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:

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

Geometric Sequences and Series

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:

$\begin{array}{cc}{u}_n=u_1\cdot r^{n-1}& .\end{array}$

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

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 $lo{g}_a\cdot b=x$

$\begin{array}{c}{a}^x=e^{x\cdot \ln a}\end{array}$

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:

• $a^m{a}^n=a^{m+n}$
• $(ab{)}^m=a^m{b}^m$
• $(a^m{)}^n=a^{mn}$
• $a^{m/n}=\sqrt[n]{a^m}$

Logarithms

Laws of Logarithms

${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$

${\log }_b(\frac{x}{y})={\log }_{b}x-{\log }_{b}y$

${\log }_b{x}^y=y{\log }_{b}x$

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

$(x+y{)}^n{=}_n{C}_0{x}^n{y}^0{+}_n{C}_1{x}^{n-1}{y}^1{+}_n{C}_2{x}^{n-2}{y}^2+...{+}_n{C}_r{x}^{n-r}{y}^r+...{+}_n{C}_n{x}^0{y}^n$

For this equation, essentially one would go through the exponents that would occur with the final product of the function ($x^n{y}^0+x^{n-1}{y}^1+x^{n-2}{y}^2+...+x^0{y}^n$). 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