Question

in all, eulers legacy included the introduction of the concept of functions, as well as the proper way to write them within a mathematical formula by using the notation f(x). before his formalization of the notation, functional relationships were referred to by just one letter, f, or with greek letters omitting the parentheses, φx. in fact, euler spelled out much of the mathematical notation we use today, including the letter \e\ for the base of the natural logarithm (also known as euler’s number), the letter \i\ to denote the imaginary unit, and the greek letter \σ\ for summations. he also encouraged the use of the greek letter \π\ to signify the ratio of a circle’s circumference to its diameter. which mathematical concepts did euler’s legacy include? check all that apply. the notation for the imaginary unit the idea of a functional relationship the formalization of function notation the definition of a circle’s circumference the notation for the base of the natural logarithm

Explanation:

• For "the notation for the imaginary unit": The text says Euler used the letter "i" to denote the imaginary unit, so this applies.
• For "the idea of a functional relationship": The text states Euler introduced the concept of functions (functional relationships), so this applies.
• For "the formalization of function notation": The text mentions he formalized the notation \( f(x) \) for functions, so this applies.
• For "the definition of a circle’s circumference": Euler encouraged using \( \pi \) for the ratio of circumference to diameter, not defining the circumference, so this does not apply.
• For "the notation for the base of the natural logarithm": The text says Euler used the letter "e" for the base of natural logarithm, so this applies.

Answer:

• the notation for the imaginary unit
• the idea of a functional relationship
• the formalization of function notation
• the notation for the base of the natural logarithm