DU1: Complex numbers and sequencesmathematics

1. COMPLEX NUMBERS.

Some second degree equations have no solution, or at least not in the real number’s group. These are the square roots of negative numbers.

𝑥2+𝑥+1=0 𝑥=−1±√−32

So we have a bigger number’s group called the COMPLEX NUMBER’s group.

1. COMPLEX NUMBERS.

We represent the imaginary unit as “i”. And the value of “i” is

=i = 3i

=i = 5i

We represent a complex number as a combination of two numbers. It is called the binomic form of the complex number.

𝑧=𝑎+𝑏𝑖 h𝑤 𝑒𝑟𝑒𝑎 ,𝑏∈ℝ

1. COMPLEX NUMBERS.

Opposite complex number:

𝑧=𝑎+𝑏𝑖 ;−𝑧=−𝑎−𝑏𝑖 𝑧=2−3 𝑖 ;−𝑧=−2+3 𝑖

𝑧=−1+𝑖 ;−𝑧=1−𝑖

Conjugate complex number:

𝑧=𝑎+𝑏𝑖 ; 𝑧=𝑎−𝑏𝑖 𝑧=2−3 𝑖 ; 𝑧=2+3 𝑖

𝑧=−1+𝑖 ; 𝑧=−1−𝑖

1. COMPLEX NUMBERS.

The opposite complex number; is the complex number that has the real and the imaginary members with the signs changed.

𝑧=𝑎+𝑏𝑖 ;−𝑧=−𝑎−𝑏𝑖 𝑧=2−3 𝑖 ;−𝑧=−2+3 𝑖

𝑧=−1+𝑖 ;−𝑧=1−𝑖

Conjugate of a complex number:

𝑧=𝑎+𝑏𝑖 ; 𝑧=𝑎−𝑏𝑖 𝑧=2−3 𝑖 ; 𝑧=2+3 𝑖

𝑧=−1+𝑖 ; 𝑧=−1−𝑖

1. COMPLEX NUMBERS.

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane.

The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical .These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.

2. Graphic representation of the complex numbers.

Ardatz erreala

Ardatz irudikaria

Z1= 3+2i

Z2= -4-i

1. COMPLEX NUMBERS.

3.1. The summing-up and rest;

3. Operating with complex numbers:

𝑧1=𝑎+𝑏𝑖 ; 𝑧 2=𝑐+𝑑 𝑖 𝑧1+𝑧 2=(𝑎+𝑐)+(𝑏+𝑑) 𝑖

𝑧1−𝑧 2=(𝑎−𝑐)+(𝑏−𝑑)𝑖

3.2. The multiplication;

3.2. The division (using the conjugate);

𝑧1=2−3 𝑖 ; 𝑧2=−1+4 𝑖 ; 𝑧3=2 𝑖

1. COMPLEX NUMBERS.

4.1. The binomial form;

4. Complex number’s forms:

𝑧1=𝑎+𝑏𝑖

4.2. The polar form; 𝑧1=𝑟𝜑

Ardatz erreala

Ardatz irudikaria

Z= -4-i

φ

|𝑟|=√ (−4 )2+(−1 )2=√17=4,12

𝑡𝑎𝑛𝛼=14=0,25→𝛼=14,04 °

𝑆𝑜 ,𝜑=180°+14,04 °=194,04 °

𝑧1=4,12194,04 °

1. COMPLEX NUMBERS.

4.3. The trigonometric form;

4. Complex number’s forms:

𝑧1=𝑟 ∙ (𝑐𝑜𝑠𝜑+𝑠𝑖𝑛𝜑 𝑖 )

In this case, take care that:

= r

4.4. The affix form; 𝑧1=(𝑎 ,𝑏 )

2. REAL NUMBER’S SEQUENCES

In mathematics, informally speaking, a sequence is an ordered list of objects (or events).

It contains members (also called elements, or terms); a1, a2, …, an.

The terms of a sequence are commonly denoted by a single variable, say an, where

the index n indicates the nth element of the sequence.

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whose elements are related to the index n (the element's position) in a simple way

2. REAL NUMBER’S SEQUENCES

Examples;

• an=1/n is the next sequence: 1, ½, 1/3, ¼, 1/5, 1/6, …)

• If we have a1=3, and an+1= an+2, we obtein the next sequence: 3, 5, 7, 9, … where the general term is an=2n+1

• A sequence can be constant if all the terms have the same value; for instance: (-3, -3, -3, …), so in this case an=-3. See that the general term hasn’t any variable n.

2. REAL NUMBER’S SEQUENCES

Definitions:

A sequence , this can be written as, .

If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing

A sequence , this can be written as, .

If each consecutive term is strictly less than the previousterm then the sequence is called strictly monotonically decreasin

2. REAL NUMBER’S SEQUENCES

Definitions:

If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function.

Examples:

is a sequence.

, is a sequence.

2. REAL NUMBER’S SEQUENCES

Definitions:

If the sequence of real numbers is such that all the terms, after a certain one, are less than some real number M, then the sequence is said to be bounded from above. In less words, this means . Any such k is called an upper bound.

Likewise, if, for some real m, , then the sequence is bounded from below and any such m is called a lower bound.

If a sequence is both bounded from above and bounded from below then the sequence is said to be bounded.

The sequence   is bounded from above, because all the elements are less tan 1.

3. Limit of a SEQUENCE.

One of the most important properties of a sequence is convergence.

Informally, a sequence converges if it has a limit.

Continuing informally, a (singly-infinite) sequence has a limit if it approaches some value L, called the limit, as n becomes very large.

If a sequence converges to some limit, then it is convergent; otherwise it is divergent.

3. Limit of a SEQUENCE.

• If an gets arbitrarily large as n → ∞ we write

In this case the sequence (an) diverges, or that it converges to infinity.

• If an becomes arbitrarily "small" negative numbers (large in magnitude) as n → ∞ we write

and say that the sequence diverges or converges to minus infinity.

3. Limit of a SEQUENCE.

• Examples:

Divergent

Divergent

3. Limit of a SEQUENCE.

• Usual cases:

Where p(n) and q(n) are polinomies, the limit is the limit of the division of the main grade of both polinomies.

If p(n)’s grade is greater, then the limit is infinity.

If 1(n)’s grade is greater, then the limit is 0.

If both have the same grade, then the limit is the division of de coeficient of both polinomies.

3. Limit of a SEQUENCE.

• Usual cases:

If we have the rest of tow square root, we will multipicate and divide with the conjugate.

4. The “e” number.

The number e is an important mathematical constant that is the base of the natural logarithm.

It is approximately equal to 2.718281828, and is the limit of as n approaches infinity. 

It is a convergent sequence, and it is bounded from above.

4. The “e” number.

We can find some sequence’s limits knowing the e number;