Algebra

This chapter covers the following topics:

• Quadratic equation
• Logarithms
• Complex numbers
• Progression and series

Quadratic equation

A quadratic equation is a second-degree polynomial equation written in the form:

$$a{x}^2+bx+c=0$$

where $a,b,c$ are constants, $a\ne 0$, and $x$ is the variable.

You can find the roots (solutions) of a quadratic equation using the quadratic formula:

$$x=(-b\pm \surd (b^2-4ac))/2a$$

Discriminant ($\Delta$)

The expression under the square root is called the discriminant:

$$\Delta =b^2-4ac$$

The discriminant tells you what kind of roots the equation has:

If $\Delta >0$: The equation has two distinct real roots.

If $\Delta =0$: The equation has one real root (a repeated root).

If $\Delta <0$: The equation has two complex conjugate roots.

Sum and product of roots

For a quadratic equation $a{x}^2+bx+c=0$, the sum and product of its roots are:

Sum of roots:

$$r_1+r_2=-b/a$$

Product of roots:

$$r_1\cdot r_2=c/a$$

Logarithms

The logarithm of $x$ to the base $b$ is defined by:

$$lo{g}_b(x)=c$$

where:

$$b^c=x$$

Special definitions for $b=e$ or $b=10$ are:

$$lnx,Base=e$$

$$logx,Base=10$$

To change from one base to another, use the change-of-base formula:

$$lo{g}_{b}x=(lo{g}_{a}x)/(lo{g}_{a}b)$$

e.g.,

$$lnx=\frac{lo{g}_{10}x}{lo{g}_{10}e}=2.302585(lo{g}_{10}x)$$

Identities

$$lo{g}_b{b}^n=n$$

$$log{x}^c=clogx;x^c=antilog(c\ast logx)$$

$$logxy=logx+logy$$

$$lo{g}_{b}b=1;log1=0$$

$$logx/y=logx-logy$$

Complex numbers

Definition of complex numbers

A complex number is written in the form:

$$z=a+bi$$

where $a$ is the real part ($Re(z)$), $b$ is the imaginary part ($Im(z)$), and $i$ is the imaginary unit satisfying $i^2=-1$.

Conjugate of a complex number

The conjugate of a complex number $z=a+bi$ is:

$$\bar{z}=a-bi$$

Modulus (magnitude) of a complex number

The modulus of $z=a+bi$ is:

$$|z|=\surd (a^2+b^2)$$

Argument (angle) of a complex number

The argument (or phase) of $z$ is the angle $\theta$ it makes with the positive real axis:

$$\theta =ta{n}^{-1}(b/a)$$

Operations with complex numbers

Addition and subtraction

For two complex numbers $z_1=a+bi$ and $z_2=c+di$:

$$z_1+z_2=(a+c)+(b+d)i$$

$$z_1-z_2=(a-c)+(b-d)i$$

Multiplication

$$z_1\cdot z_2=(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$

Division

$$z_1/z_2=(a+bi)(c-di)/(c^2+d^2)=(ac+bd)/(c^2+d^2)+(bc-ad)/(c^2+d^2)i$$

Polar form of a complex number

A complex number can also be represented in polar form:

$$z=r(cos\theta +isin\theta )$$

where $r=|z|=\surd (a^2+b^2)$ is the modulus and $\theta$ is the argument of $z$.

Euler’s formula and identity

Euler’s formula states:

$$e^{i\theta }=cos\theta +isin\theta$$

So the polar form can also be written as:

$$z=r{e}^{i\theta }$$

Euler’s identity

A special case of Euler’s formula occurs when $\theta =\pi$:

$$e^{i\pi }+1=0$$

This is known as Euler’s identity. It connects five fundamental numbers:

$e,i,\pi ,1,and,0$

De Moivre’s theorem

For any integer $n$, De Moivre’s Theorem states:

$$(cos\theta +isin\theta {)}^n=cos(n\theta )+isin(n\theta )$$

or equivalently in exponential form:

$$(e^{i\theta }{)}^n=e^{in\theta }$$

Progression and series

Arithmetic progression

To check whether a finite sequence is an arithmetic progression, subtract each term from the next term. If the differences are equal, the sequence is arithmetic.

1. The first term is $a$.
2. The common difference is $d$.
3. The number of terms is $n$.
4. The last or $n$th term is $l$.
5. The sum of $n$ terms is $S$.

$$l=a+(n-1)d$$

$$S=n(a+l)/2=n[2a+(n-1)d]/2$$

Geometric progression

To check whether a finite sequence is a geometric progression (G.P.), divide each term after the first by the preceding term. If the quotients are equal, the sequence is geometric.

1. The first term is $a$.
2. The common ratio is $r$.
3. The number of terms is $n$.
4. The last or nth term is $l$.
5. The sum of $n$ terms is $S$.

$$l=a{r}^{(n-1)}$$

$$S=a(1-r^n)/(1-r);r\ne 1$$

$$S=(a-rl)/(1-r);r\ne 1$$

$$S_n=a/(1-r);r<1$$

A G.P. converges if $|r|<1$ and it diverges if $|r|>1$.