Word: Algebraic Number
Definition: An algebraic number is a type of number that can be a solution (or "root") to an algebraic equation. An algebraic equation is a mathematical sentence that includes numbers and letters (variables) combined using addition, subtraction, multiplication, and division, and it has rational coefficients. Rational coefficients are numbers that can be expressed as fractions, where both the numerator (top number) and denominator (bottom number) are whole numbers.
Usage Instructions: - Use "algebraic number" when discussing mathematics, especially topics related to algebra and numbers. - When referring to numbers that can be solutions to equations, you can say, "The number is an algebraic number."
Example: 1. The number ( \sqrt{2} ) is an algebraic number because it is a solution to the equation ( x^2 - 2 = 0 ). 2. The number ( 3 ) is also an algebraic number because it is the solution to the equation ( x - 3 = 0 ).
Advanced Usage: In more advanced mathematics, algebraic numbers can be classified further. For example, they can be categorized into: - Algebraic integers: A special type of algebraic number that satisfies polynomial equations with integer coefficients. For example, ( \sqrt{2} ) is an algebraic integer. - Transcendental numbers: These are numbers that are not algebraic, meaning they cannot be a solution to any algebraic equation with rational coefficients. An example of a transcendental number is ( \pi ).
Word Variants: - Algebraic (adjective): Relating to algebra or algebraic numbers. For example, "She solved the algebraic equation." - Algebra (noun): The branch of mathematics dealing with symbols and rules for manipulating those symbols.
Different Meanings: - In a broader context, "algebra" refers to the field of mathematics that studies mathematical symbols and the rules for manipulating these symbols, while "algebraic number" specifically refers to numbers that meet the criteria mentioned above.
Synonyms: - There are not many direct synonyms for "algebraic number," but you can refer to it in a more general sense as "solution" or "root" when discussing equations.
Idioms and Phrasal Verbs: - There are no specific idioms or phrasal verbs that directly relate to "algebraic number," as it is a technical term used mainly in mathematics.