IB Mathematics SL/Algebra

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Topic 1 - Algebra[edit]

Introduction[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Sum of Finite and Infinite Geometric Series[edit]

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[edit]

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[edit]

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}

Logarithms[edit]

Laws of Logarithms[edit]

 \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[edit]

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 (x^ny^0+x^{n-1}y^1+x^{n-2}y^2+...+x^0y^n). 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. 7_5=35

Pascal's Triangle[edit]

                  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