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 + 1/2 + 1/4 + 1/8 + 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, u₁ is the first term, d is the difference, and n is the number of terms

Sum of Infinite and Finite Arithmetic Sequences

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 sequence 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

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 $log_a \cdot b$

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

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

Logarithms

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

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

For this equation, essentially one would go through the exponents that would occur with the final product of the function (xⁿy⁰ + x^{n − 1}y¹ + x^{n − 2}y² + ... + x⁰yⁿ). From this C_n comes in as the coefficent, where C equals the row number of the row from Pascal's Triangle, and n is the specific number from that row.

Ex. 7₅ = 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