throughout all areas of mathematics.
Tamasheq: ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙⵏⵜ ⵜⵏⴰⵎⴽⴰⵜⵉⵏ ⴷⵖⵉ ⴳ ⵢⵉⴳⵔⴰⵏ ⵎⴰⵕⵕⴰ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.
English: This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic.
Tamasheq: ⵜⴰⴷⵖ ⵜⴰⴱⵔⵉⴷⵜ ⵏ ⵓⵙⵉⵙⵙⵏ ⵙ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴷⵜ ⵜⴰⵎⴰⵖⵓⵍⵜ, ⵙⴳ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴹⵜ, ⴷ ⴷⴰ ⵜⴳⴳⴰ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴹⵜ ⵜⴰⵎⵙⵙⴰⵔⵜⵓⵜ.
English: Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions.
Tamasheq: ⵜⴰⴱⵖⵓⵔⵜ ⵏ ⵓⵙⵉⵙⵙⵏ ⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵎⵉ ⴳⴳⵓⴷⵉⵏⵜ ⵜⵉⵏⴷⵉⵜⵉⵏ, ⴼⴰⵡⵏⵜ ⴽⵉⴳⴰⵏ ⵉⴳ ⴷⴰ ⵏⵙⴽⵙⵉⵡ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⵓⴷⴷⵉⵙⵜ, ⴷ ⴰⵔ ⵜⴳⴳⴰ ⵓⵍⴰ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⴼⴰⵔⵙⵜ.
English: Such a function is called the principal value of the function.
Tamasheq: ⵜⵢⴰⵙⵙⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⴰⴷ ⵙ ⵡⴰⵜⵉⴳ ⴰⴷⵙⵍⴰⵏ ⵏ ⵜⵡⵓⵔⵉ.
English: Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions.
Tamasheq: ⴰⵙⵖⵉⵡⵙ ⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵉⴳⴰ ⴰⵎⵢⴰ ⵏ ⵓⵙⵖⵉⵡⵙ ⵏⵏⴰ ⴳ ⵜⵍⵍⴰ ⵜⵓⵙⴽⴰ ⵏ ⵉⵖⴰⵡⴰⵙⵏ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵎⴳⴳⵉⵜⵏ ⴰⵢⵢⴰⵡⵏ ⴷⴰⵢ, ⵏⵏⴰ ⵉⵜⴳⴳⴰⵏ ⵣⵓⵏⴷ ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.
English: Except for computer-language terminology, "function" has the usual mathematical meaning in computer science.
Tamasheq: ⵙ ⵓⵙⵍⵉⴷ ⵏ ⵢⵉⵔⵎⵏ ⵏ ⵜⵓⵜⵍⴰⵢⵜ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵉⵍⵍⴰ ⵖⵓⵔ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵓⵏⴰⵎⴽ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵏⵏⴰ ⵢⴰⴷ ⵉⵍⵍⴰⵏ ⴳ ⵜⵓⵙⵙⵏⴰⵎⵙⵙⵓⴷⵙ.
English: Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation.
Tamasheq: ⴷⴰ ⵉⵜⵜⵓⵙⵓⵔⴰⵔ ⵙ ⵜⴳⵓⵔⵉⵡⵉⵏ ⵙⴳ ⵉⵜⵙⵏ ⵉⵍⴳⴰⵎⵏ (ⵜⴰⴳⴷⴰⵣⴰⵍⵜ, ⴰⵣⵓⴳⵣ, ⴰⵙⵏⴼⵍ), ⴷ ⵏⵉⵜⵏⵉ ⵉⵏⴼⵍⴰⵍⴰⵢⵏ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⴷ ⵏⵖⵢ ⴰⴷ ⵏⵙⵙⴼⵔⵓ ⵉⵙ ⵜⴳⴰ ⵉⵍⴳⴰⵎⵏ ⵏ ⵓⵙⵙⵉⵟⵏ.
English: Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century.
Tamasheq: ⵉⵙⵙⵎⵔⵙ ⵏⵉⴽⵓⵍⴰⵚ ⵜⵛⵓⴽⵉ ⵜⴰⵍⵖⴰ ⵙⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵓⵣⵎⵎⴻⵎ ⵏ ⵜⵙⵉⵍⴰ ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 15, ⴷ ⵉⵙⵡⵓⵔⵉ ⵉⵙ ⴹⴰⵕⴰⵙ ⵀⵉⵏⵔⵉⴽⵓⵙ ⴳⵔⴰⵎⴰⵜⵓⵙ ⴷ ⵎⵉⵛⵉⵍ ⵙⵜⵉⴼⵍ ⴳ ⵓⵙⴰⵙⵓ ⵡⵉⵙⵙ 16 .
English: Thus they would write polynomials, for example, as .
Tamasheq: ⵉⵎⴽⵉ ⴰⵙ ⵜⵜⵓⵔⵓⵏ ⵜⵉⵎⴳⴳⵓⴷⵢⵉⵏ ⵏ ⵉⵡⵜⵜⴰ, ⵙ ⵓⵎⴷⵢⴰ, ⵣⵓⵏⴷ.
English: The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents.
Tamasheq: ⵜⴰⵢⴰⴼⵓⵜ ⵜⴳⴰ ⴰⵀⴰ ⵓⵟⵟⵓⵏ ⵓⵎⵏⵉⴳ ⵏ ⵜⵉⴷⵜ, ⴳⵉⵎⵉⵏⵜ ⵜⵎⴰⴳⵉⵜⵉⵏ ⴷ ⵉⵏⴰⵎⴰⵥⵏ ⵉⵎⵙⵙⴼⵔⵓⵜⵏ ⴰⴼⵍⵍⴰ, ⵉⴷⵙⵍⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⵉⵎⴷⴷⴰⴷⵏ ⴰⴽⴷ ⵉⵙⵓⵙⵙⵏ ⴰⴷ ⵏ ⵉⴷⵙⵍⴰⵏ ⵏ ⵜⵉⴷⵜ.
English: This function equals the usual th root for positive real radicands.
Tamasheq: ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⴰⴷ ⴰⵢⴷ ⵉⴳⴰⵏ “ⴰⵥⵓⵕ th”, ⴳ ⴰⵀⴰ ⵍⵍⴰⵏ ⵉⵣⵖⵕⴰⵏ ⵓⵎⵏⵉⴳⵏ ⵏ ⵜⵉⴷⵜ.
English: This is the starting point of the mathematical theory of semigroups.
Tamasheq: ⵜⴰⴷⵖ ⴷ ⵜⴰⵏⵇⵇⵉⴹⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵜⵓⵙⵏⴰⴽⵜ ⵉ ⵜⵣⵏⴰⵔⴰⴱⴱⵓⵜ.
English: We can again replace the set N with a cardinal number n to get Vn, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism.
Tamasheq: ⵜⵉⴽⵍⵜ ⵢⴰⴹⵏ ⵏⵖⵢ ⴰⴷ ⵏⵙⵏⴼⵍ ⵜⴰⵔⴱⵉⵄⵜ N ⵙ ⵡⵓⵟⵟⵓⵏ ⴰⴷⵙⵍⴰⵏ n, ⵎⴰⵔ ⴰⴷ ⴰⵖⴷ ⵜⴽ Vn, ⵡⴰⵅⵅⴰ ⵓⵔ ⵏⵙⵜⵉⵢ ⴽⴰⵏ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ ⴷ ⵓⵥⵓⵕ n ⴷⴰ ⵉⵜⵜⵓⵢⴰⵙⵙⴰⵏ ⵓⵢⴰ ⴰⵔ ⴰⵎⵜⵔⴰⵏ.
English: Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
Tamasheq: ⵏⵉⴽⵓⵍⴰⵚ ⴱⵓⵔⴱⴰⴽⵉ, ⵉⴼⵕⴹⵉⵚⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⵜⵔⵓⴱⴱⴰ ⵙⴱⵔⵉⵏⵊ-ⴼⵉⵔⵍⴰⴳ, 2004, III.§3.5.
English: Iterating tetration leads to another operation, and so on, a concept named hyperoperation.
Tamasheq: ⴷⴰ ⵉⵜⵜⴰⵡⵢ ⵢⵉⵍⵙ ⵏ ⵜⵙⵏⴰⵡⴰⵢⵜ ⵖⵔ ⵜⵎⵀⵍⵜ ⵏⵏⵉⴹⵏ, ⵉⵎⴽⵉ, ⴰⵔⵎⵎⵓⵙ ⵉⴳⴰⵏ ⵜⴰⵎⵔⵏⵉⵡⵜ ⴳ ⵜⵎⵀⵍⵜ.
English: In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable.
Tamasheq: ⴳ ⵜⵙⵖⵍⵉⵏ ⵏ ⵜⵙⵏⵙⵉ, ⵜⵙⵎⴷⵢⴰ ⵜⵏⴰⵎⴽⴰⵏⵜⵉⵏ ⵏ ⵜⵙⵉⵍⴰ, ⴰⵙⵖⵏ ⵖ ⵢⴰⴽⴽⴰ ⵓⵙⵏⴼⵍ ⴰⵎⵣⴷⴰⵢ ⴳ ⵓⵙⵏⴼⵍ ⴰⵎⵥⵍⴰⵢ; ⴰⵙⵏⴼⵍ ⴰⵎⴰⵙⴰⵖ ( ⵜⵉⴳⵎⵉⴹⵉ ⵏ ⵜⵔⵏⵓⵜ ⵏⵖⴷ ⵓⴽⵓⵙ ), ⴳ ⵓⵎⵙⵏⴼⵍ ⴰⵎⴹⴼⴰⵕ.
English: This function property leads to exponential growth or exponential decay.
Tamasheq: ⴷⴰ ⵉⵜⵜⴰⵡⵢ ⵓⵏⴰⵎⴰⵥ ⵏ ⵜⵎⵔⵙⵜ ⴰⴷ ⵖⵔ ⵜⵉⴳⵎⵉ ⵏ ⵜⵙⵉⵍⴰ ⵏⵖⴷ ⴰⵍⴰⴽⴰⵢ ⵏ ⵜⵙⵉⵍⴰ.
English: Similarly, the composition of onto (surjective) functions is always onto.
Tamasheq: ⵙ ⵓⵔⵡⴰⵙ, ⴰⵙⵎⵓⵜⵜⴳ ⵏ ⵜⵎⵔⵙⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ ( ⵜⵉⵙⵡⵉⵏⴳⵎⴰⵏⵉⵏ) ⴷⴰ ⴰⵀⴰ ⵉⵜⵜⵉⵍⵉ ⴳ ⵜⵡⵓⵔⵉ.
English: Then one can form chains of transformations composed together, such as .
Tamasheq: ⴷ ⵉⵖⵢ ⵓⴼⴳⴰⵏ ⴰⴷ ⵉⵙⵙⴽⵔ ⵜⵉⴳⴼⴼⵓⵔⵉⵏ ⵜⵉⵎⵙⵏⴼⵍⵉⵏ ⵉⵎⴰⵏⴻⵏ, ⵣⵓⵏⴷ.
English: This alternative notation is called postfix notation.
Tamasheq: ⴰⵙⵏⴰⵎⴰⵜⴰⵔ ⵉⵎⴽⴽⵉⵙⵉ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵣⵎⵎⴻⵎ ⵏ ⴱⵓⵙⵜⴼⵉⴽⵙ.
English: The category of sets with functions as morphisms is the prototypical category.
Tamasheq: ⵜⴰⴳⵔⵔⵓⵎⴰ ⵏ ⵜⵔⵓⴱⴱⴰ ⴳ ⵍⵍⴰⵏⵜ ⵜⵎⵔⵙⵜⵉⵏ ⵣⵓⵏ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⴳⴰⵏ ⵜⴰⴳⵔⵔⵓⵎⴰ ⵏ ⵓⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ.
English: For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example).
Tamasheq: ⵙ ⵓⵎⴷⵢⴰ, ⴷⵉⵙⵉⴱⵍ (ⴷⴱ), ⵜⴳⴰ ⵜⴰⴳⵣⵣⵓⵎⵜ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴰⴷ ⵜⵙⵉⵡⵍ ⵖⴼ ⴰⵙⵖⵍ ⴽⵉⵍⵓⴳⴰⵔⵉⵜⵎⴰⵜ, ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⵉⴽⴽⴰⵍ ⵉ ⵜⵣⵎⵔⵜ ⵏ ⵜⵎⴰⵜⴰⵔⵜ ⴷ ⵓⴽⵜⵜⵓⵔ ( ⵉⴷⵔ ⵏ ⵉⵎⵙⵍⵉ ⴰⵢⴷ ⵡⴰⵍⴰ ⵉⵍⵍⴰⵏ).
English: They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.
Tamasheq: ⴷⴰ ⵜⵜⴰⵡⵙ ⴳ ⵓⵙⵏⵓⵎⵍ ⵏ ⵓⴳⵍⵓⴳⵍ ⵏ ⵉⵣⵎⴰⵣ ⵏ ⵓⵥⴰⵡⴰⵏ, ⴰⵔⴷ ⵜⴼⴼⴻⵖ ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵙⵙⵉⵟⵉⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵏⵖⴷ ⵜⴰⵏⵎⵉⵍⵉⵜ ⵏ ⵉⴼⵕⴹⵉⵚⵏ, ⴷ ⴰⵔ ⵜⵙⵙⵍⵎⴰⴷ ⵉⵜⵙⵏ ⵉⵎⴷⵢⴰⵜⵏ ⴳ ⵜⵉⴽⵍⵉⵙⵏⵜ ⵜⴰⴼⵉⵣⵉⴽⵜ, ⴷ ⵜⵖⵢ ⴰⴷ ⵜⴰⵡⵙ ⴳ ⵓⵙⵙⵉⵟⵏ ⵏ ⵓⵏⴱⴽⴰⴹ.
English: The next integer is 4, which is the number of digits of 1430.
Tamasheq: ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⴰⴷ ⵉⴳⴰ 4, ⴷ ⵏⵜⵜⴰ ⴰⵢⴳⴰⵏ ⵉⵎⵉⴹ ⵏ ⵜⵓⵣⵓⵏⵉⵏ 1430.
English: Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600.
Tamasheq: ⴷⴰⵜ ⵏ ⴰⴷ ⵉⵜⵜⵓⵀⵢⵢⴰ “ⵏⴰⴱⵉⵔ”, ⵍⵍⴰⵏ ⵜⵉⴽⵏⵉⵜⵉⵏ ⵏⵏⵉⴹⵏ ⵉⵍⴰⵏ ⵉⵖⵓⴼⴰⵍⵏ ⴰⵎ ⵡⵉ, ⵣⵓⵏⴷ “ⵜⵉⵎⵍⵙⵉⵜ”, ⵏⵖⴷ ⴰⵙⵙⵎⵔⵙ ⵏ ⵉⵙⵎⵢⴰⵍⵍⴰⵏⴻⵏ ⵏ ⵜⵉⴹⴼⵓⵔⵜ, ⵏⵏⴰ ⵉⵙⴱⵓⵖⵍⵍⴰ ⵊⵓⵙⵜ ⴱⵓⵔⵊⵉ ⴳ ⵓⵖⵓⴼⴰⵍ ⴰⴱⴰⵔⴰⵡ ⴰⵜⵜⴰⵢⵏ ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⵏ1600.
English: Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the “order of a number”.
Tamasheq: ⵉⴳ ⴷⴰ ⵏⵙⴰⵡⴰⵍ ⵖⴼ ⵡⵓⵟⵟⵓⵏ ⵉⵔⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ, ⵜⴳⴰ ⵜⴰⵎⴰⵜⴰⵔⵜ ⵏ ⵓⵏⵎⵉⵍⴰ ⵖⵔ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⵛⵛⵓⵔ,ⵉⵎⴽ ⵣⴰⵕⵙ ⵉⵏⵄⵜ ⴰⵔⵅⵎⵉⴷ ⵉⴷ ⵏⵜⵜⴰ ⴰⵢⴷ ⵉⴳⴰⵏ “ⴰⵙⵙⵓⴷⵙ ⵏ ⵡⵓⵟⵟⵓⵏ”.
English: Such methods are called prosthaphaeresis.
Tamasheq: ⴷⴰ ⴰⵙⵏ ⵜⵜⵉⵏⵉⵏⵜ ⵉ ⵜⴱⵔⵉⴷⵉⵏ ⴰⴷ; ⵜⵉⵏⴽⵔⴰⵎⵉⵏ.
English: For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor.
Tamasheq: ⵙ ⵓⵎⴷⵢⴰ, ⴽⵓ ⵜⴰⵎⵕⵚⵉⵜ ⴳ ⵉⴼⵔⴽⵉ ⵏ ⵏⵓⵜⵉⵍⵓⵙ, ⵜⴳⴰ ⵜⵓⵏⵖⵉⵍⵜ ⵜⴰⵏⵎⵉⵍⵉⵜ ⵏ ⵜⵎⵕⵚⵉⵜ ⵥⴰⵕⵙ ⵉⵍⵍⴰⵏ ⵜⵜⵓⵙⵖⴰⵍ ⵙ ⴰⵎⵎⴰⴽ ⵉⵣⵣⴳⴰⵏ.
English: Logarithms are also linked to self-similarity.
Tamasheq: ⵍⵓⴳⴰⵔⵉⵜⵎ ⵍⴰⵏⵜ ⴰⵡⴷ ⵜⴰⵙⵖⵏ ⵙ ⴰⵎⵔⵡⵙ ⴰⵏⵉⵎⴰⵏ.
English: It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics.
Tamasheq: ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⵎⴰⵔ ⴰⴷ ⵉⵙⵖⵍ ⵉⵙⵡⵉⵔⵏ ⵏ ⵜⵣⵎⵔⵜ ⴳ ⵢⵉⵣⵏ ⵏ ⵜⵎⵓⵍⵉ ⵜⵉⵎⵥⵥⴰⵕⵓⵕⵉⵏ, ⴰⴼⴰⴷ ⵏⵙⵏⵓⵎⵍ ⵉⵙⵡⵉⵔⵏ ⵏ ⵜⵣⵎⵔⵜ ⵏ ⵉⵎⵙⵍⵉ ⴳ ⵜⵎⵙⵍⵉⵜⵉⵏ, ⴷ ⵓⵙⵙⵓⵎ ⵏ ⵜⵔⵉⵙⵉⵏⵜⵉ ⴳ ⵢⵉⴳⵔⴰⵏ ⵏ ⵓⵡⵍⴰⴼ ⴷ ⵉⵎⵡⴰⵍⴰⵏ.
English: Vinegar typically has a pH of about 3.
Tamasheq: ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵉⵍⵉ ⴳ ⵍⵅⵍⵍ ⴰⵔ ⴰⵜⵜⴰⵢⵏ 3 ⵏ ⵜⵙⴽⴼⴰⵍ ⵏ ⵜⴰⵙⵎⵎⵉ.
English: This "law", however, is less realistic than more recent models, such as Stevens's power law.)
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴰⵣⵔⴼ ⴰⴷ ⵓⵔ ⵉⴳⵉ ⴰⵏⵉⵍⴰⵡ ⴰⵎ ⵉⵎⴷⵢⴰⵜⵏ ⵉⵜⵔⴰⵔⵏ, ⵣⵓⵏⴷ ⴰⵣⵔⴼ ⴰⵏⴱⴱⴰⴹ ⵏ ⵙⵜⵉⴼⵏⵙ.
English: When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.
Tamasheq: ⵉⴳ ⵉⴳⴰ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⵙⵏⴼⵍ ⴰⵎⵅⴱⴱⴹ ⵏ ⵉⴱⵟⵟⵓ ⴰⵖⴰⵔⴰⵏ, ⴷⴰ ⵉⵜⵢⴰⵏⵏⴰ ⵉⴷ ⴰⵎⵙⵏⴼⵍ ⵖⵓⵔⵙ ⴰⴱⵟⵟⵓ ⴰⵍⵓⴳⴰⵔⵉⵜⵎ ⵓⵏⵣⵉⵍ.
English: For such a model, the likelihood function depends on at least one parameter that must be estimated.
Tamasheq: ⴳ ⵣⵓⵏⴷ ⴰⵎⴷⵢⴰ ⴰⴷ, ⴷⴰ ⵜⴱⴷⴷⴰ ⵜⵎⵔⵙⵜ ⵏ ⵉⵖⵢ ⵖⴼ ⵢⵓⵡⵜ ⵜⵙⵖⵍⵜ, ⵙ ⵉⵍⴰⵇ ⴰⴷ ⵜⵜⵓⵡⵇⵇⵔ.
English: Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results.
Tamasheq: ⵣⵓⵏⴷ, ⴷⴰ ⵜⴳⴳⴰ ⴰⵍⴳⵓⵔⵉⵜⵎ ⴰⵙⵓⴼⵖ ⵏ ⵓⵎⵎⴽⵛⵎ, ⵙ ⵡⵓⴼⵓⵖ ⵏ ⵜⵍⴳⴰⵎⵜ ⴷ ⵓⵔ ⵉⴼⴼⵉⵖⵏ ⵙ ⵜⴱⵔⵉⴷⵜ ⵏ ⵜⵍⴳⴰⵎⵜ ⵖⴼ ⵙⵉⵏ, ⴷ ⵡⵓⴼⵓⵖ ⵏⵏⵙ ⴷⴰⵜ ⵓⵙⵎⵓⵏ ⵏ ⵜⵢⴰⴼⵓⵜⵉⵏ.
English: Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system.
Tamasheq: ⴷⴰ ⵙⵙⵎⵔⴰⵙⵏ ⵡⵉⵏⵏⴰ ⵉⵇⵔⴰⵏ ⵙ ⵍⵢⴰⴱⵓⵏⵓⴼ ⵍⵓⴳⴰⵔⵉⵜⵎ ⵎⴰⵔ ⴰⴷ ⵙⵙⵖⴰⵍⵏ ⵜⴰⵙⴽⴼⵍⵜ ⵏ ⵓⵔⵡⴰⵢ ⴳ ⵓⵏⴳⵔⴰⵡ ⴰⴷⵉⵏⴰⵎⵉⴽ.
English: The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length.
Tamasheq: ⵏⵖ ⴰⴷ ⵏⴷⵍ ⴰⵎⴽⵕⴰⴹ ⵙⵢⵉⵔⴱⵉⵏⵙⴽⵉ ( ⴳ ⵜⵡⵍⴰⴼⵜ), ⵙ ⴽⵕⴰⴹⵜ ⵜⵓⵏⵖⵉⵍⵉⵏ, ⴽⵓ ⵢⵓⵡⵜ ⴷⵉⴽⵙⵏⵜ ⵙ ⵓⵣⴳⵏ ⵏⵜⴰⵖⵣⵉ ⴰⵥⵓⵕⴰⵏ.
English: Another example is the p-adic logarithm, the inverse function of the p-adic exponential.
Tamasheq: ⴰⵎⴷⵢⴰ ⵏⵏⵉⴹⵏ ⵉⴳⴰⵜ p-adic ⵍⵓⴳⴰⵔⵉⵜⵎ ⵏ ⵜⵡⵓⵔⵉ ⵜⴰⵎⴰⵖⵓⵍⵜ ⵜⴰⵙⵉⵍⴰ ⵏ p-adic.
English: Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.
Tamasheq: ⵏⵖⵢ ⴰⴷ ⵏⴳ ⵜⴰⵎⵀⵍⵜ ⵏ ⵜⵙⵉⵍⴰ ⵙ ⵜⵣⵎⵔⵜ, ⵎⴰⴽⴰ ⴷⴰ ⵉⵜⵢⴰⵖⵉⵍ ⵉⵙ ⵉⵛⵇⵇⴰ ⵓⵙⵙⵉⵟⵏ ⵏ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⴱⴹⵉ ⴳ ⵉⵜⵙⵏⵜ ⵜⵔⵓⴱⴱⴰ.
English: Square roots of negative numbers can be discussed within the framework of complex numbers.
Tamasheq: ⵉⵖⵢ ⵓⵎⵔⴰⵔⴰ ⵏ ⵡⴰⵡⴰⵍ ⵖⴼ ⵉⵥⵓⵕⴰⵏ ⵉⵎⴽⴽⵓⵥⵏ ⵏ ⵉⵎⴹⵏ ⵓⵣⴷⵉⵔⵏ ⴰⴳⵏⵙⵓ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ.
English: In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).
Tamasheq: ⴳ ⵍⵀⵉⵏⴷ ⵜⴰⵇⴱⵓⵔⵜ ⵜⴽⴽⴰⵜ ⵜⵓⵙⵙⵏⴰ ⵙ ⵜⵙⴳⴳⵉⵏ ⵜⵉⵎⴰⴳⵓⵏⵉⵏ ⴷ ⵉⵙⴽⴽⵉⵔⵏ ⵉⵥⵓⵕⴰⵏ ⵉⵎⴽⴽⵓⵥⵏ ⴷ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵉⴳⴰ ⴰⵇⴱⵓⵔ ⵣⵓⵏⴷ ⵙⵓⵍⴱⴰ ⵙⵓⵜⵔⴰⵙ ⵏⵏⴰ ⵉⴽⴽⴰⵏ ⴰⵜⵜⴰⵢⵏ ⵏ 800 ⴰⵔ 500 ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ (ⵉⵖⵢ ⵉⴷ ⵓⴳⴳⴰⵔ).
English: The letter jīm resembles the present square root shape.
Tamasheq: ⴰⵙⴽⴽⵉⵍ jīm ⵢⴰⵖ ⴳ ⵜⴰⵍⵖⴰ ⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵏ ⴷⵖⵉ.
English: It defines an important concept of standard deviation used in probability theory and statistics.
Tamasheq: ⴷⴰ ⵉⵙⵜⵜⵉ ⴰⵔⵎⵎⵓⵙ ⵉⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⵏ ⵡⵓⵏⵓⴼ ⴰⵏⴰⵡⴰⵢ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵢⵉⵖⵉⵢ ⴷ ⵜⴰⴷⴷⴰⴷⴰⵏⵜ.
English: Most pocket calculators have a square root key.
Tamasheq: ⵜⵍⵍⴰ ⴳ ⴽⵉⴳⴰⵏ ⵜⵎⵙⵙⵉⵟⵏⵉⵏ ⵜⵙⴰⵔⵓⵜ ⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ.
English: The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.
Tamasheq: ⴰⵙⵔⵔⵓⵏⴽⵙ ⴰⵣⵎⵣⴰⵏ ⵎⴰⵔ ⴰⴷ ⵏⵙⵙⵉⵟⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵙ ⵡⵓⵟⵟⵓⵏ n ⵙ ⵓⵏⵖⴰⴷ, ⵢⴰⴽⵙⵓⵍ ⴷ ⵓⵙⵔⵔⵓⴽⵙ ⵏ ⵓⵙⴼⵓⴽⵜⵉ ⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ ⵙⴳ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ n.
English: Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
Tamasheq: ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⴱⵔⵜ ⴳⴰⵏ ⴰⴳⵏⴰⵔ ⴷ ⴽⵕⴰⴹ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵉⴼⵙⵔⵜⵏⵜ ⵓⵎⵓⵙⵏⴰⵡ ⴰⵍⵎⴰⵏⵉ ⴷⵉⴼⵉⴷ ⵀⴰⵍⴱⵔ ⴰⵙⴳⴳⵯⴰⵙ ⵏ 1900.
English: For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist.
Tamasheq: ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏⵏⵉⴹⵏ, ⵣⵓⵏⴷ ⵡⵉⵙⵙ 5 ⵎⵙⴰⵙⴰⵏ ⵉⵎⴰⵙⵙⴰⵏ ⵙ ⵜⵣⴰⵢⴽⵓⵜ ⵖⴼ ⵢⵓⵡⵏ ⵓⵙⵙⴼⵔⵓ, ⵉⵜⵜⵓⴼⴽ ⵓⴼⵙⵙⴰⵢ ⵉ ⵓⵙⵙⴼⵔⵓⵜⵏ ⵉⵜⵢⴰⵙⵜⴰⵢⵏ, ⵎⴰⴽⴰ ⵍⵍⴰⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵉⵍⴰ ⵜⴰⵣⵍⴰⵖ ⵣⴰⵕⵙ ⵓⵔ ⵜⴰ ⵜⵜⵢⴰⴼⵙⴰⵢⵏ.
English: There are two problems that are not only unresolved but may in fact be unresolvable by modern standards.
Tamasheq: ⵍⵍⴰⵏ ⵙⵏⴰⵜ ⵜⵎⵓⴽⵔⵉⵙⵉⵏ ⴷⴰⵢ ⵓⵔⵜⴰ ⵜⵢⴰⴼⵙⴰⵢⵏⵜ, ⵎⴰⴽⴰ ⴳ ⵜⵉⵏⴰⵡⵜ ⵉⵖⵢ ⵉⵙ ⵓⵔ ⵍⵉⵏⵜ ⵉⴼⵙⵙⴰⵢⵏ ⵙ ⵉⵏⴰⵡⴰⵢⵏ ⴰⴷ ⵉⵜⵔⴰⵔⵏ.
English: The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance.
Tamasheq: ⵜⵜⵓⴼⴽ ⴰⵙⵏ ⵜⴰⵖⴷⴼⵜ ⴽⵉⴳⴰⵏ ⵉ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴰⵏ ⴷ ⵉⵇⵉⵎⴰⵏ ⵙⴳ ⵓⴳⵏⴰⵔ ⴷ ⵙⵎⵎⵓⵙ, ⴳ ⵜⵢⵉⵔⵉⵡⵉⵏ ⵏ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ ⵙⵉⵎⵔⴰⵡ ⵜⵙⵓⵍ ⵜⵡⵓⵔⵉ ⵖⴼ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴰⴷ ⵜⵍⴰ ⵜⴰⵖⴹⴼⵜ.
English: Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.
Tamasheq: ⵉⴷⴷⵔ ⵀⴰⵍⴱⵔⵜ 12 ⵏ ⵓⵙⴳⴳⵯⴰⵙ, ⴹⴰⵕⵜ ⵓⴼⵙⴰⵔ ⵏ ⴽⵓⵔⵜ ⴳⵓⴷⵍ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ, ⵎⴰⴽⴰ ⵓⵔ ⵉⴱⴰⵢⵏ ⵉⵙ ⵢⴰⵔⵓ ⴽⴰⵏ ⵜⵎⵔⴰⵔⵓⵜ ⵖⴼ ⵜⵡⵓⵔⵉ ⴳⵓⴷⵍ.
English: In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.
Tamasheq: ⵉⴳ ⵉⵍⵍⴰ ⵓⵎⵔⴰⵔⴰ ⵏ ⵡⴰⵡⴰⵍ ⵖⴼ ⵜⴰⵏⵏⴰⵢⵜ ⵏⵏⵙ ⵏ ⴽⵓ ⵜⴰⵎⵓⴽⵔⵉⵙⵜ ⵜⵓⵙⵏⴰⴽⵜ ⵉⵇⵏⴻⵏ ⴰⵙ ⵢⵉⵍⵉ ⵓⴼⵙⵙⴰⵢ, ⵢⵓⴷⵊⴰ ⵀⵉⵍⴱⵔ ⵜⴰⵣⵎⵔⵜ ⵏ ⵡⵉⵙ ⵉⵖⵢ ⵓⴼⵙⵙⴰⵢ ⴰⴷ ⵉⴳ ⵉⵏⵉⴳⵉ ⵏ ⵡⴰⴳⵓⵎ ⵏ ⵜⵎⵓⴽⵔⵉⵙⵜ ⵜⴰⵥⵓⵕⴰⵏⵜ.
English: The first of these was proved by Bernard Dwork; a completely different proof of the first two, via ℓ-adic cohomology, was given by Alexander Grothendieck.
Tamasheq: ⵜⵜⵓⵡⵔ ⵜⵎⵣⵡⴰⵔⵓⵜ ⵏⵏⵙⵏ ⵙⴳ ⵖⵓⵔ ⴱⵉⵔⵏⴰⵔⴷ ⴷⵓⵔⴽ, ⵉⴼⴽ ⴰⵍⵉⴽⵙⴰⵏⴷⵔ ⴳⵔⵓⵜⵉⵏⴷⵉⴽ ⴰⵏⵥⴰ ⵓⵔ ⵢⴰⴽⵙⵓⵍⵏ ⴰⴽⴽⵯ ⴷ ⵉⵎⵣⵡⵓⵔⴰ, ⴳ ⵜⵓⵙⵙⵏⴰⵎⵢⵉⴷⵉⵔⵜ ⵉⵛⵛⴰⵔⵏ ℓ-adic.
English: However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴳⴰⵏ ⵉⵙⵡⵉⵏⴳⵉⵎⵏ ⵏ ⵡⵉⵍ; ⴳ ⵓⵙⴰⵜⴰⵍ ⵏⵏⵙⵏⵜ ⵣⵓⵏⴷ ⵜⴰⵎⵓⴽⵔⵉⵙⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵏ ⵀⵉⵍⴱⵔ, ⴷ ⵓⵔ ⵉⵏⵏⵉ ⵡⵉⵍ ⵉⵙ ⵉⴷ ⵏⵜⵜⴰⵜ ⴰⵢⴷ ⵉⵏⵏⴰⵏ ⴰⴷ ⵉⴳ ⴰⵖⴰⵡⴰⵙ ⵏ ⵜⵓⵙⵏⴰⴽⵉⵏ ⴽⵓⵍⵍⵓ.
English: Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.
Tamasheq: ⴰⴽⴽⴰⵜ ⴰⵔ ⵡⴰⵍⴰ ⵢⴰⴽⴽⴰ ⵉⵔⴷⵓⵙ ⵜⵉⵙⵎⵖⵓⵔⵉⵏ ⵏ ⵉⴷⵔⵉⵎⵏ, ⴰⵜⵉⴳ ⵏ ⵜⵙⵎⵖⵓⵔⵜ ⵖⴼ ⵛⵇⵇⵉⵢⵜ ⵏ ⵜⵎⵓⴽⵔⵉⵙⵜ ⵉⵜⵢⴰⵏⵏⴰⵢⵏ.
English: At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute.
Tamasheq: ⵡⴰⵅⵅⴰ ⴰⵢⵏⵏⴰⵖ ⴳ ⵉⵎⴰⵙⵙⵏ ⵏ ⵓⵙⵎⵎⴰⵍ ⵉⴷⵙⵍⴰⵏ, ⵜⴳⴰ ⵜⴰⵔⵡⵙⴰⵏⵜ ⵜⵉⵎⴳⴳⵉⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⵍⴱⵔⵜ ⴳ ⵓⵣⵎⵣ ⵡⵉⵙⵙ 21, ⵜⴰⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵜⵙⵎⵖⵓⵔⵜ ⵏ ⵜⵉⴼⴹⵜ ⵜⵉⵙⵙ ⵙⴰ ⵉⵙⵜⵉ ⵓⵙⵉⵏⴰⴳ ⴽⵍⴰⵢ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 2000.
English: The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
Tamasheq: ⵜⵓⵔⴷⴰ ⵏ ⵔⵉⵎⴰⵏ ⵜⵔⴰ ⴰⵙⵏⵉⵖⵙ ⴰⵛⴽⵓ ⵜⴱⴰⵢⵏⴷ ⴳ ⵜⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⴱⵔⵜ, ⴷ ⵜⵍⴳⴰⵎⵜ ⵏ ⵙⵎⵉⵍ, ⴷ ⵜⵍⴳⴰⵎⵜ ⵏ ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵓⵙⵎⵖⵔ ⵏ ⵜⵉⴼⴹⵏⵜ, ⵓⵍⴰ ⴰⵡⴷ ⴳ ⵉⵙⵡⵉⵏⴳⵉⵎⵏ ⵏ ⵡⵉⵍ ⵙⴳ ⵜⴰⵍⵖⴰ ⵏⵏⵙ ⵜⴰⵏⵣⴳⴰⵏⵜ.
English: 1931, 1936 3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?
Tamasheq: 1931, 3 ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1936 ⵉⴳ ⵏⵓⵙⵢ ⵙⵉⵏ ⵉⴼⵔⵖⴰⵙ ⵢⴰⴽⵙⵓⵍⵏ ⴳ ⵓⴽⵙⴰⵢ, ⵉⵙ ⵏⵥⴹⴰⵕ ⴰⴷ ⴰⵀⴰ ⵏⴱⵟⵟⵓ ⴰⵎⵣⵡⴰⵔⵓ ⴰⵔ ⴷ ⵉⴳ ⵜⴳⵣⵣⵓⵎⵉⵏ ⵜⵉⵎⵥⵍⴰⵢ ⵉⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⴷⵍⴰⵙⵏ, ⵏⵏⴰ ⵙ ⵏⵖⵢ ⴰⴷⵜⵏ ⵏⵙⵎⵓⵏ ⴰⵔ ⴷ ⴰⵖ ⴷ ⴽⵉⵏ ⵜⵉⵙⵙ ⵙⵏⴰⵜ?
English: — 12th Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field.
Tamasheq: __12 ⵉⵙⵙⴰⵔⵡ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⵡⵉⴱⵔ ⵜⵢⵓⵔⵎ ⵅⴼ ⵉⵣⴷⴰⵎⵏ ⵏ ⴰⴱⵉⵍⵢⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ ⵖⵔ ⴽⴰ ⵉⴳⴰⵜ ⵉⴳⵔ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⴷⵙⵍⴰⵏ.
English: 1959 15th Rigorous foundation of Schubert's enumerative calculus.
Tamasheq: 15 ⴳ 1959 ⵜⴰⵙⵉⵍⴰ ⵜⵓⵇⵊⵉⵕⵜ ⵏ ⵓⵙⵙⵉⵟⵏ ⵛⵓⴱⵔⵜ ⴰⵎⵉⴹⴰⵏ.
English: 1927 18th (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?(b) What is the densest sphere packing?
Tamasheq: 18 ⴳ 1927 (ⴰ) ⵉⵙ ⵉⵍⵍⴰ ⴱⵓ ⵉⵎⵢⴰⵏⴰⵡⵏ ⵉⴷⵍⴰⵙⵏ ⵉⵜⵜⴰⴷⵊⴰⵏ ⴰⴽⴼⴰⴼ ⵎⵉ ⵜⵢⴰⴽⵣⵏ ⵉⴷⵍⴰⵙⵏ ⵏⵏⵙ ⴳ ⴽⵕⴰⴹ ⵡⵓⴳⴳⵓⴳⵏ? (ⴱ) ⵎⴰⵜⵜⴰ ⵉⴳⵔⴰⵏ ⴳ ⵜⴳⴳⵓⴷⵢ ⵜⴰⵏⵥⵥⵉ ⴳ ⵓⵙⵎⵓⵙⵙⵓ?
English: A number is a mathematical object used to count, measure, and label.
Tamasheq: ⵓⵟⵟⵓⵏ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵎⵖⵏⴰⵡ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⴳ ⵓⵙⵙⵉⵟⵏ ⴷ ⵓⵙⵖⴰⵍ ⴷ ⵓⵙⴻⵙⵙⴰⵖ.
English: More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five.
Tamasheq: ⵙ ⵓⵎⴰⵜⴰ, ⵉⵖⵢ ⵓⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵎⵥⵍⴰⵢ ⵙ ⵜⵎⴰⵜⴰⵔⵉⵏ ⵉⴳⴰⵏ ⵉⵏⵎⴹⴰⵏ, ⵙ ⵓⵎⴷⵢⴰ “5” ⵜⴳⴰ ⵉⵎⵉⴹ ⴰⵖ ⵢⴰⴽⴽⴰⵏ ⵓⵟⵟⵓⵏ ⵙⵎⵎⵓⵙ.
English: Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation.
Tamasheq: ⴰⵙⵙⵉⵟⵏ ⵙ ⵡⵓⵟⵟⵓⵏ ⴷⴰ ⵜⵡⴰⴼⴽⴰ ⵙⴳ ⵜⵎⵀⵍⵉⵏ ⵏ ⵓⵙⵙⵉⵟⵏ, ⵜⵉⵏⵏⴰ ⵡⴰⵍⴰ ⵉⵍⵍⴰⵏ; ⴰⵎⴰⴳⵓⵜ ⴷ ⵜⵓⴽⴽⵙⴰ, ⴷ ⵓⵙⴼⵓⴽⵜⵉ ⴷ ⵜⵓⵟⵟⵓⵜ ⴷ ⵜⴰⵙⵉⵍⴰ.
English: Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24, 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, Nov 30, 1992.
Tamasheq: ⴳⵉⵍⵙⴷⵓⵔⴼ, ⵜⵓⵎⴰⵙ ⵉⵢ, ⵜⴰⵎⵙⵙⵏⴽⴷⵜ ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⵜⴰⴷⵍⵙⴰⵏⵜ, ⴳ ⵜⵣⵔⴰⵡⵜ ⵏ ⵓⵟⵓⵎⵢⵉ ⴷ ⵉⵏⴽⴰⵙ, ⵊⵓⵏ ⵡⵉⵍⵉⵢ & ⵙⵓⵏⵙ, 24 ⵉⴱⵔⵉⵔ 2012. ⵔⵉⵙⵜⵉⴼⵓ , ⵙ. ⵜⵓⵙⵏⴰⴽⵜ ⴳ ⵓⵖⵔⴼ ⴷ ⵓⵎⵣⵔⵓⵢ, ⵙⴱⵔⴰⵏⴳⵔ ⵙⴰⵢⵙ, ⵉⵎⴰⵙⵙⵏ ⵏ ⵓⵙⵏⵖⵎⵙ ⴰⵙⴱⴱⴰⴱ, 30 ⵏⵓⵡⴰⵏⴱⵉⵔ 1992.
English: During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept.
Tamasheq: ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 19, ⵙⵙⵏⵜⵉⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⴰⵙⴱⵓⵖⵍⵓ ⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵡⵏⴳⵉⵎⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ ⵓⵛⵓⵔⵏⵜ ⵉⵜⵙⵏ ⵉⵏⵥⵍⴰⵢⵏ ⵏ ⵡⵓⵟⵟⵓⵏ, ⵏⵖ ⴰⴷ ⵏⵉⵏⵉ ⴰⵙⵙⵉⵔⵡ ⵏ ⵓⵙⵉⵙⵙⵏ.
English: A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers.
Tamasheq: ⴰⵏⴳⵔⴰⵡ ⵏ ⵓⵙⵎⵣⴰⵣⴰⵍ ⵓⵔ ⴷⵉⴽⵙ ⴰⵙⵉⵙⵙⵏ ⵏ ⵡⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ (ⵉⵎⴽ ⵉⴳⴰ ⵡⴰⴷⴷⴰⴷ ⵏ ⵓⵣⵎⵎⴻⵎ ⴰⵎⵔⴰⵡⵉ ⴰⵜⵔⴰⵔ), ⵉⵎⴽ ⵉⵡⴷⴰ ⴳ ⵓⵙⵎⴷⵢⴰ ⵏ ⵉⵎⴹⴰⵏ ⵉⵅⴰⵜⴰⵔⵏ.
English: Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero.
Tamasheq: ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵟⴰ ⵙⴳ ⴱⵔⴰⵀⵎⴰⵙⴼⵓⵟⴰⵙⵉⴷⴰⵏⵟⴰ; ⴰⴷⵍⵉⵙ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵏⵏⴰⵏ ⴰⵎⵢⴰ ⵉⴳⴰ ⵓⵟⵟⵓⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵉⴳⴰ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵟⴰ ⴰⵎⵣⵡⴰⵔⵓ ⵉⵙⵔⵙⵏ ⴰⵙⵉⵙⵙⵏ ⵏ ⵓⵎⵢⴰ.
English: In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).
Tamasheq: ⴳ ⵓⵙⴰⵔⴰ ⴷⵖ ⵏⵏⵉⴽ ⵉⵙⵙⵎⵔⵙ ⴱⴰⵏⵉⵏⵉ ( ⴰⵙⴰⵜⵓ ⵡⵉⵙⵙ 5 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⴰⵎⵡⵓⵔⵉ ⵉⵅⵡⴰⵏ (ⴰⵎⵢⴰ), ⴳ ⴰⵛⵜⴰⴷⴰⵢⴰⵢⵉ, ⵉⴳⴰⵏ ⴰⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ ⵏ ⵉⵍⴳⴰⵎⵏ ⵍⵊⵉⴱⵔ ⵏ ⵜⵓⵜⵍⴰⵢⵜ ⵜⴰⵙⴰⵏⵙⵉⴽⵔⵉⵜ (ⵥⵕ ⴰⵡⴷ ⴱⵉⴳⴰⵍⴰ).
English: By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals.
Tamasheq: ⴷⴷⴰⴳ ⵢⵓⵡⴹ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 130 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵉⴽⴽⴰⵜ ⴱⴰⵟⵍⵉⵎⵓⵙ ⵉⴹⵉⵚ ⵙ ⴱⵀⵉⴱⴰⵔⵅⵓⵙ ⴷ ⵍⴱⴰⴱⵉⵍⵢⵢⵉⵏ, ⵉⵙⵡⵓⵔⵉ ⵉ ⵜⵎⴰⵜⴰⵔⵜ 0 ( ⵢⴰⵜ ⵜⵡⵔⴻⵔⵔⴰⵢⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ ⵉⵍⴰⵏ ⵢⴰⵏ ⵉⴼⵉⵍⵓ ⴰⵎⴰⴼⵍⵍⴰ ⴰⵖⵣⵣⴰⴼ ), ⴰⴳⵏⵙⵓ ⵏ ⵓⵏⴳⵔⴰⵡ ⴰⵏⵇⴹ ⴰⵡⵙⵢⴰⵏ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵢⵓⵏⴰⵏⵉⵢⵏ ⵉⴳⵎⵎⴰⵢⵏ.
English: Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today.
Tamasheq: ⵉⵜⵜⵓⵎⵔⴰⵔⴰ ⵡⴰⵡⴰⵍ ⵖⴼ ⵓⵙⴰⵖⵓⵍ ⴷⵢⵓⴼⴰⵏⵜⵓⵙ ⵢⴰⴷ ⵉⵜⵜⵓⵙⵔⵙⵏ ⵙ ⵓⵙⵙⴼⵔⵓ ⵏ ⵓⵎⵓⵙⵏⴰⵡ ⵏⵜⵓⵙⵏⴰⴽⵜ ⴰⵀⵉⵏⴷⵉ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵜⴰ ⴳ ⴱⵔⴰⵀⵎⴰⵙⴼⵓⵟⴰⵙⵉⴷⴰⵏⵟⴰ ⴳ 628, ⵉⵙⵙⵎⵔⵙⵏ ⵓⵟⵟⵓⵏ ⵓⵣⴷⵉⵔⵏ ⵎⴰⵔ ⴰⴷ ⵉⵙⵏⴼⵍⴻⵍ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴽⴽⵓⵥⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ, ⵉⵙⵓⵍⵏ ⵙ ⴰⵙⵙⴰ ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙ.
English: At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.
Tamasheq: ⴳ ⵜⵉⵣⵉ ⵏⵏⴰⵖ ⵏⵏⵉⴽ, ⵍⵍⴰⵏ ⵉⵚⵉⵏⵉⵢⵏ ⵙⵏⵄⴰⵜⵏ ⵙ ⵓⵟⵟⵓⵏⴻⵏ ⵓⵣⴷⵉⵔⵏ ⵙ ⵜⴱⵔⵉⴷ ⵏ ⵡⵓⵏⵓⵖ ⵏ ⵉⵣⵔⵉⵔⵉⴳ ⴰⵡⵓⵎⴰⵏ ⵙⴳ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵢⴼⴼⴰⵙ ⵓⵔ ⵉⴳⵉⵏ ⴰⵎⵢⴰ, ⵙⴳ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵉⴹⴰⵏ ⵓⵎⵏⵉⴳ ⵉⵍⵍⴰⵏ ⵏⵉⵍ ⴰⵙ.
English: Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory.
Tamasheq: ⵙⴽⵔⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵉⵢⵓⵏⴰⵏⵉⵢⵏ ⴷ ⵉⵀⵉⵏⴷⵉⵢⵏⵉⴽⵍⴰⵙⵉⴽⵉⵢⵏ, ⵜⵉⵣⵔⴰⵡⵉⵏ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⵣⵓⵏⴷ ⵉⵎⵉⴽ ⴳ ⵜⵣⵔⴰⵡⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵎⴹⴰⵏ.
English: The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem.
Tamasheq: ⵉⵣⵍⵖ ⵓⵔⵎⵎⵓⵙ ⵏ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵔⴰⵡⵏ, ⴰⵣⵍⴰⵖ ⵉⵜⵣⵎⵎⴰⵎⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⴰⵎⵔⴰⵡ, ⵍⴰⵏ ⵙⵙⵉⵏ ⴰⴱⵓⵖⵍⵓ ⵉⵎⴰⵏⴻⵏ.
English: However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers.
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵉⴼⴼⵓⵍⵙ ⴼⵉⵜⴰⵖⵓⵔⵙ ⵜⵉⵍⵉⵜ ⵏ ⵉⵎⴹⴰⵏ, ⴷ ⵓⵔ ⵉⵣⴹⴰⵕ ⴰⴷ ⵉⵙⵢⴰⵀⴰ ⵏ ⵜⵉⵍⵉⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⵎⴳⵉⵏⴻⵏ.
English: By the 17th  century, mathematicians generally used decimal fractions with modern notation.
Tamasheq: ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 17, ⵙⵙⵎⵔⵙⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵙ ⵓⵎⴰⵜⴰ ⵉⵎⵜⵡⴰⵍⵏ ⵉⵎⵔⴰⵡⵏ ⵙ ⵓⵣⵎⵎⴻⵎ ⴰⵜⵔⴰⵔ.
English: In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine, Georg Cantor, and Richard Dedekind was brought about.
Tamasheq: ⴳ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1872 ⵜⵜⵓⴼⵙⴰⵔⵏⵜ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⴽⴰⵕⵍ ⵡⵢⵢⵉⵔⵙⵜⵔⴰⵙ ( ⵙⴳ ⵖⵓⵔ ⵓⵏⵍⵎⴰⴷ ⵏⵏⵙ ⵉⵢ ⴽⵓⵙⴰⴽ), ⴷ ⵉⴷⵡⴰⵕⴷ ⵀⴰⵢⵏⴰ ⴷ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ ⴷ ⵔⵉⵜⵛⴰⵔⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ.
English: Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties.
Tamasheq: ⴰⵔ ⵉⵙⴽⴰⵏ ⵡⵉⵢⵉⵙⵜⵔⴰⵙ ⴷ ⴽⴰⵏⵜⵓⵔ ⴷ ⵀⵉⵏ ⵜⵉⵎⴳⵓⵏⵉⵏ ⵏⵏⵙⵏ ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵜⴰⵔⵜⵎⵉ, ⵉⵙⴱⴷⴷ ⴷⵉⴷⴽⵉⵏ ⵜⴰⵡⵏⴳⵉⵎⵜ ⵏ ⵜⵓⴱⵓⵢⵜ (ⵙⵛⵏⵉⵜ) ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵉⴱⴹⴰ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ ⴽⵓⵍ ⵅⴼ ⵙⵏⴰⵜ ⵜⵔⵓⴱⴱⴰ ⵎⵉ ⵖⵓⵔ ⵍⵍⴰⵏ ⴽⴰⵏ ⵉⴼⵕⴹⵉⵚⵏ ⵉⵎⵥⵍⴰⵢⵏ.
English: Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations).
Tamasheq: ⵙⴳ ⵓⵢⴰ ⵉⵇⵏⴻⵏ ⵓⵅⵣⵣⵔ ⴳ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴰⵎⵓⵣⵣⵓⵜ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵍⵊⵉⴱⵔ ( ⵉⴼⵙⵙⴰⵢⵏ ⵎⴰⵕⵕⴰ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵎⵉ ⴳⴳⵓⴷⵉⵏ ⵉⵡⵜⵜⴰ).
English: Aristotle defined the traditional Western notion of mathematical infinity.
Tamasheq: ⵉⵙⵏⵎⵍ ⴰⵔⵉⵙⵟⵓ ⴰⵔⵎⵎⵓⵙ ⴰⵖⵔⴱⵉ ⴰⵣⴰⵢⴽⵓ ⵖⴼ ⵜⴰⵔⵜⵎⵉ ⵜⵓⵙⵏⴰⴽⵜ.
English: But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis.
Tamasheq: ⵎⴰⴽⴰ ⴰⵣⵣⵉⴳⵣ ⴰⴷ ⵎⵇⵇⵓⵕⵏ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵉⵙⴽⵔⵜ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ; ⴳ 1895 ⵉⴼⵙⵔ ⴰⴷⵍⵉⵙ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ ⵜⴰⵎⴰⵢⵏⵓⵜ ⵉ ⵜⵔⴰⴱⴱⵓⵜ, ⵉⵙⵙⵏⴽⴷ ⴰⴽⴷ ⵜⵖⴰⵡⵙⵉⵡⵉⵏ ⵏⵏⵉⴹⵏ, ⵉⵎⴹⴰⵏ ⵉⵣⵔⵔⵉⵏ ⵉⵡⵜⵜⴰ, ⴷ ⵉⵙⵙⴽⵔ ⵜⵓⵔⴷⴰ ⵜⴰⵙⵓⵍⵜ.
English: A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction.
Tamasheq: ⵜⵓⵏⵖⵉⵍⵜ ⵜⴰⵏⵣⴳⴰⵏⵜ ⵜⴰⵜⵔⴰⵔⵜ ⵙⴳ ⵡⴰⵔⵜⵎⵉ ⴷ ⵉⵜⵜⵓⴼⴽⴰⵏ ⵙⴳ ⵜⴰⵏⵣⴳⴰⵏⵜ ⵜⴰⵙⵜⵓⵜⵜⵉⵜ, ⵏⵏⴰⴷ ⵢⴰⴽⴽⴰⵏ ⵜⵉⵏⵇⵇⴰⴹ ⵢⴰⵜⵜⵓⵢⵏ ⵙⴳ ⵖⵓⵔ ⵡⴰⵔⵜⵎⵉ, ⵜⴰⵏⵇⵇⵉⴹⵜ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵏⵉⵍⴰ ⵜⴰⴷⵖⴰⵔⴰⵏⵜ.
English: The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De algebra tractatus.
Tamasheq: ⵜⴱⴰⵢⵏⴷ ⵜⵡⵏⴳⵉⵎⵜ ⵏ ⵓⵙⵎⴷⵢⴰ ⴰⵡⵏⵖⴰⵏ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵙⴳ 1685 ⴳ ⵓⴷⵍⵉⵙ “ⵡⴰⵍⵉⵙ” “ⵉⵙⴰⴽⴰⵜⵏ ⵏ ⵍⵊⵉⴱⵔ”.
English: In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.
Tamasheq: ⴳ 240 ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ, ⵉⵙⵙⵎⵔⵙ ⵉⵔⴰⵜⵓⵙⵜⴰⵏⵙ ⴰⵔⴽⴽⵓⵜ ⵏ ⵉⵔⴰⵜⵓⵙⵜⴰⵏⵙ ⵏ ⵓⵥⵍⴰⵢ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵙ ⵣⵣⵔⴰⴱⵉⵜ.
English: Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes.
Tamasheq: ⵜⵉⵢⴰⴼⵓⵜⵉⵏ ⵏⵏⵉⴹⵏ ⵏ ⵓⴱⵟⵟⵓ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷⵉⴽⵙ ⵜⵓⵣⴳⵉⵜ ⵏ ⵓⵍⵔ ⵉⵜⵜⵉⵏⵉⵏ ⵎⴰⵙⴷ ⴷⴰ ⵜⵜⵎⵄⵔⴰⵇⵏⵜ ⵜⵔⴱⵉⵄⵉⵏ ⵏ ⵜⵎⴳⴳⵉⵜⵉⵏ ⵉⵜⵜⵎⴼⴽⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷ ⵓⵙⵡⵉⵏⴳⵎ ⵏ ⴳⵓⵍⴷⴱⴰⵛ ⵏⵏⴰ ⵉⵜⵜⵉⵏⵉ ⵎⴰⵙⴷ ⴽⴰ ⵉⴳⴰⵜ ⵉⵎⵉⴹ ⴰⵅⴰⵜⴰⵔ ⵉⴳⴰⵏ ⵙⵉⵏ ⴳ ⵜⵍⵍⴰ ⵜⵓⴳⴷⵓⵜ ⵉⴳⴰ ⵜⴰⵎⵓⵏⵜ ⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ ⵙⴳ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ.
English: Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.)
Tamasheq: ⵙ ⵜⵣⴰⵢⴽⵓⵜ, ⵜⵙⵙⵏⵜⵉ ⵜⵙⵏⵙⵍⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏ ⵙ ⵡⵓⵟⵟⵓⵏ 1 (0 ⵓⵔ ⴷⵊⵓⵏ ⵉⴳⵉ ⵓⵟⵟⵓⵏ ⵏ ⵍⵢⵓⵏⴰⵏ ⵉⵇⴱⵓⵔⵏ).
English: In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.
Tamasheq: ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵜⵙⵉⵍⴰ ⴰⴷ 10, ⴷⴰ ⵉⵜⵜⵉⵍⵉ ⵉ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵜⵙⴳⴰ ⵏ ⵓⵢⴼⴼⴰⵙ ⵡⴰⵍⴰ ⵏ ⵡⵓⵟⵟⵓⵏ ⴰⵖⴰⵔⴰⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵉⴳⴰⵏ 1, ⴷ ⵉ ⵡⵓⵟⵟⵓⵏ ⵢⴰⴹⵏ ⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵎⵔⴰⵡ ⵏ ⵜⵉⴽⴽⴰⵍ ⵖⴼ ⵡⴰⵜⵉⴳ ⴰⴷⵖⴰⵔⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵉⵍⵍⴰⵏ ⴳ ⵓⵢⴼⴼⴰⵙ ⵏⵏⵙ.
English: Negative numbers are usually written with a negative sign (a minus sign).
Tamasheq: ⴷⴰ ⵡⴰⵍⴰ ⵜⵢⴰⵔⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵓⵣⴷⵉⵔⵏ ⵙ ⵜⵎⴰⵜⴰⵔⵜ ⵜⵓⵣⴷⵉⵔⵜ ( ⵜⴰⵎⴰⵜⴰⵔⵜ ⵜⴰⵏⴰⴽⵜⴰⵎⵜ).
English: Here the letter Z comes .
Tamasheq: ⴷⴰⴷⵖ ⴷⴰⴷ ⵉⵜⴷⴷⵓ ⵓⵙⴽⴽⵉⵍ ⵥ.
English: Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0.
Tamasheq: ⵉⵖⵢ ⴰⴷ ⴳⵉⵏ ⵉⵎⵜⵡⴰⵍⵏ; ⵏⵏⵉⴳ ⵏ, ⵏⵖⴷ ⴷⴷⴰⵡ ⵏ, ⵏⵖ ⴰⴽⵙⵓⵍⵏ ⴷ 1, ⴷ ⵉⵖⵢ ⴰⴷ ⴳⵉⵏ ⵓⵎⵏⵉⴳⵏ ⵏⵖⴷ ⵓⵣⴷⵉⵔⵏ ⵏⵖⴷ 0.
English: The following paragraph will focus primarily on positive real numbers.
Tamasheq: ⵜⵜⵡⴰⵏ ⵏ ⵜⵙⴷⴷⴰⵔⵜ ⴰⴷ ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵖⴼ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ ⵉⴳⵏ ⵓⵎⵏⵉⴳⵏ.
English: Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
Tamasheq: ⵉⵎⴽⵉ, ⵙ ⵓⵎⴷⵢⴰ, ⴰⵣⴳⵏ ⵉⴳⴰ 0.5, ⵜⵉⵙⵙ ⵙⵎⵎⵓⵙ ⵜⵡⴰⵍ ⵜⴳⴰ 0.2, ⵜⵉⵙⵙ ⵎⵔⴰⵡⵜ ⵜⵡⴰⵍ ⵜⴳⴰ 0.1, ⴷ ⵢⴰⵏ ⵅⴼ ⵙⵎⵎⵓⵙ ⵎⵔⴰⵡ ⵉⴳⴰ 0.02.
English: Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral.
Tamasheq: ⵓⵔ ⵉⴷ ⵖⴰⵙ ⵉⵎⴷⵢⴰⵜⵏ ⴰⴷ ⴰⵢⴷ ⵉⵜⵢⴰⵙⵙⵏ ⴷⴰⵢ, ⵎⴰⴽⴰ ⵎⴰⵕⵕⴰ ⵓⵟⵟⵓⵏ ⵏ ⵜⵉⴷⵜ ⵓⵔ ⵡⴰⵍⴰ ⴳⵉⵏ ⵓⵎⴳⵉⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵓⵔ ⵍⵉⵏ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵜⵜⵢⴰⵍⵙⵏ, ⴰⵖⴼ ⵓⵔ ⵉⵍⵍⵉ ⵡⵓⵟⵟⵓⵏ ⴰⵎⵔⴰⵡ ⴰⵎⴰⴽⵙⴰⵍ.
English: Since not even the second digit after the decimal place is preserved, the following digits are not significant.
Tamasheq: ⵎⴰⵛⴽⵓ ⵓⵔ ⵉⵍⵍⵉ ⵓⵃⵟⵟⵓ ⵏ ⵡⵉⵙⵙ ⵙⵉⵏ ⴹⴰⵕⵜ ⵜⵉⵙⴽⵔⵜ ⵜⴰⵎⵔⴰⵡⵜ, ⴰⵖⴼ ⵓⵔ ⵢⴰⴷ ⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⵡⵓⵟⵟⵓⵏ ⴷ ⵉⴽⴽⴰⵏ ⴷⴰⵜ.
English: For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1.
Tamasheq: ⵙ ⵓⵎⴷⵢⴰ, 0.999..., 1.0, 1.00, 1.000, ..., ⴽⵓⵍⵍⵓⵜⵏ ⵙⵎⴷⵢⴰⵏ ⵙ ⵡⵓⵟⵟⵓⵏ ⴰⵖⴰⵔⴰⵏ 1.
English: Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place.
Tamasheq: ⵜⵉⵢⵉⵔⴰ, ⵉⴳ ⵍⵍⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵎⴰⵕⵕⴰ ⴳ ⵡⵓⵟⵟⵓⵏ 0, ⴷⴰ ⵉⵜⴳⴳⴰ ⵡⵓⵟⵟⵓⵏ 0, ⴷ ⵎⴽ ⴳⴰⵏ ⵡⵓⵟⵟⵓⵏ ⴽⵓⵍ ⴳ ⵡⵓⵟⵟⵓⵏ ⵏ ⵜⴳⴼⴼⵓⵔⵜ ⵓⵔ ⵉⵜⴼⵓⴽⴽⵓⵏ ⴳ 9, ⵜⵖⵉⴷ ⴰⴷ ⵜⵙⵓⴳⵣⵣⴷ ⵜⵥⴰ ⴳ ⵓⵢⴼⴼⴰⵙ ⴳ ⵓⴷⵖⴰⵔ ⴰⵎⵔⴰⵡ, ⵜⵔⵏⵓⴷ ⵢⴰⵏ ⵉ ⵜⴳⴼⴼⵓⵔⵜ ⴳ ⵍⵍⴰⵏ 9 ⵏ ⵜⵉⵙⵏⴰⵜⵉⵏ ⴳ ⵓⵥⵍⵎⴰⴹ ⵏ ⵡⴰⵏⵙⴰ ⴰⵎⵔⴰⵡ.
English: Thus the real numbers are a subset of the complex numbers.
Tamasheq: ⴷ ⴳⵉⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ; ⵜⴰⵔⴱⵉⵄⵜ ⵜⴰⵢⵢⴰⵡⵜ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ,
English: The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a root in the complex numbers.
Tamasheq: ⵜⵙⵍⴽⴰⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⴷⵙⵍⴰⵏⵜ ⵏ ⵍⵊⵉⴱⵔ ⵖⴼ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵉⵜⵜⴳⴳⴰⵏ ⵉⴳⵔ ⵏ ⵍⵊⵉⴱⵔ ⵉⴳⵏ ⴰⵎⴰⵖⵓⵏ, ⴰⵖ ⵉⵜⵜⵉⵏⵉⵏ ⵉⵙⴷ ⴰⵎⴳⴳⵓⴷⵢ ⵏ ⵉⵡⵜⵜⴰ ⴰⴽⴷ ⵉⵔⵡⵉⵏ ⵖⵓⵔⵙ ⴰⵥⵓⵕ ⴳ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ.
English: The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered.
Tamasheq: ⵜⵜⵓⴳⴰ ⵜⵣⵔⴰⵡⵜ ⵖⴼ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⴳ ⵓⴼⵓⵖⴰⵍ ⴰⵎⵉⵔⵉⵡ ⵓⴳⴳⴰⵔ ⵏ 2000 ⵏ ⵓⵙⴳⴳⵯⴰⵙ, ⵢⵓⵡⵉⵏ ⵖⵔ ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⵇⵙⵉⵜⵏ, ⵜⵜⵓⴼⴽⴰ ⵜⵎⵔⴰⵔⵓⵜ ⵉ ⵉⵜⵙⵏ ⴷⴰⵢ.
English: Real numbers that are not rational numbers are called irrational numbers.
Tamasheq: ⴷⴰ ⵜⵙⵎⵎⴰⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵏⵏⴰ ⵓⵔ ⵉⴳⵉⵏ ⵓⵟⵟⵓⵏⴻⵏ ⵓⵎⴳⵉⵏⴻⵏ; ⵉⵎⴹⴰⵏ ⵓⵔ ⵉⴳⵉⵏ ⵓⵎⴳⵉⵏⴻⵏ.
English: The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers.
Tamasheq: ⵓⵟⵟⵓⵏ ⵉⵜⵜⵓⵙⵉⵟⵏ; ⵡⵔⵏ ⵖⴼ ⵜⵉⴳⴳⵉⵜⵉⵏ ⵏ ⵓⵙⵙⵉⵟⵏ ⵉⵜⵜⵓⵡⴰⵍⴼⵏ, ⴳ ⵉⵍⵍⴰ ⵓⵙⵙⵉⵟⵏ ⵏ ⵓⵥⵓⵕ ⵏ ⴱⵓ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ, ⴰⵢⴰ ⴰⵖⴼ ⵜⵙⴽⴰⵔ ⵉⴳⵔ ⴰⵎⴰⵇⵇⴰⵏ ⵏ ⵜⵉⴷⵜ ⴳ ⵍⵍⴰⵏ ⵡⵓⵟⵟⵓⵏ ⵏ ⵍⵊⵉⴱⵔ ⵏ ⵜⵉⴷⵜ.
English: One reason is that there is no algorithm for testing the equality of two computable numbers.
Tamasheq: ⵢⴰⵏ ⴳ ⵉⵎⵏⵜⵉⵍⵏ ⵉⴳⴰⵜ ⵓⵔ ⵜⵍⵍⵉ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵏ ⵢⵉⵔⵎ ⵏ ⵜⵏⴳⵉⴷⴷⵉⵜ ⵉⵏⴳⵔ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ ⵉⴳⵏ ⵡⵉⵏ ⵓⵙⵙⵉⵟⵏ.
English: The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.
Tamasheq: ⴷⴰ ⵉⵙⴽⵓⵜⵜⵓ ⵓⵏⴳⵔⴰⵡ ⵏ ⵡⵓⵟⵟⵓⵏ ⴷ ⵉⵜⵜⵓⴼⴽⴰⵏ ⵙ ⵓⵙⵉⵍⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵉ ⵡⵓⵟⵟⵓⵏ, ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵍⴳⴰⵎⵜ ⵉⵖⵉⵏ ⴰⴷ ⵢⵉⵍⵉ, ⵎⴰⴽⴰ ⵜⴰⵍⴳⴰⵎⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ, ⴷⴰ ⵜⴰⴽⴽⴰ ⵉⵎⵥⵍⴰⵢ ⵉⵖⵓⴷⴰⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.
English: The former gives the ordering of the set, while the latter gives its size.
Tamasheq: ⴷⴰ ⵢⴰⴽⴽⴰ ⵓⵎⵣⵡⴰⵔⵓ ⴰⵙⵙⵓⴷⵙ ⵏ ⵜⵔⴱⵉⵄⵜ, ⴰⵔ ⵢⴰⴽⴽⴰ ⵓⵎⴳⴳⴰⵔⵓ ⴰⴽⵙⴰⵢ ⵏⵏⵙ.
English: This standard basis makes the complex numbers a Cartesian plane, called the complex plane.
Tamasheq: ⴰⵙⵉⵍⴰ ⴰⴷ ⴰⵎⵙⵖⴰⵍ ⴷⴰ ⵉⵜⴳⴳⴰ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⴳ ⵓⵙⵡⵉⵔ ⴰⴷⵉⴽⴰⵔⵜⵉⵢ ⴷⴰ ⵉⵜⵙⵎⵎⴰ ⴰⵙⵡⵓⵔ ⵓⴷⴷⵉⵙ.
English: The complex numbers of absolute value one form the unit circle.
Tamasheq: ⴷⴰ ⵜⴳⴳⴰⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ ⵏ ⵡⴰⵜⵉⴳ ⴰⵎⴳⴳⴰⵔⵓ, ⵢⴰⵜ ⵜⵣⴳⵓⵏⵜ ⵉⴳⵏ ⵢⵓⵡⵜ.
English: In domain coloring the output dimensions are represented by color and brightness, respectively.
Tamasheq: ⴳ ⵢⵉⴳⵔ ⵏ ⵓⵙⵓⵖⵏ, ⴷⴰ ⵜⵜⵓⵙⵎⴷⵢⴰⵏ ⵡⵓⴳⴳⵓⴳⵏ ⵏ ⵓⵙⵙⵓⴼⵖ ⵙ ⵓⴽⵯⵍⵉ ⴷ ⵓⵣⵣⵏⵥⵕ ⴰⵎⵣⴷⴰⵢ.
English: Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.
Tamasheq: ⵢⵓⵡ ⵓⵙⵡⵓⵔⵉ ⵖⴼ ⵜⵎⵓⴽⵔⵉⵙⵉⵏ ⵏ ⵜⵎⴳⴳⵓⴷⵉⵏ ⵏ ⵉⵡⵜⵜⴰ ⵜⵉⵎⴰⵜⵜⵓⵜⵉⵏ ⴳ ⵜⵢⵉⵔⴰ ⵖⵔ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵙⵉⵍⴰⵏⵜ ⵏ ⵍⵊⵉⴱⵔ, ⵏⵏⴰ ⵉⵙⵙⴼⵔⵓⵏ ⴰⴽⴷ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ, ⵉⵍⵍⴰ ⵓⴼⵙⵙⴰⵢ ⵏ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵉⵍⴰ ⵉⵡⵜⵜⴰ ⵉⴳⴳⵓⴷⵉⵏ; ⵙⴳ ⵜⵙⴽⵯⴼⵍⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵏⵖⴷ ⵏⵏⵉⴳ ⴰⵙ.
English: Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.
Tamasheq: ⴱⴰⵢⵏⴷ ⵜⵙⵎⴽⵜⵉⵜⵉⵏ ⵏ ⵡⵉⵙⵍ ⴳ ⵜⵉⵡⵉⵏ ⵜⵉⴽⴰⴷⵉⵎⵉⵜⵉⵏ ⵣⵓⵏⴷ “ⴽⵓⴱⵏⵀⴰⴳⵏ”, ⵎⴰⴽⴰ ⵜⵣⵔⵉ ⵓⵔ ⵜⵜ ⵢⴰⵏⵏⴰⵢ ⴰⵡⴷ ⵢⴰⵏ ⴽⵉⴳⴰⵏ.
English: Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Tamasheq: ⵙⴳ ⵉⵎⴰⵔⵔⴰⵜⵏ ⵉⴽⵍⴰⵙⵉⴽⵏ ⴳ ⵜⵉⵣⵉ ⵢⴰⴹⵏ ⵖⴼ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵏ ⵔⵉⵜⵛⴰⵔⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ, ⵏⵖⴷ ⵓⵜⵓ ⵀⵉⵍⴷⵔ,ⴷ ⴼⵉⵍⵉⴽⵙ ⴽⵍⵉⵏ, ⴷ ⵀⵉⵏⵔⵉ ⴱⵡⴰⵏⴽⵉⵔ, ⴷ ⵀⵉⵔⵎⴰⵏ ⵛⵡⴰⵔⵜⵣ, ⴷ ⴽⴰⵔⵍ ⵡⵉⵔⵙⵜⵔⴰⵙ ⴷ ⵡⵉⵢⵢⴰⴹ.
English: The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).
Tamasheq: ⵓⵔ ⵉⵜⵜⵓⵙⵢⴰⵀⴰ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵎⴹⴰⵏ ⵉⵡⵏⴳⵉⵎⵏ ⴳ ⵓⵖⵓⴼⴰⵍ ⴰⵎⵉⵔⵉⵡ, ⴰⵔ ⵜⴰⵡⵓⵔⵉ ⵏ ⵍⵢⵓⵏⴰⵔⴷ ⵓⵍⵔ (1707-1783), ⴷ ⴽⴰⵕⵍ ⴼⵔⵉⴷⵔⵉⵜⵛ ⴳⴰⵡⵙ (1777-1855).
English: The integers form the smallest group and the smallest ring containing the natural numbers.
Tamasheq: ⴷⴰ ⵜⴳⴳⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ; ⵜⴰⵔⴱⵉⵄⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ, ⴷ ⵜⵅⵔⵙⵜ ⵜⴰⵎⵥⵥⴰⵏⵜ ⴳ ⵍⵍⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵖⴰⵔⴰⵏⴻⵏ.
English: It is the prototype of all objects of such algebraic structure.
Tamasheq: ⵉⴳⴰ ⴰⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ ⵏ ⵎⴰⵕⵕⴰ ⵉⵎⵖⵏⴰⵡⵏ, ⵣⵓⵏⴷ ⵜⵓⵙⴽⵉⵡⵜ ⵜⴰⵊⵉⴱⵔⵉⵜ.
English: Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
Tamasheq: ⴷⴰ ⵜⵜⵓⴱⴷⴰⵔⵏ ⵡⴰⵏⴰⵡⵏ ⵏ ⵉⵏⵎⵎⴰⵍⵏ ⵏ ⵓⵙⵏⵎⵉⵍⵉ ⵏ ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⵉⵍⴰⵏ ⵜⴰⵖⵣⵉ ⵉⵡⵔⵏ (ⵏⵖⴷ ⵜⵉⵔⴱⵉⵄⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ), ⵉⴳⴰⵏ ⵙ ⵜⵎⴰⵜⴰⵔⵜ “ int “ ⵏⵖⴷ “Integer” ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⵓⵜⵍⴰⵢⵉⵏ ⵏ ⵓⵙⵖⵉⵡⵙ ( ⵣⵓⵏⴷ Algol68, ⴷ C, ⴷ Java,ⴷ Delphi, ⴷ ⵜⵉⵢⵢⴰⴹ).
English: These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.
Tamasheq: ⵉⵎⵥⵍⴰⵢ ⴰⴷ ⵉⵖⵢ ⴰⴷ ⵜⵏⵜ ⵏⵙⵡⵔ ⵉ ⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⴷ ⵉⵏⴳⵔⴰⵡⵏ ⵉⵎⵉⴹⴰⵏ ⵉⵎⵙⵓⵔⵙⵏ, ⴷ ⵓⵔⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙ ⴰⵎ ⵉⵙⵉⵙⵙⵏ.
English: Since the triangle is isosceles, a = b).
Tamasheq: ⵎⴰⵢⴷ ⵉⴳⴰ ⵡⴰⵎⴽⵕⴰⴹ ⴰⵎⵙⴰⵙⴽⵍ ⵏ ⵉⴳⴰⵍⴰⵍⵏ, a = b).
English: Since c is even, dividing c by 2 yields an integer.
Tamasheq: ⵎⴰⵢⴷ ⵉⴳⴰ c ⴰⵎⵙⵉⵏ, ⴷⴰ ⴰⵖⴷ ⵜⴰⴽⴽⴰ ⵜⵓⴱⴹⵓⵜ ⵏ c ⵅⴼ 2; ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ.
English: Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.
Tamasheq: ⴰⵙⵏⴼⵍ ⵏ 4y2 ⵖⵔ c2, ⴳ ⵜⴳⴰⴷⴰⵣⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ (c2 = 2b2), ⴰⵖ ⵢⴰⴽⴽⴰⵏ 4y2= 2b2.
English: Since b2 is even, b must be even.
Tamasheq: ⵎⴰⵢⴷ ⵉⴳⴰ b2 ⴰⵎⵙⵉⵏ, ⵉⵇⵏⴻⵏ ⴰⴷ ⵉⴳ b ⴰⵎⵙⵉⵏ.
English: However this contradicts the assumption that they have no common factors.
Tamasheq: ⵎⴰⴽⴰ ⴰⵢⴰ ⵉⵜⵎⴳⵍⴰ ⴷ ⵓⵎⵔⴷⵓ ⵉⵜⵜⵉⵏⵉⵏ ⵓⵔ ⵍⵍⵉⵏ ⵉⵎⴳⴳⵉⵜⵏ ⵉⵎⵛⵛⵓⵔⵏ ⴳⵔⴰⵙⵏ.
English: Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ , ⵉⵍⵍⴰ ⵢⵉⵍⵖ ⵏ ⵀⵉⴱⴰⵙⵓⵙ ⵖⴼ ⵜⵣⵎⵎⴰⵔ ⵏⵏⵙ, ⴰⵢⴷ ⵉⵏⵏⴰ ⵢⴰⵏ ⵡⵓⵎⵉⵢ, ⵢⵓⴽⵣ, ⵉⴽⵉⵣ ⵏⵏⵙ ⴰⵢⵏⵏⴰ ⵉⵍⵍⴰ ⴳ ⵢⵉⵍ, ⴳⵔⵏⵜ ⵉⵎⴷⴷⵓⴽⴰⵍ ⵏⵏⵙ ⵉⴼⵉⵜⴰⵖⵓⵔⵙⵏ ⴳ ⵜⵉⵣⵉ ⵢⴰⴹⵏ,..., ⴰⵛⴽⵓ ⵙⵏⴼⵍⵏ ⵢⴰ ⵓⴼⵕⴹⵉⵚ ⴳ ⵉⵖⵣⵡⵔ ⵓⵔ ⵉⵔⵉⵏ... ⵜⴰⵖⴰⵍⵜ ⵉⵜⵜⵉⵏⵉⵏ ⵉⵖⵢ ⵓⵣⴳⵣⵍ ⵏ ⵜⵓⵎⴰⵏⵉⵏ ⴳ ⵉⵖⵣⵡⵔ ⵖⵔ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⴷ ⵓⵙⵖⴰⵍ ⵏⵏⵙⵏ.
English: For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on.
Tamasheq: ⵙ ⵓⵎⴷⵢⴰ, ⴳ ⵜⵡⵏⴳⵉⵎⵜ ⵏⵏⴽ ⵜⴰⴳⵣⵣⵓⵎⵜ ⵉⵏⵎⵏ: ⵉⵖ ⴰⴷ ⵜⵜⵓⴱⴹⵓ ⵖⵔ ⵙⵏⴰⵜ, ⴷ ⵓⵣⴳⵏ ⵏ ⵓⵣⴳⵏ ⵖⵔ ⵙⵉⵏ, ⴷ ⵓⵣⴳⵏ ⵏ ⵓⵣⴳⵏ ⵖⴼ ⴰⵣⵏⴳ, ⵉⵎⴽⵉ.
English: This is just what Zeno sought to prove.
Tamasheq: ⴰⵢⴰ ⴰⵢⴷ ⵉⵜⵏⴰⵖ ⵣⵉⵏⵓⵏ ⴰⴷ ⵜⵜ ⵉⵙⵡⵔ.
English: In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.
Tamasheq: ⴳ ⵜⵡⵏⴳⵉⵎⵉⵏ ⵏ ⵍⵉⵖⵔⵉⵇ, ⵓⴽⵓⵙ ⵏ ⵓⵎⴷⴷⴰⴷ ⵏ ⴽⴰⵏ ⵜⴰⵏⵏⴰⵢⵜ ⵓⵔ ⴷⴰ ⵉⵙⵡⴰⵔ ⴰⵎⴷⴷⴰⴷ ⵏ ⵜⴰⵏⵏⴰⵢⵜ ⵏⵏⵉⴹⵏ, ⵅⴼ ⵓⵢⴰ ⴰⵙⵏ ⵉⵇⵇⵏ ⵓⵣⵔⵔⵓ ⵓⴳⴳⴰⵔ.
English: A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously.
Tamasheq: ⴰⴽⵙⴰⵢ”...ⵓⵔ ⵉⴳⵉ ⵉⵎⵉⴹ, ⵎⴰⴽⴰ ⵉⵙⵎⴷⵢⴰ ⵙ ⵉⵏⵎⴰⵍⵍⴰⵜⵏ ⵣⵓⵏⴷ: ⵉⵎⵣⵔⴰⵢⵏ ⵏ ⵓⵣⵔⵉⵔⵉⴳ, ⴷ ⵜⵖⵎⵔⵉⵏ, ⴷ ⵜⵉⵊⵉⵎⵎⴰ, ⴷ ⵉⴽⵙⴰⵢⵏ, ⴷ ⵜⵉⵣⵉ ⵉⵖⵉⵏ ⴰⴷ ⵜⵎⵣⵉⵔⵉⵢ, ⵉⵎⴽ ⵏⵜⵜⵉⵏⵉ ⵙ ⵜⴰⵍⵖⴰ ⵉⵣⴷⵉⵏ.
English: Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios.
Tamasheq: ⴰⵛⴽⵓ ⵓⵔ ⵉⵜⵢⴰⴽⵣ ⵡⴰⵜⵉⴳ ⵏ ⵡⴰⵏⵛ ⵉⵍⵍⴰⵏ ⴳ ⵓⴽⵙⴰⵢ, ⵉⵖⵢ ⵉⴷⵓⴽⵙⵓⵙ ⴰⴷ ⵉⵙⵙⵉⵟⵏ ⴰⵙⵖⴰⵍⵏ ⵉⴳⴰⵏ ⵡⵉⵏ ⵓⵙⵇⵇⵓⵍ, ⴷ ⵜⵉⵏⵏⴰ ⵓⵔ ⵉⴳⵉⵏ ⵜⵉⵏ ⵓⵙⵇⵇⵓⵍ ⵙⴳ ⵓⵙⵜⴰⵢ ⵏ ⵓⵙⵖⵍ ⵙⴳ ⵓⴽⵙⴰⵢ, ⴷ ⵓⵙⵖⵍ ⵣⵓⵏⴷ ⴰⵙⵙⵉⴽⵙⵍ ⵏ ⵙⵉⵏ ⵉⵙⵖⴰⵍ.
English: This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.
Tamasheq: ⴷⴰ ⵉⵜⵜⵓⴳⴰ ⵉ ⵡⵉⵏⵏⴰ ⵓⵔ ⵉⵥⴹⴰⵕⵏ ⵉ ⵓⵙⵇⵇⵓⵍ ⴳ ⵉⴼⵕⴹⵉⵙⵏ ⵏ ⵉⵇⵍⵉⴷⵙ, ⴰⴷⵍⵉⵙ ⵎⵔⴰⵡ, ⴰⵙⵓⵎⵔ 9.
English: In fact, in many cases algebraic conceptions were reformulated into geometric terms.
Tamasheq: ⴳ ⵜⵉⵏⴰⵡⵜ; ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⴷⴷⴰⴷⵏ ⵉⵜⵜⵓⵢⴰⵍⵍⵙ ⵜⵉⵀⴽⵛⵜ ⵉ ⵉⵔⵎⵎⵓⵙⵏ ⵏ ⵍⵊⵉⴱⵔ ⵖⵔ ⵉⵔⵎⴰⵏ ⵉⵏⵣⴳⴰⵏⴻⵏ.
English: The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory.
Tamasheq: ⵉⵇⵏⴻⵏ ⵡⵓⵙⵓⵏ ⵏ ⵉⵜⵙⵏ ⵉⵔⵎⵎⵓⵙⵏ ⵉⴷⵙⵍⴰⵏ ⴰⴳⵏⵙⵓ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⴷⵖⵉ, ⵉⵎⵣⴰⵔⴰⵢ ⵏ ⵜⵉⵍⵉⵜ ⵏ ⵜⵉⴳⴳⵉ ⴰⵣⵔⵔⵓ ⴰⴽⴽⵯ ⵉⴳⵏ ⴰⵖⵣⵓⵔⴰⵏ ⵉ ⵜⵉⴳⴳⵉⵡⵉⵏ ⴷ ⵉⵙⵡⵉⵏⴳⵎⵏ ⵉⵍⵍⴰⵏ ⴹⴰⵕⵜ ⵜⵎⴰⴳⵓⵏⵜ ⴰⴷ.
English: However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".
Tamasheq: ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵢⴰⵔⴰ ⵓⵏⵎⵣⵔⵓⵢ ⴽⴰⵕⵍ ⴱⵏ ⵊⴰⵎⵉⵏ ⴱⵡⵉⵔ ⴰⵏ; “ⵣⵓⵏⴷ ⵜⴰⴼⴰⴽⵓⵍⵜ ⴰⴷ ⵓⵔ ⵉⵍⵉⵏ ⵉⵙⵡⵔⵏ ⵉⵖⵓⴷⴰⵏ, ⵓⵔ ⵜⵏⵏⵉ ⴰⴷ ⵜⴳ ⵜⴰⵎⴷⴷⴰⴷⵜ”.
English: Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.
Tamasheq: ⴽⴰⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵣⵓⵏⴷ ⴱⵔⴰⵀⴰⵎⴰⴳⵓⵜⴰ (ⴰⵙⴳⴳⵯⴰⵙ ⵏ 628 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⴷ ⴱⴰⵙⴽⴰⵔⴰ (ⴳ 629 ⴹⴰⵕⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ), ⵉⵎⵢⵉⵡⴰⵙⵏ ⴳ ⵢⵉⴳⵔ ⴰⴷ, ⵎⴽ ⵙⴽⵔⵏ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵢⴰⴹⵏ ⵉⴹⴼⴰⵕⵏ ⵎⴰⵢⴰⵏ.
English: The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind.
Tamasheq: ⴳ ⵓⵙⴳⴳⵯⴰⵙ 1872 ⵜⵢⴰⴼⵙⴰⵔⵏⵜ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⴽⴰⵕⵍ ⵡⵉⵢⵔⵙⵜⵔⴰⵙ ( ⵙⴳ ⵖⵓⵔ ⵓⵏⵍⵎⴰⴷ ⵏⵏⵙ ⵉⵔⵏⵙⵜ ⴽⵓⵙⴰⴽ) ⴷ ⵉⴷⵡⴰⵕⴷ ⵀⴰⵢⵏ ( ⵜⴰⵙⵖⵏⵜ ⵏ ⴽⵕⵉⵍ, 74), ⴷ ⵊⵓⵕⵊ ⴽⴰⵏⵜⵓⵔ (ⴰⵏⴰⵍⵉⵏ, 5), ⴷ ⵔⵉⵜⵛⴰⵕⴷ ⴷⵉⴷⵉⴽⵉⵏⴷ.
English: Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain characteristic properties.
Tamasheq: ⵉⵙⴽⴰ ⵡⵉⵢⵔⵙⵜⵔⴰⵙ ⴷ ⴽⴰⵏⵜⵓⵔ ⴷ ⵀⵉⵏ; ⵜⵉⵎⴰⴳⵓⵏⵉⵏ ⵏⵏⵙⵏ ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵜⴰⵔⵜⵎⵉ, ⵉⵙⴱⴷⴷ ⴷⵉⴷⴽⵉⵏ ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏⵏⵙ ⵖⴼ ⵜⵡⵏⴳⵉⵎⵜ ⵏ ⵡⵓⴱⵓⵢ (ⵙⵉⵛⵏⵉⵜ), ⴳ ⵓⵏⴳⵔⴰⵡ ⵏ ⵎⴰⵕⵕⴰ ⵓⵟⵟⵓⵏ ⵓⵎⴳⵉⵏⴻⵏ, ⴰⵔⵜⵏ ⵢⴰⵟⵟⵓ ⵖⴼ ⵙⵏⴰⵜ ⵜⵔⵓⴱⴱⴰ ⵉⵍⴰⵏ ⵉⵜⵙⵏ ⵉⵎⵥⵍⴰⵢⵏ ⵉⵙⵜⵉⵏ.
English: Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
Tamasheq: ⵉⵔⵏⴰ ⴰⵡⴷ ⴷⵉⵍⵉⵛⵍⵉⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ, ⵉⵎⴽ ⴳⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⵉⵏⵏⴰ ⵢⵓⵡⵙⵏ ⵏ ⵜⵙⵏⵙⵉⵜⵉⵏ ⵏ ⵓⵙⵏⵜⵍ.
English: This asserts that every integer has a unique factorization into primes.
Tamasheq: ⴰⵢⴰ ⵉⵙⵍⴽⴰⵏ ⵏ ⵡⵉⵙ ⴷ ⴽⵓ ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⵖⴰⵔⵙ ⵉⵎⴳⴳⵉ ⴰⵎⵥⵍⴰⵢ ⴳ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ.
English: To show this, suppose we divide integers n by m (where m is nonzero).
Tamasheq: ⵎⴰⵔ ⴰⴷ ⵏⵙⴱⴰⵢⵏ ⴰⵢⴰⴷ ⵢⵓⵔⴷⴰ ⵉⵙ ⴷⴰ ⵏⴰⵟⵟⵓ ⵉⵎⴹⴰⵏ ⴰⵎⴷⴷⴰⴷⵏ n ⵅⴼ ⵎ ( ⴰⵛⴽⵓ m ⵓⵔ ⵜⴳⵉ ⵜⴰⵎⵢⴰⵜ).
English: If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once.
Tamasheq: ⵉⴳ ⵓⵔ ⴰⴽⴽⵯ ⵉⵜⵜⵓⴳⴰ 0, ⵜⵖⵢ ⴰⴷ ⵜⵙⵡⵓⵔⵉ ⴰⵍⴳⵓⵔⵉⵜⵎ ⵖⴼ ⴽⵉⴳⴰⵏ ⵏ ⵜⵙⵓⵔⵉⴼⵉⵏ m - 1, ⴱⵍⴰ ⵏⵙⵙⵎⵔⵙ ⴰⵡⴷⵢⴰⵏ ⵓⵎⴰⴳⵓⵔ ⵡⴰⵀⵍⵉ ⵏ ⵜⵉⴽⴽⴰⵍ.
English: In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
Tamasheq: “ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵓⵟⵟⵓⵏⴻⵏ ⵉⵖⴰⵔⴰⵏⴻⵏ ⴳⴰⵏⵜⵏ ⵜⵉⵏⵏⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵓⵙⵙⵉⵟⵏ ( ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ : ⵍⵍⴰⵏⵜ ⵙⴹⵉⵚ ⵏ ⵉⴷⵔⵉⵎⵏ ⴰⴼⵍⵍⴰ ⵏ ⵜⴷⴰⴱⵓⵜ), ⴷ ⵓⵙⵙⵓⴷⵙ ( ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ : ⵡⴰⴷ ⴰⵖⵔⵎ ⵡⵉⵙⵙ ⴽⵕⴰⴹ ⴳ ⵜⵎⵓⵔⵜ).”
English: These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Tamasheq: ⴷⴰ ⵜⴳⴳⴰⵏⵜ ⵜⴳⴼⴼⵓⵔⵉⵏ ⵏ ⵉⵣⴷⴰⵡⵏ ⵏ ⵡⵓⵟⵟⵓ ⵉⵖⴰⵔⴰⵏ ⴳ ⵍⵍⴰⵏ ( ⵉⵥⵍⵉⵏ), ⴳ ⵉⵎⴰⴳⵔⴰⵡⵏ ⵏ ⵡⵓⵟⵟⵓⵏ ⵢⴰⴹⵏ.
English: The first major advance in abstraction was the use of numerals to represent numbers.
Tamasheq: ⴰⵣⵡⴰⵔ ⴰⵎⵣⵡⴰⵔⵓ ⵏ ⵜⴰⵏⵣⵖⵜ, ⵉⴳⴰⵜ ⵓⵙⵙⵎⵔⵙ ⴰⵎⴰⵟⵟⵓⵏ ⵉ ⵓⵙⵎⴷⵢⴰ ⵏ ⵡⵓⵟⵟⵓⵏ.