Chapter 1 Algebra

1.1 Real numbers

1.1.1 Number line

1.1.2 Operator precedence

PEMDAS

Please Excuse My Dear Aunt Sally

• • Parentheses

• • Exponents

• • Multiplication/Division

• • Addition/Subtraction.

1.1.3 Parenthesis

\begin{align*} a \cdot (b + c) &= a\cdot b + a\cdot c\\ (a + b) \cdot c &= a\cdot c + b\cdot c\\ (a\cdot b)\cdot c &= a \cdot b \cdot c \end{align*}

1.1.4 Sign

\begin{align*} +\cdot + &= +\\ -\cdot - &= +\\ -\cdot + &= -\\ +\cdot - &= - \end{align*}

1.1.5 Parenthesis and sign

\begin{align*} -(a\cdot b) &= - a\cdot b\\ &\overset {\text {\normalsize or}}{=} a \cdot (-b)\\ +(\pm a \pm b) &= \pm a+ \pm b\\ -(\pm a \pm b) &= \mp a \mp b \end{align*}

1.1.6 Fractions

\begin{align*} \dfrac {a+b}{c} &= \dfrac {a}{c} + \dfrac {b}{c}\\ \dfrac {a}{b} + \dfrac {c}{d} &= \dfrac {ad + bc}{bd}\\ a \cdot \dfrac {1}{b} &= \dfrac {a}{b}\\ a \cdot \dfrac {b}{c} &= \dfrac {ab}{c}\\ \dfrac {a}{b} \cdot \dfrac {c}{d} &= \dfrac {ac}{bd}\\ \dfrac {ac}{bc} &= \dfrac {a}{b}\\ -\dfrac {a}{b} &= \dfrac {-a}{b} = \dfrac {a}{-b}\\ \left . \dfrac {a}{b} \middle / \dfrac {c}{d} \right . &= \dfrac {a}{b}\cdot \dfrac {d}{c} = \dfrac {ad}{bc} \end{align*}

1.2 Notable products

1.2.1 Notable products

NP1

\begin{align*} (a+b)^2 &= a^2 + 2ab + b^2\\ (a-b)^2 &= a^2 - 2ab + b^2 \end{align*}

NP2

\begin{align*} (a+b)(a-b) &= a^2 - b^2 \end{align*}

NP3

\begin{align*} (a+b)^3 &= a^3 + 3a^2b + 3ab^2 + b^3\\ (a-b)^3 &= a^3 - 3a^2b + 3ab^2 - b^3 \end{align*}

NP4

\begin{align*} (a-b)(a^2+ab+b^2) &= a^3-b^3\\ (a+b)(a^2-ab+b^2) &= a^3+b^3 \end{align*}

1.2.2 Newton’s binomial

\[(a+b)^n=\sum ^n_{k=0}{n \choose k}a^{n-k}b^k\]

\[\qquad \qquad \qquad \qquad \qquad \scriptsize \text {met }{n \choose k} = \frac {n!}{k!\cdot \left (n-k\right )!}\]

1.3 Polynomial

1.3.1 Zero polynomial

Form

\[0\]

Remark Degree of the zero polynomial is undefined

1.3.2 First degree polynomial

Form

\[ax+b\]

Root

\[x_0=\dfrac {-b}{a}\]

1.3.3 Second degree polynomial

Form

\[ax^2+bx+c\]

Roots

Calculate \(D = b^2-4ac\)

• • \(D>0 \Rightarrow \) two different roots \(\in \mathbb {R}\)

  \(\seteqnumber{0}{1.}{0}\)

  \begin{align*} x_1 &= \dfrac {-b-\sqrt {D}}{2a}\\ x_2 &= \dfrac {-b+\sqrt {D}}{2a}\\ \end{align*}

• • \(D=0 \Rightarrow \) two equal roots \(\in \mathbb {R}\)

  \[x_1=\dfrac {-b}{2a}\]

• • \(D<0 \Rightarrow \) no roots \(\in \mathbb {R}\)

Properties

• • Factors: Are \(x_1\) and \(x_2\) roots

  \[\Rightarrow ax^2+bx+c=a(x-x_1)(x-x_2)\]

1.3.4 Polynomials of higher degree

Form

\[p(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\]

• • \(x\) : unknown

• • \(a_ix^i\) : terms

• • \(a_i\) : coefficients

• • \(n\) : degree

• • \(a_0\) : constant term

Properties

• • The value of \(p\) in \(\alpha \in \mathbb {R}\) is

  \[p(\alpha )=a_n\alpha ^n + a_{n-1}\alpha ^{n-1} + \cdots + a_1\alpha + a_0\]

• • \(a\in \mathbb {R}\) is a root of \(p(x)\)

  \[\Leftrightarrow p(a)=0\]

  Synonyms: zero value of zero point

• • \(d(x)\) is a factor of \(p(x)\)

  \[\Leftrightarrow p(x)=d(x) \cdot q(x)\]

  with \(q(x)\) a polynomial

• • Link root and factors

  \[p(a) = 0 \Leftrightarrow p(x) = (x-a) \cdot q(x)\]

  with \(q(x)\) a polynomial

1.4 Exponents

1.4.1 Definition exponents

\begin{align*} a^n &= \underbrace {a \cdot a \cdot \ldots \cdot a}_{n \text { factors}}\\ a^1 &= a \\ a^0 &= 1 \end{align*}

1.4.2 Negative exponent

\[a^{-n} = \dfrac {1}{a^n}\]

1.4.3 Sign of exponent

\begin{align*} (\pm a)^{\overbrace {2k}^{\text {\normalsize even}}} &= + a^{2k} \\ (\pm a)^{\overbrace {2k+1}^{\text {\normalsize odd}}} &= \pm a^{2k+1} \end{align*}

1.4.4 Properties exponents

\begin{align*} a^m\cdot a^n &= a^{m+n} \\\dfrac {a^m}{a^n} &= a^{m-n} \\\left (a^m\right )^{^n} &= a^{m\cdot n} \\\left (ab\right )^m &= a^mb^m \\\left (\dfrac {a}{b}\right )^m &= \dfrac {a^m}{b^m} \end{align*}

1.4.5 Rationale exponent

\[a^{n/m} = \sqrt [m]{a^n}\]

1.5 Square roots

1.5.1 Definition square root

\(x\) is a square root of \(a\)

\[\Leftrightarrow x^2=a\]

1.5.2 Properties square root

\begin{align*} \sqrt {ab} &= \sqrt {a}\sqrt {b}\\ \sqrt {\dfrac {a}{b}} &= \dfrac {\sqrt {a}}{\sqrt {b}}\\ \sqrt {a^m} &= \sqrt {a}^{^m}\\ \sqrt {a^2} &= |a|\\ \sqrt {a} &= \sqrt [2]{a} \end{align*}

1.6 Roots

1.6.1 Definition roots

\(x\) is a \(n\)-th root of \(a\)

\[\Leftrightarrow x^n=a\]

1.6.2 Properties roots

\begin{align*} \sqrt [n]{ab} &= \sqrt [n]{a}\sqrt [n]{b}\\ \sqrt [n]{\dfrac {a}{b}} &= \dfrac {\sqrt [n]{a}}{\sqrt [n]{b}}\\ \sqrt [n]{a^m} &= \sqrt [n]{a}^{^m}\\ \sqrt [n]{\sqrt [m]{a}} &= \sqrt [m\cdot n]{a}\\ \sqrt [n \cdot p]{a^{m\cdot p}} &= \sqrt [n]{a^m}\\ \sqrt [n]{a^n} &= \begin{cases} |a| & (n \text { is even})\\ a & (n \text { is odd})\end {cases} \end{align*}

1.7 Logarithms

1.7.1 Definition logarithm

\[\log _a x = y \Leftrightarrow a^y=x\]

1.7.2 Notable logarithms

\[ \begin {array}{l|c} \text {Name} & \text {Notation}\\ \hline \text {Briggse logarithm} & \log x = \log _{10} x\\ \text {Neperian logarithm} & \ln x = \log _e x\\ \text {Binary logarithm} & \lg x= \log _2 x\\ \end {array} \]

1.7.3 Properties logarithms

Same base

\begin{align*} \log _a(x \cdot y) &= \log _a x + \log _a y\\ \log _a \dfrac {x}{y} &= \log _a x - \log _a y\\ \log _a \dfrac {1}{y} &= - \log _a y\\ \log _a(x^p) &= p \cdot \log _a x \end{align*}

Different base

\begin{align*} \log _b x &= \dfrac {\log _a x}{\log _a b}\\ \log _b x &= \dfrac {1}{\log _x b} \end{align*}

1.8 Absolute value

1.8.1 Absolute value

Definition

\[|a| = \begin {cases} a & (a \geq 0)\\ -a & (a \leq 0) \end {cases}\]

Corollary

\[|x| = \sqrt {x^2}\]

1.8.2 Properties absolute value

\begin{align*} |a| &\geq 0\\ |a| &= 0 \Leftrightarrow a = 0\\ |a \cdot b| &= |a| \cdot |b|\\ |a + b| &\leq |a| + |b|\\ \Big | | a | \Big | &= |a|\\ |-a| &= |a|\\ |a-b| &= 0 \Leftrightarrow a = b\\ |a-b| &\leq |a-c| + |c-b|\\ \left |\dfrac {a}{b}\right | &= \dfrac {|a|}{|b|} \quad (b\neq 0)\\ |a-b| &\ge \Big | |a| - |b| \Big |\\ |a| &\leq b \Leftrightarrow -b \leq a \leq b\\ |a| &\geq b \Leftrightarrow a \leq -b \text { of } a \geq b \end{align*}

1.9 Divisibility

1.9.1 Divisibility

Definition: \(d\) divides \(a\)

\begin{align*} &\Leftrightarrow d | a \\ &\Leftrightarrow d \in \mathbb {Z}_0 \land \exists q \in \mathbb {Z} : a = qd \end{align*}

• • \(d \in \mathbb {Z}_0\) : divisor of \(a\)

• • \(a \in \mathbb {Z}\) : multiple of \(d\)

• • \(\text {del}(a)=\{d|a : d \in \mathbb {Z}_0\}\)

Properties:

• • \(d | a \Rightarrow -d | a\)

• • \(d | a \Rightarrow d | -a\)

• • \(\forall a \in \mathbb {Z} : \pm 1 | a\)

• • \(\forall d \in \mathbb {Z}_0 : d | 0\)

• • \(\forall a \in \mathbb {Z} : 0 \nmid a\)

• • \(\forall d \in \mathbb {Z}_0 : d | d\)

Linear Combinations:

\[ \left . \begin {aligned} d &| a\\ d &| b \end {aligned} \right \} \Rightarrow d | xa + yb,\; \forall x,y \in \mathbb {Z} \]

1.9.2 Common multiples/divisors

Least Common Multiple: \(m\) is the smallest natural number that is a multiple of \(a\) and \(b\)

\begin{align*} &\Leftrightarrow m = \text {lcm(a,b)}\\ &\Leftrightarrow \begin{cases} m \in a\mathbb {Z} \cap b\mathbb {Z}\\ \forall c \in a\mathbb {Z} \cap b\mathbb {Z}: c \geq m \wedge c \neq 0 \end {cases} \end{align*}

Greatest Common Divisor: \(d\) is the greatest natural number that is a divisor of \(a\) and \(b\)

\begin{align*} &\Leftrightarrow d = \text {gcd(a,b)}\\ &\Leftrightarrow \begin{cases} d \in \text {del}(a) \cap \text {del}(b)\\ \forall c \in \text {del}(a) \cap \text {del}(b): c \leq d \end {cases} \end{align*}

Property:

\[\text {lcm}(a,b) \cdot \text {gcd}(a,b) = a \cdot b\]

1.9.3 Euclidean Division

\[\forall a \in \mathbb {Z}, b \in \mathbb {Z}_0 : \exists ! q \in \mathbb {Z}, \exists ! r \in \mathbb {Z} :\]

\[a = bq + r \qquad (0 \leq r \lt |b|)\]

• • \(q\) : quotient

• • \(r\) : remainder

1.9.4 Prime Numbers

Definition: A prime number \(p \in \mathbb {N}\setminus \{0,1\}\) has only \(1\) and itself as positive divisors

\begin{align*} &\Leftrightarrow p \text { is prime}\\ &\Leftrightarrow \text {div}(p) \cap \mathbb {N} = \{1,p\} \end{align*}

Fundamental Theorem of Arithmetic: Every \(n \in \mathbb {N}\setminus \{0,1\}\) can be factored into prime factors

\[n = p_1^{\alpha _1} \cdot p_2^{\alpha _2} \cdot ... \cdot p_k^{\alpha _k}\]

• • \(p_1, ..., p_k\) : distinct prime factors

• • \(\alpha _1, ..., \alpha _k \in \mathbb {N}_0\)

• • \(\#\text {div}(n) = (\alpha _1 + 1)(\alpha _2 + 1) ... (\alpha _k + 1)\)

1.10 Summations

1.10.1 Definition

\[\sum ^n_{i=0} a_i = a_0 + a_1 + \cdots + a_n\]

• • \(\Sigma \) : summation-symbol

• • \(i\) : summation-index

• • \(a_i\) : terms

1.10.2 Operations on summations

Group sums

\[\sum _{i=0}^n a_i + \sum _{i=0}^n b_i = \sum _{i=0}^n(a_i+b_i)\]

Group difference

\[\sum _{i=0}^n a_i - \sum _{i=0}^n b_i = \sum _{i=0}^n(a_i-b_i)\]

Split sum

\[\sum _{i=0}^n a_i = \sum _{i=0}^m a_i + \sum _{i=m+1}^n a_i \qquad (m \lt n)\]

Scalair product sum

\[\sum _{i=0}^n (k \cdot a_i) = k \cdot \sum _{i=0}^n a_i\]

Sum of constants

\[\sum _{i=0}^n k = (n+1)k\]

1.10.3 Summation identities

Sum of natural numbers

\[\sum _{i=1}^n i = \dfrac {n \cdot (n+1)}{2}\]

Sum of squares

\[\sum _{i=1}^n i^2 = \dfrac {n \cdot (n+1) \cdot (2n+1)}{6}\]

Geometric sum

\[\sum _{i=1}^n x^i = \dfrac {x - x^{n+1}}{1 - x} \qquad \text { met } x \neq 1\]

Telescopic sum

\[\sum _{i=1}^n (a_{i+1} - a_i) = a_{n+1} - a_1\]