Core Pure 1 Chapter 1: Complex Numbers

19 flashcards · Mathematics · by TheoTyson

1. What is the set of all natural numbers?

   N=0, 5\N = { 0, \, 5}N=0,5

2. What is the set of all integers?

   Z=−13, 0, 5\Z = {-13, \, 0, \, 5}Z=−13,0,5

3. What is the set of all rational numbers?

   Q=−13, 0, 5, 73\mathbb{Q} = {-13, \, 0, \, 5, \, \frac{7}{3}}Q=−13,0,5,37​

4. What is the set of all real numbers?

   R=−13, 0, 5, 73, 2, π, e\reals = {-13, \, 0, \, 5, \, \frac{7}{3}, \, \sqrt{2}, \, \pi, \, e}R=−13,0,5,37​,2 ​,π,e

5. What is the set of all complex numbers?

   C=−13, 0, 5, 73, 2, π, e, 1+2i\cnums = {-13, \, 0, \, 5, \, \frac{7}{3}, \, \sqrt{2}, \, \pi, \, e, \, 1+2i}C=−13,0,5,37​,2 ​,π,e,1+2i

6. How are imaginary numbers produced?

   Roots of negative numbers.

7. How are complex numbers produced?

   Sums of real and imaginary numbers.

8. What is the form of an imaginary number?

   bbb i, where b∈Rb\in\Realsb∈R

9. What is the form of a complex number?

   a+ba+ba+b i, where a,b∈Ra,b\in\Realsa,b∈R

10. What is i?

    −1\sqrt{-1}−1 ​

11. What is i2?

    -1

12. What is the complex number notation for z=a+biz = a +biz=a+bi ?

    Re(z)=aRe(z) = aRe(z)=a is the real part. Im(z)=bIm(z) = bIm(z)=b is the imaginary part.

13. What is the complex conjugate of z=a+biz=a+biz=a+bi ?

    z∗=a−biz^*=a-biz∗=a−bi

14. What is a complex conjugate pair?

    z=a+bi, z∗=a−biz=a+bi, \,z^*=a-biz=a+bi,z∗=a−bi

15. What kind of number is the result of the product of a complex conjugate pair?

    Real number.

16. What are the possible combinations of roots for a quadratic?

    Two real roots. Complex conjugate pair.

17. If the roots of an equation are α\alphaα and β\betaβ . The equation can be rewritten as...

    (z−α) (z−β)=0(z - \alpha) \, (z - \beta) = 0(z−α)(z−β)=0 z2−(α+β) z+αβ=0z^2 - (\alpha + \beta) \, z + \alpha \beta = 0z2−(α+β)z+αβ=0

18. What are the possible combinations of roots for a cubic?

    Three real roots. One real root, complex conjugate pair.

19. What are the possible combinations of roots for a quartic?

    Four real roots. Two real roots, complex conjugate pair. Two complex conjugate pairs.