[REFERENCES](#page-7-0)

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REFERENCES

• 1. Asimov, Isaac. Asimov on Numbers. Bell Publishing, New York, 1982.
• 2. Calinger, R., ed. Classics of Mathematics. Moore Publishing, Oak Park, IL, 1982.
• 3. Hogben, Lancelot. Mathematics in the Making. Doubleday, New York, 1960.

• 4. Cajori, Florian. A History of Mathematics, 4th ed. Chelsea, New York, 1985.

• 5. Encyclopaedia Britannica. Micropaedia IV, 15th ed., vol. 11, p. 1043. Chicago, 1982.
• 6. Singh, Jagjit. Great Ideas of Modern Mathematics. Dover, New York, 1959.
• 7. Dunham, William. Journey Through Genius. Wiley, New York, 1990.

PROBLEMS

• B.1-1 Given a complex number w = x + jy, the complex conjugate of w is defined in rectangular coordinates as w^∗ =x−jy. Use this fact to derive complex conjugation in polar form.
• B.1-2 Express the following numbers in polar form:
  • (a) w^a = 1+j
  • (b) ^{wb = 1}+ej
  • (c) w^c = −4+j3
  • (d) w^d = (1+j)(−4+j3)
  • (e) ^{we =} ejπ/^{4 +}2e−jπ/⁴

(f)

$$w_f = \frac{1+j}{2j}$$

(g)

$$w_g = (1+j)/(-4+j3)$$

(h)

$$w_h = \frac{1-j}{\sin(j)}$$

• B.1-3 Express the following numbers in Cartesian (rectangular) form:
  • (a) ^{wa = j}+ej
  • (b) ^{wb = 3}ejπ/⁴
  • (c) ^{wc = 1}/ej
  • (d) w^d = (1+j)(−4+j3)

(e)

$$w_e = e^{j\pi/4} + 2e^{-j\pi/4}$$

• (f) ^{wf =} ej ⁺¹
• (g) ^{wg = 1}/2*^j*
• (h) w^h = j j j (j raised to the j raised to the j)
• B.1-4 Showing all work and simplifying your answer, determine the real part of the following numbers:
  • (a) ^{wa = 1 j} (j−5e2−3*^j* )
  • (b) w^b = (1+j)ln(1+j)
• B.1-5 Showing all work and simplifying your answer, determine the imaginary part of the following numbers:
  • (a) ^wa = −jejπ/⁴

(b)

$$w_b = 1 - 2je^{2-4j}$$

• (c) w^c = tan(j)

• B.1-6 For complex constant w, prove:

  • (a) Re(w) = (w +w∗)/2
  • (b) Im(w) = (w −w∗)/2j

• B.1-7 Given w = x −jy, determine: (a) Re(ew)

  • (b) Im(ew)

• B.1-8 For arbitrary complex constants w¹ and w2, prove or disprove the following:

  • (a) Re(jw1) = −Im(w1)
  • (b) Im(jw1) = Re(w1)
  • (c) Re(w1)+Re(w2) = Re(w¹ +w2)
  • (d) Im(w1)+Im(w2) = Im(w¹ +w2)
  • (e) Re(w1)Re(w2) = Re(w1w2)
  • (f) Im(w1)/Im(w2) = Im(w1/w2)

• B.1-9 Given ^{w1 = 3}+j4 and ^{w2 = 2}ejπ/4.

  • (a) Express w¹ in standard polar form.
  • (b) Express w² in standard rectangular form.
  • (c) Determine |w1| ² and |w2| 2.