Rules and Formulas in Mathematics

Study Notes

Rules and Formulas

Formulas are mathematical equations used to solve problems and express relationships between quantities. They can also represent physical laws or principles. There are several types of rules and formulas that we encounter in various contexts in our lives. Some common ones include:

Arithmetic Operators

In math, there are four basic arithmetic operators: addition (+), subtraction (-), multiplication (*), and division (/). These operators follow certain rules when they're combined to create expressions. For example, addition is commutative, so changing the order of numbers being added does not change their sum. On the other hand, multiplication is associative, meaning that it doesn't matter which grouping you multiply first: (a*b)*c = a*(b*c).

Order of Operations (PEMDAS)

The 'Order of operations', commonly known by its acronym PEMDAS (Parentheses Exponents Multiplication Division Addition Subtraction), dictates how we perform calculations involving more than one operation. According to this rule, if a calculation involves multiple operations, you must carry out the following sequence of steps:

Calculate within parentheses from left to right.
Evaluate exponentials.
Multiply from left to right.
Divide from left to right.
Add from left to right.
Subtract from left to right.

For instance, given the expression [7 - 9 \div 3 + 5], the correct answer is [7 - 3 + 5] due to the strict application of the rules.

Changing Percentages into Decimals

To convert percentage values to decimals, simply remove the percent symbol (%):

[10% = 0.1]

This method works because one hundredth of something is equal to ( \frac{1}{10} ) of itself. Therefore, to get 1% of something, divide it by 100; to get x% of some quantity, divide it by x.

Algebraic Equations

An algebraic equation features variables like letters placed in place of unknown numerical values. To solve these equations, you manipulate them using rules called properties of equality, such as adding or subtracting the same number on both sides of the equals sign, dividing both sides by the same nonzero value, or multiplying both sides by the same factor. Solving these requires understanding what each variable represents and properly applying the appropriate property of equality.

Linear Inequalities

Linear inequalities involve two linear functions, or lines on a graph. These inequalities have different forms depending on whether the signs before the inequality operator are positive or negative. Each inequality can be solved visually on a graph by finding where the line crosses the axis or comparing slopes. Once crossed with the function expression, you determine the solution set based on the comparator.

Graphical Methods

Graphical methods utilize diagrams to help us visualize data points and trends in graphs. By observing patterns in the arrangement of dots or squares on a grid, we can often draw conclusions about the underlying relationship between the input and output variables without having explicit access to any formulae. This approach is particularly useful in fields where data analysis plays a crucial role, such as science and mathematics.

Remember, while some rules have exceptions, understanding and practicing these general guidelines will improve your ability to tackle most real-world problems involving rules and formulas.

Description

Explore common rules and formulas used in mathematics, such as arithmetic operators, order of operations (PEMDAS), converting percentages into decimals, algebraic equations, linear inequalities, and graphical methods. Enhance your problem-solving skills by understanding and applying these fundamental principles.