Notes, summaries, assignments, exams, and problems for Mathematics

1. Simplifying Algebraic Expressions

How to Simplify Algebraic Expressions

• Multiply numbers and letters together.
• Combine like terms (terms with the same letters and powers).

Example:
−2ac × 4bd = −8abcd

Example:
5ab − 8b²a + ba = 5ab − 8ab² + ab = −8ab² + 6ab

2. Expanding Brackets (Distributive Law)

How to Expand Brackets

• Multiply everything inside the bracket by what is outside.

Example:
−4(x + 7) = −4x − 28

Example:
5(x − 3) + 2(6 − x) = 5x − 15 + 12 − 2x = 3x − 3

3. Solving Equations

How to Solve Equations

• Get all variables (e.g., x's) on one side and numbers on the other.
• Perform the same operation on both sides of the equation.

Example:
3x + 2 = 17
3x = 15
x = 5

Example:
9x − 8 = 4x + 7
5x = 15
x = 3

4. Solving Inequalities

How

Understanding Inheritance in Python

Inheritance allows a class to use and extend the properties and methods of another class. It promotes code reusability and reduces duplication. Think of it as “child learns from parent.” 👨‍👦‍💻

Code Duplication: Program Without Inheritance

When classes share common attributes or methods (such as first_name, last_name, and get_age), implementing them separately leads to redundant code, as shown below:

import datetime

class TennisPlayer:
    def __init__(self, fname, lname, birth_year):
        self.first_name = fname
        self.last_name = lname
        self.birth_year = birth_year
        self.aces = []

    def get_age(self):
        now = datetime.datetime.now()
        return now.year -

System Types
The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate or compatible system when it admits an infinite set of solutions.
Fitting and classification:

Calculating the rank of a matrix for determining

1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
• A line is proportional to another.
• A line is a linear combination of others.
Delete the third column because it is a linear... Continue reading "Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem" »

Part I: Complete the Following Table

Part II: Complete the Following Sentences

1. She is from the United States. She is American.
2. She is from Nigeria. She is Nigerian.
3. She is from Germany. She is German.
4. They are from Egypt. They are Egyptian.
5. We are from Canada. We are Canadian.

Part III: Select the Correct Sentence

1. Select the correct sentence.
   • a) She lives with Carlos, and she works on Saturdays.
2. Select the correct sentence.
   • c) They are running in the park.
3. Select the correct sentence.
   • b) She is 20 years old.
4. Select the correct sentence.
   • a) We always

1. Operations with Integers

Addition Rules

• Same signs: Add the numbers and keep the original sign.
  Example: -4 + (-6) = -10
• Different signs: Subtract the smaller number from the larger number and keep the sign of the larger number.
  Example: 7 + (-3) = 4

Subtraction Rules

To subtract, turn the operation into addition by changing the sign of the second number:

a - b = a + (-b)

Step-by-Step Example:
Solve: 5 - 8
Step 1: Rewrite as addition → 5 + (-8)
Step 2: Different signs, so subtract → 8 - 5 = 3
Step 3: Keep the sign of the larger number → -3

Multiplication and Division Sign Rules

Step-by-Step Example:
Solve: -3 × -4
Step 1: Same signs result in a positive product.
Step 2: 3 ×... Continue reading "Mastering Integers, Indices, and Scientific Notation" »

Understanding Statistical Concepts

Rank Correlation

Rank correlation measures the degree of association between the ranks of two variables instead of their actual values. It is used when data is ordinal or not normally distributed. Spearman’s rank correlation coefficient is a common method.

Variance Deviation

Variance deviation refers to the squared deviation of data values from their mean. The variance is the average of these squared deviations and represents the spread or dispersion of a dataset.

Binomial Distribution

A Binomial distribution is a discrete probability distribution of the number of successes in a fixed number of independent trials, each with the same probability of success p.

Formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k)

Moment

Fundamental Accounting Principles and Practices

Core Accounting Definitions

Assets = Liabilities + Owner’s Equity: This is the fundamental accounting equation, representing the balance of a company's financial position.

Double-Entry Accounting

The recording of both debit and credit parts of every transaction, ensuring the accounting equation remains balanced.

Objective Evidence Principle

Requires that a source document (e.g., invoice, receipt) be prepared for every entry in a journal, providing verifiable proof of a transaction.

File Maintenance

The process of arranging accounts in a general ledger, assigning account numbers, and keeping records current.

Opening an Account

Writing an account title and number on the heading of an account in the ledger.... Continue reading "Essential Accounting Concepts and Financial Reporting" »

Chapter 1: Core Accounting Fundamentals

Key Financial Statements Equations

• Balance Sheet (BS) Equation: Assets = Liabilities + Stockholders' Equity (SHE)
• Income Statement (IS) Equation: Revenues - Expenses = Net Income (NI)
• Retained Earnings (RE) Equation: Beginning RE + Net Income - Dividends = Ending RE

Debit and Credit Rules

• A Debit increases: Expenses, Assets, Dividends (DEAD)
• A Credit increases: Liabilities, Revenues, Equity (CLEAR)

Objective of Financial Reporting

1. Provide information useful to equity investors and creditors.
2. Maintain the Entity Perspective (separating the company from its owners/people).
3. Ensure Decision Usefulness.

Adjusting Entries: Deferrals and Accruals

Deferrals (Cash Paid/Received Before Recognition)

Prepaid Expenses

Meaning: Paid... Continue reading "Core Financial Accounting Concepts and Principles" »

Math Reference Cheat Sheet

Rules for Multiplying and Dividing Integers

Special Rules for Integers

• Even amount of negative integers: Product/quotient is positive.
• Odd amount of negative integers: Product/quotient is negative.

Classifying Decimals

• Terminating decimals: Have a remainder of zero; the decimal stops.
• Non-terminating decimals: Continue infinitely.
  • Repeating: Continues infinitely and repeats a pattern (denoted by bar notation). Examples: 0.333..., 0.49090..., 0.166...
  • Non-repeating: Continues infinitely but does not repeat a pattern. Examples: Pi (π) and Euler's number (e).

Calculating Percent Error

% Error = [(Estimated Value... Continue reading "Essential Math Cheat Sheet: Integers, Decimals, and Properties" »

Digital Signal Processing

Load the data file: load('exam.mat');

Parameters

• x = xBPFI; % Input signal
• Fs = 10e3; % Sampling frequency
• Ts = 1/Fs;
• t = (0:length(x)-1) * Ts; % Time vector
• n_f = 10000;
• f = (0:n_f-1)*Fs/n_f; % Frequency vector

Input Signal Analysis

Plotting the input signal in the time and frequency domains:

plot(t,x); xlabel('t (s)'); ylabel('x(t) (V)'); title('Input signal in TIME');
X = fft(x,n_f)/length(x);
figure(); subplot(2,1,1); plot(f,abs(X)); xlabel('f (Hz)'); ylabel('|X(f)| (V)'); title('Input signal in FREQUENCY');
subplot(2,1,2); plot(f,angle(X)); xlabel('f (Hz)'); ylabel('\angleX(f) (rad)');

Bandpass Filter Design

Configuring the filter parameters and applying a Hamming window:

• Tbw = 150; fpl = 2400; fcl = (fpl-Tbw/2)/Fs; wcl = 2*