IB Mathematics SL/Algebra

=Topic 1 - Algebra=

Introduction
The aim of this section is to introduce candidates to some basic algebraic concepts and applications. Number systems are now in the presumed knowledge section.

Sequences and Series
A series is a sum of numbers. For example,

$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$$

A sequence is a list of numbers, usually separated by a comma. The order in which the numbers are listed is important, so for instance,

$$1, 2, 3, 4, 5, ...$$

Finite and Infinite Sequences
A more formal definition of a finite sequence with terms in a set S is a function from {1,2,...,n} to $$S$$ for some n ≥ 0.

An infinite sequence in S is a function from {1,2,...} (the set of natural numbers)

Arithmetic
Arithmetic series or sequences simply involve addition.

1, 2, 3, 4, 5, ...

Is an example of addition, where 1 is added each time to the prior term.

The formula for finding the nth term of an arithmetic sequence is:


 * $$\ u_n = u_1 + (n - 1)d.$$

Where $$ u_n $$ is the nth term, $$ u_1 $$ is the first term, d is the difference, and n is the number of terms

Sum of Infinite and Finite Arithmetic Series
An infinite arithmetic series is an infinite series whose terms are in an arithmetic progression. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·.

The sum (Sn) of a finite series is:

$$ S_n=\frac{n}{2} \cdot ( 2u_1+(n-1)d)=\frac{n}{2} \cdot ( u_1+u_n)$$.

Sum of Finite and Infinite Geometric Series
A geometric series is a series with a constant ratio between successive terms. Each successive term can be obtained by multiplying the previous term by 'r'

The nth term of a geometric sequence:



\begin{align} u_n = u_1\cdot r^{n - 1}&. \end{align} $$

$$ S_n=\frac{u_1( r^n-1)}{r-1}=\frac{u_1( 1-r^n)}{1-r}$$.

The sum of all terms (an infinite geometric sequence): If -1 < r < 1, then

$$S = \frac{u_1}{1-r}$$

Exponents
$$a^x = b $$ is the same as $$ log_a \cdot b =x $$

$$ \begin{align} a^x = e^{x \cdot\ln a}\, \end{align} $$

Laws of Exponents
The algebra section requires an understanding of exponents and manipulating numbers of exponents. An example of an exponential function is $$a^c$$ where a is being raised to the $$c^{th}$$ power. An exponent is evaluated by multiplying the lower number by itself the amount of times as the upper number. For example, $$2^3 = 2 \times 2 \times 2 = 8$$. If the exponent is fractional, this implies a root. For example, $$4^\frac{1}{2}=\sqrt{4} = 2$$. Following are laws of exponents that should be memorized:
 * $$a^ma^n=a^{m+n}$$
 * $$(ab)^m = a^mb^m$$
 * $$(a^m)^n = a^{mn}$$
 * $$a^{m/n} = \sqrt[n]{a^m}$$

Laws of Logarithms
$$ \log_b(xy) = \log_bx + \log_by \,\!$$

$$ \log_b(\frac{x}{y}) = \log_bx - \log_by $$

$$ \log_bx^y = y\log_bx \,\!$$

Change of Base formula:


 * $$ \log_b(a) = \frac{\log_c(a)}{\log_c(b)}. $$

Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:


 * $$ \log_2(16) = \frac{\log(16)}{\log(2)}. $$

Binomial Theorem
The Binomial Expansion Theorem is used to expand functions like $$(x+y)^n$$ without having to go through the tedious work it takes to expand it through normal means

$$(x+y)^n = _nC_0x^ny^0+_nC_1x^{n-1}y^1 + _nC_2x^{n-2}y^2 + ... +_nC_rx^{n-r}y^r+... + _nC_nx^0y^n\,\!$$

For this equation, essentially one would go through the exponents that would occur with the final product of the function ($$x^ny^0+x^{n-1}y^1+x^{n-2}y^2+...+x^0y^n$$). From this $$C_n$$ comes in as the coefficient, where $$C$$ equals the row number of the row from Pascal's Triangle, and $$n$$ is the specific number from that row.

Ex. $$7_5=35$$

Pascal's Triangle
1                     =Row 0 1  1                    =Row 1 1  2   1                  =Row 2 1  3   3   1                =Row 3 1  4   6   4   1              =Row 4 1  5  10  10   5   1            =Row 5 1  6  15  20  15   6   1          =Row 6 1  7  21  35  35  21   7   1        =Row 7 1  8  28  56  70  56  28   8   1      =Row 8 1  9   36 84 126 126  84  36   9   1    =Row 9