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Topic 1: Number and Algebra

1.1 Numbers & Rounding

1. The Core Number Sets

Before we can do advanced algebra, we need to know the alphabet of mathematics. Here are the foundational sets of numbers:

$\mathbb{N} = \{0, 1, 2, 3, \dots\}$ Natural Numbers (Counting numbers + zero)
$\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ Integers (Whole numbers)
$\mathbb{Q} = \left\{\frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0\right\}$ Rational Numbers (Fractions)
$\mathbb{R}$ Real Numbers (Everything: Rational + Irrational like $\pi$ or $\sqrt{2}$)

Note: We use a superscript plus/minus to specify positive/negative (e.g., $\mathbb{Z}^{+}$ means positive integers only), and an asterisk to mean "non-zero" (e.g., $\mathbb{Z}^{*}$).

2. Rounding: d.p. vs s.f.

In the IB, precision matters. Let's look at how we round the number 123.4567

Decimal Places (d.p.)

Count from the decimal point.

1 d.p. $\rightarrow$ 123.5
2 d.p. $\rightarrow$ 123.46
3 d.p. $\rightarrow$ 123.457

Significant Figures (s.f.)

Count from the first non-zero digit.

5 s.f. $\rightarrow$ 123.46
3 s.f. $\rightarrow$ 123
2 s.f. $\rightarrow$ 120

⚠️ The Golden IB Exam Rule

Unless a question specifically asks otherwise, your final numerical answers in the IB exams must always be given exactly, or rounded to exactly 3 significant figures.

1.2 Sequences & Series

Before jumping into formulas, we need to establish the difference between a sequence and a series.

The Sequence ($u_n$)

A sequence is an ordered list of numbers. We use $u_n$ to describe the $n$-th term.

$u_1, u_2, u_3, u_4, \dots, u_n$

Example: $2, 4, 6, 8, 10$

The Series ($S_n$)

A series is the sum of the terms in a sequence. We use $S_n$ for the sum of the first $n$ terms.

$S_n = u_1 + u_2 + u_3 + \dots + u_n$

Example: $S_3 = 2 + 4 + 6 = \mathbf{12}$

Understanding Sigma ($\Sigma$) Notation

Writing out long series gets exhausting. We use the Greek letter Sigma ($\Sigma$) as a mathematical shorthand for "sum up everything."

$$\sum_{n=1}^{k} u_n$$

"Plug in every integer starting from $n=1$, all the way up to $k$, into the formula $u_n$, and add all the results together."

Example Expansions

$\sum_{n=1}^{3} 2^n = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = \mathbf{14}$

$\sum_{n=3}^{6} (2n+1) = 7 + 9 + 11 + 13 = \mathbf{40}$ (Notice you don't always start at 1!)

1.3 Arithmetic Sequences

If I give you a starting number ($u_1 = 5$) and ask you to constantly add a fixed value ($d = 3$) to find the next term, you generate an arithmetic sequence: $5, 8, 11, 14, 17 \dots$

In an arithmetic sequence, the difference between any two consecutive terms is constant.

The Arithmetic Formulas

Finding the $n$-th Term

$$u_n = u_1 + (n-1)d$$

Use this to find a specific term in the list.

Finding the Sum ($S_n$)

$$S_n = \frac{n}{2}(u_1 + u_n)$$

Use this if you know the first and last term.

$$S_n = \frac{n}{2}[2u_1 + (n-1)d]$$

Use this if you only know $u_1$ and $d$.

Working with Consecutive Terms

If you are told that three unknown terms (let's call them $a, x, b$) are consecutive terms in an arithmetic sequence, you can always find the middle term because the common difference is constant:

$x - a = b - x$ $\Rightarrow$ $2x = a + b$ $\Rightarrow$ $$x = \frac{a+b}{2}$$

The middle term is always the exact mean (average) of the two terms around it.

EXAMPLE: Consecutive Terms

Let $x+1$, $3x$, and $6x-5$ be consecutive terms of an Arithmetic Sequence. Find $x$.

Because the difference is constant, we know:
$(3x) - (x+1) = (6x-5) - (3x)$
$2x - 1 = 3x - 5$
$\mathbf{x = 4}$

(If you plug 4 back in, you'll see the sequence is 5, 12, 19... which works perfectly!)

Topic 1