# 1.2 Audio files

**The following files accompany the exercise in Section 4.2**

Clicking on the link below opens an extract from Section 4.2 of the course (PDF, 1.7 MB) which accompanies the audio clips, also below. Listen to each of them in turn with the extracted pages open (you may like to print them out). Work on the problems at the appropriate places – you'll find the answers at the foot of this page.

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#### Transcript: Track 2

*z = 1*, and that has equation

*X = 1*On the right of the Frame is the straight line with equation

*X + 2y = 3*, which crosses the axes at the points shown

*x = 1*can be described as: the set of complex numbers

*z = x + iy*such that X is equal to 1

*z =X + iy*such that

*X + 2y*equals 3, or, equivalently, the set of all z such that the real part of z plus twice the imaginary part of z equals 3

#### Transcript: Track 3

*x = 1*, not including the line itself In curly bracket notation, this is: the set of z such that the real part of z is strictly greater than 1 On the right of Frame 2 is: the set of z such that the imaginary part of z is less than or equal to 2 This consists of all z which lie on or below the line

*y = 2*This time, the boundary line is included in the set

*z = 1*is excluded from the first set, but

*z = 2i*is included in the second set,

#### Transcript: Track 4

*X + 2y = 3*divides the plane into two halves - in the shaded half,

*x + 2y*is greater than 3, whereas in the other half, it's less than 3. For example, the point

*z = 0*doesn't lie in the shaded set because

*X + 2y*is less than 3 at this point. On the right of the Frame, the equation of the boundary is

*x - y = -1*, and here the point

*z = 0*does lie in the shaded set because

*X - y*is greater than -1 at that point.

*ax + by = c*, using either~a broken line or unkoken line depending on the inequality. Then you choose which side to shade by checking just one pojnt which is not on the line - for example, the point

*z = 0*if that's not on the lime

#### Transcript: Track 6

*x2 + y2 = 1*, which can be written in complex notation as lzl2 = 1: that is,

*lzl = 1*or for shore mod

*z = 1*. Thus the circle is: the set of z such that

*IzI= 1*

*lz - il = 'lz*.

*lz - il = 112*, is more concise than the

*(x,y)*-version,

*xz + (y - 1)2 = 114*

#### Transcript: Track 7

*z = 0*A disc is just the set of points inside a circle, and it may also include the circle itself. The disc on the left is the set ofz such that lzl is strictly

*< 1*. This consists of all points inside the circle centred at 0 and with radius 1, excluding the circle itself, which is the boundaxy of the disc. Since the boundary is excluded, this is called an open disc. The disc on the light of Frame 6 includes its boundary, the circle

*izl = 2*, so it's a closed disc.

#### Transcript: Track 8

*z = 0*. On the left is the set of points whpse distance to

*1 + i*is strictly less than 1. Since the distance from

*z*to

*1 + i*can be written as

*iz - 1 - il*, as in the cloud, thus disc is: the set of z such that

*lz - 1 - il*is strictly less than 1.

*z = 1*and

*z = i*are not in the set. The example on the right of the Frame is: the set of z such that

*iz - il*, the distance from

*z to i*, is less than or equal to

*1/2*

*z = 1/2i*and

*z = 3/2i*are both in the set.

#### Transcript: Track 10

*the set of z such that lzl is strictly > 1*

*z = 1 + i*. This time, the inequality is weak, and so the boundary, the circle fz - 1 - il = 1, is included. Notice that, in these figures, we've indicated

*z = 0 and z = 1 + i*to show that they're the centres of the boundary circles - they're not included in these sets

#### Transcript: Track 11

*z = i*. This time, both circles are included in the set, so that it's a closed annulus. Notice that, sometimes, one of the boundary circles may be included and the other excluded: in that case the annulus is neither open nor closed.

#### Transcript: Track 12

*0 strictly < Izl*" for the left-hand disc, and "

*0 strictly < lz + il*" for the one on the right.

#### Transcript: Track 14

*1 + i*. The culy bracket notation for the set on the right is the same as the curly bracket notation for the set on the left, except that z is replaced by

*z-(1+i)*.

#### Transcript: Track 15

*3x/4*. The boundary of this set is excluded and so we call this an open sector. Notice the empty circle at

*z = 0*to emphasize that this point is excluded.

*x/6*in modulus: that is, the principal argument of z lies between

*-x/6*and

*x/6*inclusive. At first sight, you might guess that this set is closed because it includes its boundary, but this isn't so. The boundary point

*z = 0*is excluded because Arg 0 is not defined, and so this sector is neither open nor closed.

#### Transcript: Track 16

*z = 0*to the appropriate point. For instance, the left-hand sector can be obtained from the set of z with the modulus of Arg z less than

*3x/4*by translating a distance of 1 to the right. For the definition of a general open sector, at the bottom of Frame 15, we need to specify the location of the vertex a of the sector, and the values a and b associated with the two rays which form the boundary of the sector.

#### Transcript: Track 18

#### Transcript: Track 19

*Re z > 1 for A, and lz - 11 <= 1*for B Thus A u B is the set of z such that either

*Re z > 1 or lz - 11 <= 1*, or both On the other hand, A n B is the set of z such that both the

*Re z > 1 and lz-lI<=1*. Finally, A-B is the set of z such that

*Re z >1*but

*lz - lI*is not less than or equal to 1. This condition can be written more concisely as

*Re z > 1*and

*lz - 11 > 1*, giving the form for A -B shown at the bottom of the Frame. Some examples to illustrate these operations are given in Frame 18 Each of the sets A and B is

#### Transcript: Track 20

*C - A*consists of all points

*z*which do not lie in

*A*. You've already seen examples of complements of discs in Frame 9, and two further examples are shown in this Frame On the left is the complex plane with the single point

*1 + i*deleted - this is the complement of the single point set

*[l + i)*. On the right is the complex plane with the negative real axis and zero deleted.

Click on the link below to open the answers to the audio section (PDF, 0.3 MB).