Table of Contents
Overview
Arithmetic is the part of mathematics that deals with numbers and basic calculations. Long before people developed advanced topics like algebra or calculus, they were already adding, subtracting, multiplying, and dividing quantities in daily life: counting animals, sharing food, measuring lengths, or trading goods. In this course, arithmetic acts as the practical foundation you will stand on when you later meet algebra, geometry, and beyond.
In this chapter, we describe what makes arithmetic special, how it is used, and what kinds of skills you will build in the later arithmetic sections of this course. Detailed “how‑to” procedures for each operation—addition, subtraction, and so on—are saved for their own chapters.
What Arithmetic Is About
Arithmetic focuses on:
- Numbers: such as $1, 2, 3$, fractions like $\frac{1}{2}$, decimals like $0.75$, and percentages like $25\%$.
- Basic operations on numbers: adding, subtracting, multiplying, dividing.
- Comparing quantities: which is bigger, smaller, or equal.
- Simple relationships between quantities: like ratios (e.g. $2:3$) and proportions (e.g. “$2$ is to $3$ as $4$ is to $6$”).
While the course later introduces many types of numbers in detail, in arithmetic you mainly learn how to work with them efficiently and accurately.
Why Arithmetic Matters
Arithmetic skills support almost everything else in mathematics:
- In algebra, you will manipulate symbols that behave like numbers. If arithmetic is shaky, algebra becomes confusing.
- In geometry, you use arithmetic to compute lengths, areas, and volumes.
- In statistics, you use arithmetic to find averages and to interpret data.
- In everyday life, you rely on arithmetic whenever you handle money, cook with recipes, compare prices, or interpret percentages in news and reports.
Good arithmetic is not only knowing “what button to push” on a calculator; it is understanding what a calculation means and being able to judge whether an answer is reasonable.
Core Ideas of Basic Operations
The later chapters in this arithmetic section each focus on a specific operation or type of number. Here we highlight, at a high level, what each operation is for, without going into full procedures.
Addition
Addition combines quantities. If you have $3$ apples and get $2$ more, the total is
$$3 + 2 = 5.$$
Addition answers questions like:
- “How many in total?”
- “What is the sum of these numbers?”
It is also closely connected to counting: adding $1$ is like moving to the next number in the counting sequence.
Subtraction
Subtraction is the opposite of addition. It removes or compares quantities. If you have $7$ apples and give away $3$, then
$$7 - 3 = 4.$$
Subtraction answers questions like:
- “How many are left?”
- “How much more does one quantity have than another?”
Thinking of subtraction as “finding the difference” will be important later, especially in algebra and statistics.
Multiplication
Multiplication is repeated addition in a structured way. If you have $4$ bags, each with $3$ apples, the total number of apples is
$$4 \times 3,$$
which means “$4$ groups of $3$.”
Multiplication answers questions like:
- “What is the total if each of several groups has the same size?”
- “What is the product of these numbers?”
Multiplication is central for later ideas such as powers, area calculations, and proportional reasoning.
Division
Division splits a quantity into equal parts or asks how many times one number fits into another. If you have $12$ apples and want to share them equally between $3$ people, you are asking
$$12 \div 3,$$
which means “$12$ shared equally into $3$ groups.”
Division answers questions like:
- “If I share equally, how much does each get?”
- “How many times does this number go into that number?”
Division prepares you for fractions, ratios, and many real‑world comparisons.
Beyond Whole Numbers: Fractions, Decimals, and Percentages
Basic operations are first learned with whole (counting) numbers, but real situations often involve parts of a whole. In the arithmetic section of this course, you will learn to handle three closely related ways of describing parts: fractions, decimals, and percentages.
Fractions
Fractions describe parts of a whole divided into equal pieces. A fraction like $\frac{3}{4}$ represents $3$ equal parts out of $4$ in total.
In the fractions chapters, you will learn:
- Different types of fractions (such as “proper” and “improper”).
- How to interpret mixed numbers like $1\frac{1}{2}$.
- How to add, subtract, multiply, and divide with fractions.
Understanding fractions helps make sense of measurements, sharing quantities, and many practical tasks.
Decimals and Percentages
Decimals provide another way to write fractional amounts, using the decimal point. For example,
$$0.5 = \frac{1}{2}, \quad 0.25 = \frac{1}{4}.$$
Percentages express “out of $100$.” For example, $25\%$ means $\frac{25}{100}$, which reduces to $\frac{1}{4}$.
In the later chapters on decimals and percentages, you will:
- Convert between fractions and decimals.
- Convert between decimals and percentages.
- Use percentages to solve everyday problems, such as discounts, tax, and interest.
Seeing the connections among fractions, decimals, and percentages gives you flexibility in choosing the most convenient form in different situations.
Powers, Roots, Ratios, and Proportions
Some arithmetic topics go beyond the four basic operations but are still grounded in working with numbers directly.
Powers and Roots
Powers use repeated multiplication. For example,
$$3^4 = 3 \times 3 \times 3 \times 3.$$
Roots undo powers. The square root of $9$ is a number which, when multiplied by itself, gives $9$:
$$\sqrt{9} = 3.$$
The later chapters on exponents, square roots, and laws of exponents will show how these ideas work and how they connect with multiplication and division.
Ratios and Proportions
Ratios compare two quantities, such as $2:3$ (read “two to three”). Proportions state that two ratios are equal, such as
$$\frac{2}{3} = \frac{4}{6}.$$
In the chapters on ratios, proportions, and variation, you will learn to:
- Interpret and simplify ratios.
- Use proportions to solve problems like scaling recipes or reading maps.
- Understand direct and inverse variation, which describe how one quantity changes in relation to another.
These concepts are crucial for many applied problems, such as speed, density, and unit conversions.
Mental and Written Calculation
Arithmetic is practiced in different ways:
- Mental calculation: doing steps in your head, often using shortcuts or patterns (such as using known multiplication facts).
- Written methods: organizing calculations on paper (or a screen) in clear steps, such as arranging numbers in columns for multi‑digit addition.
- Estimation: getting a rough but sensible answer quickly, to check whether a detailed calculation is reasonable.
As you work through the arithmetic chapters, you should aim not only to get correct answers, but also to:
- Recognize patterns that make calculations simpler.
- Check results using a different method (for example, using estimation to see if the answer is in the right range).
- Develop confidence in choosing an efficient method for a given problem.
Arithmetic as a Foundation
Everything you learn in the arithmetic section is meant to be usable later:
- When reading an equation in algebra, each step will often rely on correct arithmetic.
- When interpreting a graph in later chapters, you will use arithmetic to read scales and compute values.
- When solving real‑world problems in probability, statistics, and beyond, arithmetic will underpin nearly every calculation.
As you move into the detailed chapters on basic operations, fractions, decimals and percentages, powers and roots, and ratios and proportions, remember that the goal is not just to memorize procedures, but to understand how numbers behave and how to use that understanding to solve meaningful problems.