ECE4330 Lecture 2: Math Review (continued) Prof. Mohamad Hassoun Trigonometric Identities (continued): 2휋 2 sin 푥 cos 푥 = sin 2푥 cos(휔푡) → 휔 = 2휋푓, 휔 = (rad/sec) 푇 1 (sin 푥)2 + (cos 푥)2 = 1 푇 = , 푓 is the Hz frequency; 푇 is the period 푓 (cos 푥)2− (sin 푥)2 = cos 2푥 1 (cos 푥)2 = (1 + cos 2푥) Note: (cos 푥)2 ≡ cos2 푥 2 1 (sin 푥)2 = (1 − cos 2푥) 2 1 (cos 푥)3 = (3 cos 푥 + cos 3푥) 4 1 (sin 푥)3 = (3 sin 푥 − sin 3푥) 4 sin(푥 ± 푦) = sin 푥 cos 푦 ± cos 푥 sin 푦 휋 휋 휋 [Example: sin(푥 ± ) = sin(푥) cos ( ) ± cos(푥) sin ( ) = ±cos (푥)] 2 2 2 cos(푥 ± 푦) = cos 푥 cos 푦 ∓ sin 푥 sin 푦 [Example: cos(푥 ± 휋) = cos(푥) cos(휋) ∓ sin(푥) sin(휋) = −cos (푥) tan 푥 ± tan 푦 tan(푥 ± 푦) = 1 ∓ tan 푥 tan 푦 1 sin 푥 sin 푦 = [cos(푥 − 푦) − cos(푥 + 푦)] 2 1 cos 푥 cos 푦 = [cos(푥 − 푦) + cos(푥 + 푦)] 2 1 sin 푥 cos 푦 = [sin(푥 − 푦) + sin(푥 + 푦)] 2

Example. Find sin(15°) using trigonometric identities (no calculator) sin(15°) = sin(45° − 30°) θ cos θ sin θ = sin(45°) cos(30°) − cos(45°) sin(30°) 휋 1 2 3 2 1 6 − 2 (30°) √3 √ √ √ √ √ 6 = − = ≈ 0.259 2 2 2 2 2 2 4 휋 (45°) √2 √2 4 2 2

Example. Express cos4(푥) as the sum of simple sinusoids. 1 (cos 푥 )4 = cos 푥 (cos 푥)3 = cos 푥 [ (3 cos 푥 + cos 3푥)] 4 3 1 = (cos 푥)2 + cos 푥 cos 3푥 4 4 3 1 1 1 = ( ) (1 + cos 2푥) + ( ) [cos(−2푥) + cos(4푥)] 4 2 4 2 3 3 1 1 = + cos 2푥 + cos 2푥 + cos 4푥 8 8 8 8 3 1 1 = + cos 2푥 + cos 4푥 8 2 8

Note that the result includes a constant term, a sinusoid at 휔0 = 2 and a sinusoid at frequency 4. These are called harmonic (multiple) frequencies.

Formulas for exponentiated sinusoids as sum of pure sinusoids:  n even:

푛 −1 2 푛 1 푛 2 푛 (cos 푥) = (푛) + ∑ ( ) cos[(푛 − 2푘)푥] 2푛 2푛 푘 2 푘=0 푛 −1 2 푛 푛 1 푛 2 ( −푘) 푛 (sin 푥) = (푛) + ∑(−1) 2 ( ) cos[(푛 − 2푘)푥] 2푛 2푛 푘 2 푘=0  n odd:

푛−1 2 2 푛 (cos 푥)푛 = ∑ ( ) cos[(푛 − 2푘)푥] 2푛 푘 푘=0 푛−1 2 푛−1 2 ( −푘) 푛 (sin 푥)푛 = ∑(−1) 2 ( ) sin[(푛 − 2푘)푥] 2푛 푘 푘=0

With the binomial coefficient: 0! = 1 푛 푛! ( ) = 푓표푟 0 ≤ 푘 ≤ 푛 n! = (1)(2)(3)…(n) 푘 푘! (푛 − 푘)!

Example. Express cos4(푥) as a sum of pure sinusoids. 1 4! 2 4 4 (cos 푥)4 = ( ) + [( ) cos((4 − 0)푥) + ( ) cos((4 − 2)푥)] 24 2! (4 − 2)! 16 0 1 3 1 3 1 1 = + [cos 4푥 + 4 cos 2푥] = + cos 2푥 + cos 4푥 8 8 8 2 8

Your turn: Expand the following expressions as a sum of pure sinusoids, cos5 푥 = ? sin4 푥 = ? (a) Employing Trig identities and (b) Employing the formulas. Your turn: Use Mathcad to plot 푓(푥) = cos푛(푥) for 푛 = 20 and 200. Your turn (challenge): Solve for 훽 as a function of 푘 and 훼 in the equation, sin(훼) − sin(훽) + 푘푠푖푛(훼 + 훽) = 0.

−2푘+(1+푘2)cos (훼) Ans. 훽 = 푐표푠−1 { } 1−2푘푐표푠(훼)+푘2

Has the same frequency Example. Show that 푎 cos 푥 + 푏 sin 푥 = 퐶 cos(푥 + 휃 ) where −푏 퐶 = √푎2 + 푏2 푎푛푑 휃 = tan−1 푎 Employing the trig identity cos(푥 + 푦) = cos 푥 cos 푦 − sin 푥 sin 푦, we may write: C cos(푥 + 휃) = (퐶 cos 휃) cos 푥 + (− 퐶 sin 휃) sin 푥 Let 푎 = 퐶푐표푠(휃) and 푏 = −퐶푠푖푛(휃), then 푎2 + 푏2 = 퐶2(cos 휃)2 + (−퐶)2(sin 휃)2 = 퐶2[(cos 휃)2 + (sin 휃)2] = 퐶2

Therefore, 퐶 = √푎2 + 푏2 −푏 퐶 푠푖푛휃 −푏 and = = 푡푎푛 휃 → 휃 = 푡푎푛−1 푎 퐶 푐표푠휃 푎

Example. Express 푓(푡) = cos(푡) + sin(푡) as a single sinusoid. Solution: a = 1 , b = 1 퐶 = √12 + 12 = √2 −푏 −1 −휋 휃 = tan−1 = tan−1 ( ) = tan−1(−1) = 푎 1 4 휋 푓(푡) = √2 cos (푡 − ) 4

−푏 Caution when evaluating tan−1 ( ) 퐰퐢퐭퐡 풂 < 0 using a calculator: 푎 If you use your calculator to find the inverse and if 푎 < 0, then you must add (or subtract) 휋 to (from) the answer.

Example. Express −2 cos(푡) − 3 sin(푡) as a single sinusoid. Here,푎 = −2; 푏 = −3. Then, −푏 3 tan−1 ( ) = tan−1 ( ) = −0.983 radians (calculator answer) 푎 −2 Since 푎 < 0, we need to add (or subtract) 휋, 휃 = −0.983 ± 3.1416 = 2.159 (or − 4.1246)

퐶 = √(−2)2 + (−3)2 = √4 + 9 = √13 푓(푡) = √13 cos(푡 + 2.159); we choose the smaller angle (in absolute value)

Matlab and Mathcad have the atan2 function that automatically adjusts the angle. Here is the syntax. Mathcad syntax: atan2(denominator, numerator) Example: atan2(-2,3) = 2.159 Matlab syntax: atan2(numerator, denominator) Example: atan2(3,-2) = 2.159

Your turn: Express the following functions as a single cosine function. Verify your answer by plotting the functions. 휋 푓(푥) = cos(푥) − sin(푥) , 푓(푥) = − cos(푥) + sin(푥) , 푓(푥) = cos(푥) + sin (푥 + ) 3

Complex Numbers Representations: Cartesian, polar, phasor, and exponential z ∈ C, 푗 ≜ √−1 (or 푗2 = −1) Re (z) = a Im (z) = b

Cartesian representation: z = a + jb We may represent the complex number in polar form as Polar representation: z = (r, θ) where, r = |z| is the magnitude of z θ = ∠푧 is the angle of z Phasor notation: z = |z| ∠푧

Employing the Pythagorean Theorem we may write,

2 2 푟 = √푎 + 푏 a 푏 푏 tan 휃 = → 휃 = tan−1 ( ) 푎 푎 푎 = 푟 cos 휃

푏 = 푟 sin 휃 Therefore, we have the following relation between the Cartesian and polar representations: z = a + jb = r (cos 휃 + j sin 휃)

Euler’s Identity: 푒푗휃 = 푐표푠 휃 + 푗 푠푖푛 휃

Leonhard Euler(1707 – 1783) was a pioneering Swiss mathematician and physicist. (Euler’s Identity will be proved in the next lecture)

푗휋 1 − −휋 −휋 Example: = 푒 2 = cos + 푗 sin = 0 + 푗(−1) = −푗 푗휋 2 2 푒 2

Exponential form of complex numbers: z = a + jb = r (cos 휃 + j sin 휃) = r푒푗θ = |푧|푒푗∠푧 Example. Express z = 2 + j3 in exponential form. 푟 = √22 + 32 = √13 3 휃 = tan−1 ( ) = 0.983 푟푎푑푖푎푛푠 2 z = (√13 , 0.983) = √13푒푗 (0.983)

Polar form exponential form

or: ∠0.983 in phasor notation √13

Conjugate of a complex number: z = a + jb = r푒푗휃 푧∗ = 푎 − 푗푏 = 푟푒−j휃

Properties of the imaginary unit (푗): 2 푗2 = (√−1) = −1 , 푗3 = 푗푗2 = −푗, 푗4 = (−1)(−1) = +1 , 푒푡푐. 1 1 푗 푗 푗 Also, 푗−1 = = = = = −푗 or, 푗−1 = 푗−1(1) = 푗−1푗4 = 푗3 = −푗 푗 푗 푗 푗2 −1

Algebra of complex numbers z = a + jb z + 푧∗= (a + jb) + (a – jb) = 2a So, we may write: z + 푧∗ = 2 Re{푧}

푧푧∗ = (푎 + 푗푏) ∙ (푎 − 푗푏) = 푎2 − 푗푎푏 + 푗푎푏 − 푗2푏2 = 푎2 + 푏2 = |푧|2 = 푟2 Therefore, we may write: 푧푧∗ = |푧|2

Dividing by the conjugate: 푧 푟푒푗휃 = = 푒푗(휃−(−휃)) = 푒2푗휃 = 1∠2휃 푧∗ 푟푒−푗휃

1 Note that we used the fact that = 푒−푗휃 푒푗휃

Matlab: Cartesian to polar conversion (built-in functions) Let 푧 = −2 + 푗

Alternative Matlab method:

Mathcad: Cartesian to polar conversion:

Summary of Matlab/Mathcad/TI-89 Complex number instructions: Assume z = a + jb (j is the imaginary unit i)

Mathcad: (use i from the menu or define j as √−1 ) Re(z), Im(z), 푧̅, |z|, arg(z) [also: angle(a,b), atan2(a,b)] Note: Conjugate operation is obtained by holding down the Shift key and then pressing the ” key

Matlab: (i is built-in) real(z), imag(z), conj(z), abs(z), angle(z), atan2(b,a), [angle, r] = cart2pol(a,b), [a, b] = pol2cart(angle, r) (angle in radians)

TI-89 (syntax is similar to Matlab): real(z), imag(z), conj(z), abs(z), angle(z) (Here, the imaginary unit is i located above the “CATALOG” key)

Exponential Form Matlab:

Computer Example CB.1, Page 10 of Lathi’s Text. Part(a):

Part(b):

Reciprocal: Given (z = a + jb) 1 1 1 (푎 − 푗푏) 푎 − 푗푏 푧∗ = = = = 푧 푎 + 푗푏 (푎 + 푗푏)(푎 − 푗푏) 푎2 + 푏2 |푧|2 1 푧∗ ∴ = 푧 |푧|2

Reciprocal with the exponential representation: 1 1 1 푧 = 푟푒푗휃 → = = 푒−푗휃 푧 푟푒푗휃 푟

Division of two complex numbers:

푗휃1 푗휃2 푧1 = 푟1푒 ; 푧2 = 푟2푒

푗휃1 푧1 푟1푒 푟1 푗(휃1− 휃2) = 푗휃 = 푒 푧2 푟2푒 2 푟2

(recall, phasors: 푟1∠휃1 , 푟2∠휃2)

푟1∠휃1 푟1 = ∠(휃1 − 휃2) 푟2∠휃2 푟2

Squaring: 2 푧2 = (푟푒푗휃) = 푟2푒푗2휃

Eulers′ Formula → 푒푗휋 = cos(휋) + 푗푠푖푛(휋) = −1 + 푗0 = −1

푒푗휋 + 1 = 0

The most beautiful formula (theorem) in mathematics! In the fall 1988 issue of Mathematical Intelligencer, a scholarly journal of mathematics (published by Springer-Verlag), there was the call for a vote on the most beautiful theorem in mathematics. The readers, consisting of mostly mathematicians, voted Euler’s formula as the most beautiful. In fact, three out of the five most beautiful theorems of mathematics in that survey were contributed by Euler (they ranked first second and fifth).

Just for fun: 푗푗 = ?

푗휋 푗 = 푒 2 (used Euler’s Identity)

푗휋 푗 푗2휋 휋 − 1 푗푗 = (푒 2 ) = 푒 2 = 푒 2 = ≈ 0.2078796 (a !) √푒휋 There are other solutions to 푗푗. Can you find one?

Your turn: Find a complex valued solution for: 푙푛(−1) =? 1휋 =? 1푒 =? 1√2 =? sin−1(2)

Your turn: Prove de Moivre’s formula (1667-1754): 푛 (cos(푥) + 푗sin(푥)) = cos(푛푥) + 푗sin(푛푥) (After you try, you may find the solutions in the article "푂푛 휋, 푒, √−1" )

Mathcad Example: Examples B.3, p.12, Lathi’s text

Example B.4, p.13, Lathi’s text

Starting from Euler’s identity,

푒푗휃 = cos 휃 + 푗 sin 휃

Show that (your turn):

1 cos 휃 = [푒푗휃 + 푒−푗휃] 2 and

1 sin 휃 = [푒푗휃 − 푒−푗휃] 2푗

Also, show that (your turn):

푥 푥 푥 푥 푥+휋 푗 푗( ) −푗( ) 푗( ) 푥 푥 푗( ) 푒푗푥 − 1 = 푒 2 (푒 2 − 푒 2 ) = 2푗 푒 2 sin ( ) = 2 sin ( ) 푒 2 2 2 and 푥 푥 푥 푥 푗( ) 푗( ) −푗( ) 푥 푗( ) 푒푗푥 + 1 = 푒 2 (푒 2 + 푒 2 ) = 2 cos ( ) 푒 2 2

Complex Numbers in AC Steady-State Circuit Analysis

Consider the following linear RLC circuit with R = 1 Ohm, L = 2 H and C = 1/8 F. Solve for the steady-state loop current and capacitor voltage.

The source voltage is first expressed as the sum of a dc component and sinusoidal harmonics:

3 1 1 푣 (푡) = [cos(푡)]4 = + cos(2푡) + cos(4푡) 푠 8 2 8

Then, phasor analysis is used to solve three circuits for dc (휔 = 0), 휔 = 2 and 휔 = 4. The steady-state response is the sum of the three responses. The component inputs to the three circuits in phasor form are 푉푠1 = 3 1 1 (푤푖푡ℎ 휔 = 0), 푉 = ∠0 (푤푖푡ℎ 휔 = 2), and 푉 = ∠0 (푤푖푡ℎ 휔 = 4), 8 푠2 2 푠3 8 respectively. ac steady-state phasor circuit:

dc steady-state circuit with 휔 = 0 (inductor is a short circuit and the capacitor acts as an open circuit)

3 This leads by inspection to 푖 = 0 Amp and 푣 = Volt. 푑푐 푑푐 8

Therefore, the solution is 1 푖 (푡) = 0 + cos(2푡) + 0.0205 cos(4푡 − 1.41) 푠푠 2 Neglecting the third term (because of its negligible contribution), gives 1 푖 (푡) ≅ cos(2푡) 푠푠 2 The steady-state phasor voltage is: 3 푣 (푡) ≅ + 푣 (푡) 푆푆 8 1푠푠 For 휔 = 2, the capacitor phasor voltage is, 1 1 1 π π 푉 = (I ) = ∠ 0°(−푗4) = ( ∠ 0°) (4∠ − ) = 2 ∠ − 1 1 푗휔퐶 2 2 2 2

The corresponding time-domain capacitor voltage steady-state response is: 휋 푣 (푡) = 2 cos (2푡 − ) = 2 sin(2푡) 1푆푆 2 Therefore, 3 푣 (푡) = 푣 + 푣 ≅ + 2 sin(2푡) 푉표푙푡푠 푆푆 푑푐 1푆푆 8 So, the system is basically allowing the small frequencies (and dc) to go through, as is conceptually depicted below.

It can be shown that the complete response (including transients) for the capacitor voltage is:

Verification plot [complete (blue) and approximate steady-state (red) responses]:

Limits

1 is undefined 0 But, 1 lim = ∞ 푥→0 푥 Similarly, 1 lim = 0 푥→∞ 푥

We will next look at finding the imits of expressions that lead to:

0 ∞ 표푟 0 ∞ Example:

2푥 − 1 푓(푥) = 푥 − 2

1 ∞ 푓(0) = , 푓(∞) = 2 ∞

2푥−1 1 2 − 푥 푥 2 − 0 lim 푓(푥) = lim ( 푥−2 ) = lim 2 = = 2 푥→∞ 푥→∞ 푥→∞ 1 − 1 − 0 푥 푥

In the above, we divided the numerator and denominator of 푓(푥) by x. This is not a problem since x is not evaluated at zero.

L’Hospital Rule (see last slide for some historical perspectives) for finding 0 ∞ limits of indetermined expressions of the form or . The limit is equal to the 0 ∞ limit of the expression where the numerator is replaced by its first and same for the denominator.

For the previous example, applying L’Hospital’s Rule we arrive at

푑 (2푥 − 1) 2푥 − 1 푑푥 2 lim = lim 푑 = lim = 2 푥→∞ 푥 − 2 푥→∞ (푥 − 2) 푥→∞ 1 푑푥

In some limits, repeated application of L’Hospital’s rule might be necessary.

Additional applications of L’Hospital’s rule :

sin (푥) cos (푥) lim = lim = cos(0) = 1 푥→0 푥 푥→0 1

sin(푎푥) 푎 cos(푎푥) lim = lim = 푎 cos(0) = 푎 푥→0 푥 푥→0 1

1 √푥 ( ) 푥 √푥 lim = lim 2√푥 = lim = lim = ∞ 푥→∞ ln(푥) 푥→∞ 1 푥→∞ 2√푥 푥→∞ 2 푥

Note: √푥 푎푝푝푟표푐ℎ푒푠 푖푛푓푖푛푖푡푦 푎푡 푎 푚푢푐ℎ 푓푎푠푡푒푟 푟푎푡푒 푡ℎ푎푛 푙푛(푥)

푥2 2푥 2 2 lim 푥2푒−푥 = lim = lim = lim = = 0 푥→∞ 푥→∞ 푒푥 푥→∞ 푒푥 푥→∞ 푒푥 ∞

lim sin(푥) is undefined. However, we know that the value is finite and is 푥→∞ bonded between -1 and +1.

For fun! Three students were asked to determine the following limit:

Sin(푛) lim =? 푛→1 n

The first, second and third students gave the following answers, respectively, ≅ 0.8415, ≅ 0.0175 and ≅ 0.9461. Justify the answers (Hint: the third answer is valid if you allow your imagination to go wild).

Who Invented ?

Pierre de Fermat (1607–1665) was a French lawyer at the Parliament of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to calculus. The Italian-born French mathematician Joseph Lagrange (1736-1813) who had, along with Swiss mathematician and physicist (1707–1783), developed the modern approach to the calculus of variations wrote: “One may regard Fermat as the first inventor of the new calculus.” Also, the French scholar Pierre-Simon de Laplace (1749–1827) declared: “Fermat should be rgarded … as the true discoverer of differential calculus.” However, modern historians disagree and credit the German mathematician and philosepher Gottfried Wilhelm von Leibniz (1646–1716) and the English physicist and mathematician Sir Isaac Newton (1642 – 1726) for developing the breadth of calculus. But, Newton himself stated his dept to Fermat for the invention of the differential calculus: “I had the hint of this method from Fermat’s way of drawing and by applying it to abstract equations …” Leibniz was the first to publish his investigations; however, Newton had started his work several years prior to Leibniz. The derivative notation 푓̇ is due to 푑푦 Newton. Leibniz introduced the symbol ∫ for integration and for differentiation. 푑푥

The recognition for publishing the first calculus book, however, goes to the French mathematician Guillaume-Francois-Antoine de L’Hospital (1661-1704). L’Hospital was able to absorb the discoveries of others and then present them in a coherent manner for a wide audience. He hired the brightest students of Leibniz, Johann Bernoulli (1667-1748) to teach him calculus. By 1696, de L’Hospital had sufficient material on hand from Bernoulli to publish a book, Analyse des Infiniment Petits (Analysis of the Infinitely Small).

Your turn: As a way to refersh your calculus skills, show that (without using a calculator) 푒휋 > 휋푒. Hint: First, show that the maximum of the ln(푥) function ℎ(푥) = occurs at 푥 = 푒. 푥

(Wikipedia is the source for the following biographies)

Pierre de Fermat (1607–1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of ' Arithmetica.

Gottfried Wilhelm (von) Leibniz (1646–1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy, having developed differential and integral calculus independently of Isaac Newton. Leibniz's notation has been widely used ever since it was published. It was only in the 20th century that his and Transcendental Law of Homogeneity found mathematical implementation (by means of non-standard analysis). He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of virtually all digital computers.

Sir Isaac Newton (1642–1726/27) was an English mathematician, astronomer, and physicist who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Newton's Principia formulated the laws of motion and universal gravitation that dominated scientists' view of the physical universe for the next three centuries. Newton also built the first practical reflecting telescope and developed a sophisticated theory of color of the visible spectrum. Newton's work on light was collected in his highly influential book Opticks, first published in 1704. In addition to his work on calculus, as a mathematician Newton contributed to the study of power series, generalised the to non-integer exponents, developed a method for approximating the roots of a function, and classified most of the cubic plane curves.

Guillaume François Antoine, Marquis de l'Hôpital (1661–1704) was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞. Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his treatise on the infinitesimal calculus, entitled Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. This book was a first systematic exposition of differential calculus.

Johann Bernoulli (also known as Jean or John; 1667–1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth.

Leonhard Euler (1707–1783) was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics like infinitesimal calculus and graph theory while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for , such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field. He spent most of his adult life in St. Petersburg, Russia, and in Berlin, then the capital of Prussia.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."

Watch this video about Euler’s life and his contributions to math: https://youtu.be/h-DV26x6n_Q