Lily K.

asked • 07/24/21# PLSSS HELPPP. The following equation is often referred to Euler's formula. e^πi+1=0

**a.**

** Identify three important mathematical constants in the formula. Explain why each constant is important.**

**b.**

** Use what you know about complex numbers to show that this equation is true. In other words, show that **

**eπi+1=0**

## 1 Expert Answer

Raymond B. answered • 07/24/21

e^ix = cosx + isinx

e^ipi = cospi + isinpi

e^ipi = -1 + i(0)

e^ip = -1

eip +1 = 0

pi is a constant = about 3.14159

e is a constant = about 2.732

i is a constant = square root of -1, an imaginary number

geometry may help. plot imaginary numbers on the vertical axis, and real numbers on the horizontal axis draw a unit circle, with radius =1 and center at the origin. the point on the far left side of the circle is (-1,0)

polar coordinates are (1,pi) cospi= -1, sinpi=0 isinpi=0 cospi+ isinpi = -1

The usual proof of Euler's identity is with Taylor series expansions

e^x = 1 + x + x^2/2! + x^3/3! + ... = sum of an infinite number of terms

sinx = x - x^3/3! + x^5/5! - x^7/7! ....

cosx = 1 -x^2/2 + x^4/4! - x^6/6 + x^8/8!

substitute ix for x in each of the e^x Taylor expansion and multiply the sinx expansion by i

then e^ix = cosx + isinx

(a theological amusing historical note: legend has it Empress Catherine the Great of Russia invited Euler and an atheist French philosopher/mathematician to her court for amusement. She liked to have intellectuals debate for entertainment. Euler went first and did a proof of his identity, but he added another line, "therefore God exists" QED. the atheist had no response and left the court in disgrace. There is a pun involved, i is an exponent, a power, so an imaginary power has real existence. Score 2 points if you know the name of the disgraced French atheist. Sort of like Prof. Sheldon Cooper after he has no answer to Leslie Winkle's math.) At the physics bowl showdown as Sheldon is stumped, Leslie says Denis Diderot as team PMS beats AA in the tournament as Howard goes wild. A Russian is on Sheldon's team and has the answer, but Sheldon won't approve it and loses it in more ways than one)

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Mark M.

Do you know the meaning of constant? What do you know about complex numbers?07/24/21