This document discusses complex numbers and sequences of real numbers. It begins by introducing complex numbers, representing them using binomial form as a real part plus an imaginary part multiplied by i. It describes opposite, conjugate, and graphical representation of complex numbers. It also covers basic operations and different forms of complex numbers. Next, it defines sequences as ordered lists with members indexed by integers. It provides examples and defines monotonic, bounded, and convergent/divergent sequences. Finally, it introduces the limit of a sequence and the mathematical constant e, defined as the limit of (1 + 1/n)n as n approaches infinity. It lists some common limits that can be determined using the value of e.

1. DU1: Complex numbers
and sequences
mathematics

2. 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.
푥2 + 푥 + 1 = 0 푥 =
−1 ± −3
2
So we have a bigger number’s group called the COMPLEX NUMBER’s group.

3. 1. COMPLEX NUMBERS.
We represent the imaginary unit as “i”. And the value of “i” is 푖 = −1
−9 = 9 ∙ −1 = 9 ∙i = 3i
−25 = 25 ∙ −1 = 25 ∙i = 5i
We represent a complex number as a combination of two numbers. It is called the
binomic form of the complex number.
푧 = 푎 + 푏푖 푤ℎ푒푟푒 푎, 푏 ∈ ℝ
푎 푖푠 푡ℎ푒 푟푒푎푙 푝푎푟푡 표푓 푡ℎ푒 푐표푚푝푙푒푥 푛푢푚푏푒푟
푏 푖푠 푡ℎ푒 푖푚푎푔푖푛푎푟푦 푝푎푟푡 표푓 푡ℎ푒 푐표푚푝푙푒푥 푛푢푚푏푒푟

4. 1. COMPLEX NUMBERS.
Opposite complex number:
푧 = 푎 + 푏푖 ; −푧 = −푎 − 푏푖 푧 = 2 − 3푖; −푧 = −2 + 3푖
푧 = −1 + 푖; −푧 = 1 − 푖
Conjugate complex number:
푧 = 푎 + 푏푖 ; 푧 = 푎 − 푏푖 푧 = 2 − 3푖; 푧 = 2 + 3푖
푧 = −1 + 푖; 푧 = −1 − 푖

5. 1. COMPLEX NUMBERS.
The opposite complex number; is the complex number that has the real and the
imaginary members with the signs changed.
푧 = 푎 + 푏푖 ; −푧 = −푎 − 푏푖 푧 = 2 − 3푖; −푧 = −2 + 3푖
푧 = −1 + 푖; −푧 = 1 − 푖
Conjugate of a complex number:
푧 = 푎 + 푏푖 ; 푧 = 푎 − 푏푖 푧 = 2 − 3푖; 푧 = 2 + 3푖
푧 = −1 + 푖; 푧 = −1 − 푖

6. 1. COMPLEX NUMBERS.
2. Graphic representation of the 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.
Ardatz erreala
Ardatz irudikaria
Z1= 3+2i
Z2= -4-i

7. 1. COMPLEX NUMBERS.
3. Operating with complex numbers:
3.1. The summing-up and rest;
푧1 = 푎 + 푏푖 ; 푧2 = 푐 + 푑푖 푧1 + 푧2 = (푎 + 푐) + (푏 + 푑)푖
푧1 − 푧2 = (푎 − 푐) + (푏 − 푑)푖
3.2. The multiplication;
3.2. The division (using the conjugate);
푧1 = 2 − 3푖; 푧2 = −1 + 4푖; 푧3 = 2푖

8. 1. COMPLEX NUMBERS.
4. Complex number’s forms:
4.1. The binomial form;
푧1 = 푎 + 푏푖
4.2. The polar form; 푧1 = 푟휑
Ardatz erreala
Ardatz irudikaria
Z= -4-i
φ
푟 = −4 2 + −1 2 = 17 = 4,12
푡푎푛훼 = 1
4 = 0,25 → 훼 = 14,04°
푆표, 휑 = 180° + 14,04° = 194,04°
푧1 = 4,12194,04°

9. 1. COMPLEX NUMBERS.
4. Complex number’s forms:
4.3. The trigonometric form;
푧1 = 푟 ∙ 푐표푠휑 + 푠푖푛휑 푖
In this case, take care that: 푧1 = 푟 ∙ 푐표푠휑 + 푠푖푛휑 푖 = r 푐표푠휑 + 푟 푠푖푛휑 푖 = a + bi
4.4. The affix form; 푧1 = 푎, 푏

10. 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

11. 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.

12. 2. REAL NUMBER’S SEQUENCES
Definitions:
A sequence 푎푛 푛∈ℕi s said to be monotonically increasing if each term is greater
than or equal to the one before it. For a sequence 푎푛 푛∈ℕ, this can be written as
푎푛 ≤ 푎푛+1, ∀푛 ∈ 푁.
If each consecutive term is strictly greater than (>) the previous term then the
sequence is called strictly monotonically increasing
A sequence 푎푛 푛∈ℕi s said to be monotonically decreasing if each term is less than
or equal to the previous one. For a sequence 푎푛 푛∈ℕ, this can be written as
푎푛 ≥ 푎푛+1, ∀푛 ∈ 푁.
If each consecutive term is strictly less than the previous
term then the sequence is called strictly monotonically
decreasin

13. 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:
푎푛 = 2푛 + 1, (3, 5, 7, 9 … ) is a monotonically increasing sequence.
푎푛 =
1
푛
, (1, 1
2 , 1
3 , 1
4 … ) is a monotonically decreasing sequence.

14. 2. REAL NUMBER’S SEQUENCES
Definitions:
If the sequence of real numbers 푎푛, 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 푎푛 ≤ 푀, ∀푀 ∈ 푁. Any such k is called an upper bound.
Likewise, if, for some real m, 푎푛 ≥ 푚, ∀푚 ∈ 푁, 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 푎푛 =
1
푛
is bounded from above, because all the
elements are less tan 1.

15. 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.
lim
푛→∞
푎푛 = 퐿
If a sequence converges to some limit, then it is convergent; otherwise it is
divergent.

16. 3. Limit of a SEQUENCE.
• If an gets arbitrarily large as n → ∞ we write
lim
푛→∞
푎푛 = ∞
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
lim
푛→∞
푎푛 = −∞
and say that the sequence diverges or converges to minus infinity.

17. 3. Limit of a SEQUENCE.
• Examples:
푎푛 = 푛3 lim
푛→∞
푎푛 = lim
푛→∞
푛3 = + ∞ Divergent
푎푛 = −푛2 + 1 lim
푛→∞
푎푛 = lim
푛→∞
−푛2 + 1 = − ∞ Divergent

18. 3. Limit of a SEQUENCE.
• Usual cases:
lim
푛→∞
푘 = 퐾 ∀푘 ∈ ℜ
lim
푛→∞
푘
푛푝 = 0
∀푘 ∈ ℜ
∀푝 ∈ ℕ
lim
푛→∞
푝(푛)
푞(푛)
=
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.

19. 3. Limit of a SEQUENCE.
• Usual cases:
lim
푛→∞
1
푛
= 0
lim
푛→∞
1
푛 + 푎
= 0
lim
푛→∞
1
푛 + 푛 + 푎
= 0
lim
푛→∞
푛 − 푛 + 푎 = If we have the rest of tow square root, we will
multipicate and divide with the conjugate.

20. 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 1 +
1
푛
푛
as n approaches
infinity.
푒 = lim
푛→∞
1 +
1
푛
푛
It is a convergent sequence, and it is bounded from above.

21. 4. The “e” number.
We can find some sequence’s limits knowing the e number;
lim
푛→∞
1 +
1
푛
푘푛
= 푒푘 ∀푘 ∈ ℤ
lim
푛→∞
1 +
1
푛 + 푘
푛
= 푒 ∀푘 ∈ ℝ
lim
푛→∞
1 +
1
푛
푘+푛
= 푒 ∀푘 ∈ ℤ