Welcome to Number and Algebra: The Foundation of AI Math!

Hello future Mathematicians! This chapter, Number and algebra, is where we build the essential toolkit for the rest of the Applications and Interpretation course. While it might sound theoretical, this section is incredibly practical—it is the language we use to model financial growth, population changes, and error analysis in the real world.

Don't worry if algebra hasn't been your favorite in the past. We focus heavily on using your technology (the GDC) efficiently, allowing you to concentrate on setting up the problem correctly and, most importantly, interpreting the results.

Section 1: Accuracy and Error in Applied Mathematics

In AI, we deal with real data, which is rarely perfectly clean. Understanding how to handle numbers accurately and quantify measurement mistakes is essential for credible modeling.

1.1 Working with Approximate Numbers

We often need to round numbers. The two main ways to express precision are:

  • Decimal Places (dp): This counts the digits after the decimal point. E.g., \(4.5678\) rounded to 2 dp is \(4.57\).
  • Significant Figures (sf): This relates to the reliability of a value, counting all non-zero digits, and any zeros that are not just placeholders.
    Example:
    • \(105.2\) has 4 sf.
    • \(0.0034\) has 2 sf (leading zeros don't count).
    • \(5000\) could be 1, 2, 3, or 4 sf, depending on context (assume 1 sf unless a decimal point is present, e.g., \(5000.\) is 4 sf).

Common Mistake to Avoid: When rounding multiple times in a calculation, only round the final answer. Keep intermediate values stored in your GDC!

1.2 Percentage Error

How big is our mistake? Percentage error tells us the relative size of the error compared to the true value, usually expressed as a percentage. This is a critical skill in AI for evaluating the success of models.

The Formula: \[ \text{Percentage Error} = \frac{| \text{Approximate Value} - \text{Exact Value} |}{\text{Exact Value}} \times 100\% \]

  • The vertical bars \(|\dots|\) mean we take the absolute value (the error is always treated as positive).
  • Real-World Example: If you estimate the height of a building (Approximate = 24m) but the actual height is 25m (Exact), the percentage error is:
    \( \frac{|24 - 25|}{25} \times 100\% = \frac{1}{25} \times 100\% = 4\% \)

Key Takeaway: Accuracy matters. Use the correct rounding methods (sf or dp), and remember the percentage error formula to quantify modeling success or failure.

Section 2: Sequences and Series – Modeling Growth

Sequences are ordered lists of numbers, and series are the sum of those numbers. We look at two main types: those that grow linearly (arithmetic) and those that grow exponentially (geometric).

2.1 Arithmetic Sequences and Series

What they are: Sequences where you add or subtract the same number (the common difference, \(d\)) to get the next term.
Analogy: This is like getting a consistent pay raise of $100 every month. Linear, steady growth.

Formulas (Found in your booklet!):

  • \(n^{th}\) term (\(u_n\)): \( u_n = u_1 + (n-1)d \)
    (Where \(u_1\) is the first term, \(n\) is the term number.)
  • Sum of \(n\) terms (\(S_n\)): \( S_n = \frac{n}{2} (2u_1 + (n-1)d) \)
    or \( S_n = \frac{n}{2} (u_1 + u_n) \)

2.2 Geometric Sequences and Series

What they are: Sequences where you multiply or divide by the same number (the common ratio, \(r\)) to get the next term.
Analogy: This is like compound interest—your growth is based on the current amount, leading to rapid (exponential) increase.

Formulas (Found in your booklet!):

  • \(n^{th}\) term (\(u_n\)): \( u_n = u_1 r^{n-1} \)
  • Sum of \(n\) terms (\(S_n\)): \( S_n = \frac{u_1 (r^n - 1)}{r - 1} \), where \(r \ne 1\)

2.3 Geometric Series: The Infinite Case (HL Only)

HL Students Note: If the common ratio \(r\) is between \(-1\) and \(1\) (i.e., \(|r| < 1\)), the terms eventually become so small that the sum approaches a finite limit. This is the sum to infinity, \(S_{\infty}\).

Sum to Infinity: \[ S_{\infty} = \frac{u_1}{1 - r} \quad \text{for } |r| < 1 \] Did you know? This concept is used in economics to calculate the total economic effect of government spending (the multiplier effect).

Key Takeaway: Arithmetic is linear, geometric is exponential. Geometric growth (multiplication) is crucial for understanding finance.

Section 3: Financial Mathematics – The Heart of AI

Financial applications are the most frequent real-world use of geometric sequences in AI. You must be comfortable with the terminology and using the GDC’s Finance Solver (TVM Solver).

3.1 Simple vs. Compound Interest

Simple Interest: Interest is calculated only on the original principal amount. Linear growth (Arithmetic).

Compound Interest: Interest is calculated on the principal plus any accrued interest. Exponential growth (Geometric). This is standard for almost all savings and loans.

Compound Interest Formula: \[ FV = PV \left( 1 + \frac{r}{100k} \right)^{nk} \] Where:

  • FV: Future Value (the final amount)
  • PV: Present Value (the initial principal)
  • r: Annual nominal interest rate (%)
  • n: Number of years
  • k: Number of compounding periods per year (e.g., k=4 for quarterly)

3.2 Using the GDC (TVM Solver)

The TVM solver is your best friend. It manages complex financial calculations, especially when dealing with annuities (regular payments).

Important TVM Variables:

  • N: Total number of payments/compounding periods (\(N = n \times k\)).
  • I%: Annual Interest Rate (as a percentage).
  • PV: Present Value (initial deposit or loan amount). Usually entered as negative if it is money leaving your wallet.
  • PMT: Payment amount (zero for standard compound interest).
  • FV: Future Value (target savings or balance remaining).
  • P/Y and C/Y: Payments per Year and Compounding Periods per Year. For AI, these are usually the same number! (e.g., monthly means 12).

Pro Tip: Sign Convention! Money leaving you (deposits, loan principal) is negative. Money you receive (future savings, loan payout) is positive. Consistency is key!

3.3 Depreciation (HL/SL)

Depreciation is the loss of value over time. Mathematically, it works exactly like compound interest, but the rate \(r\) is negative.

  • If a car depreciates by 15% annually, its value is multiplied by \( (1 - 0.15) = 0.85 \) each year.
3.4 Annuities and Amortization (HL Focus)

Annuities involve regular, fixed payments (PMT). Loans, mortgages, and regular retirement savings schemes are annuities.

HL Students: You must be able to use the TVM solver to calculate monthly payments (PMT) on a loan, determine the total interest paid, and potentially construct a simple amortization schedule (showing how much of each payment goes to principal vs. interest).

Key Takeaway: Financial math is all about geometric sequences. Master the TVM Solver and the sign conventions (positive/negative money flow).

Section 4: Solving Linear Equations and Systems (HL Extension)

Sometimes, our real-world problems involve several variables that must satisfy multiple conditions simultaneously—this leads to a system of linear equations.

4.1 Systems of Linear Equations (SL/HL)

For 2x2 or 3x3 systems, the easiest approach in AI is using your GDC to solve them:
Example: A bakery sells muffins (x) and croissants (y).
Equation 1 (Cost): \( 2x + 3y = 15 \)
Equation 2 (Quantity): \( x + y = 6 \)

  • GDC Method 1 (Graphical): Input equations as \(Y=\) and find the intersection point. (Best for 2x2).
  • GDC Method 2 (Solver): Use the built-in linear equation solver application (often found under 'Apps' or 'Solve'). (Best for 3x3).
4.2 Matrices and Systems of Equations (HL Only)

For larger systems, matrices provide an efficient way to organize and solve.

A system of equations can be written in the matrix form: \( AX = B \)
Where:

  • A: The coefficient matrix (the numbers in front of \(x, y, z\)).
  • X: The variable matrix (e.g., \( \begin{pmatrix} x \\ y \end{pmatrix} \)).
  • B: The constant matrix (the numbers on the right side of the equals sign).

To find the solution (the variable matrix \(X\)), we must use the inverse matrix, \(A^{-1}\): \[ X = A^{-1} B \]

Step-by-Step GDC Process (HL):

  1. Define matrix A (coefficients) and matrix B (constants) in your GDC’s matrix editor.
  2. Calculate the inverse of A: \( A^{-1} \).
  3. Multiply: \( A^{-1} \times B \). The resulting matrix is the solution \(X\).

Important Concept: A matrix system has a unique solution only if the determinant of matrix A is non-zero. If the determinant is zero, the matrix is singular, and there is either no solution or infinitely many solutions.

Key Takeaway: Use technology to solve linear systems. HL students must be proficient in using the inverse matrix method \(X = A^{-1} B\).

Quick Review: Essential Formulas & Concepts


  • Percentage Error: \( \frac{|\text{Approximate} - \text{Exact}|}{\text{Exact}} \times 100\% \)
  • Arithmetic Sequence: \( u_n = u_1 + (n-1)d \)
  • Geometric Sequence: \( u_n = u_1 r^{n-1} \)
  • Compound Interest/Depreciation: \( FV = PV \left( 1 + \frac{r}{100k} \right)^{nk} \)
  • HL Matrix Solution: \( X = A^{-1} B \)

Remember: In AI, the set-up and interpretation are just as important as the calculation. Always check if your answer makes sense in the real-world context! Good luck!