Remember that teacher? The one who insisted that math exists in EVERYTHING around you? At the time, perhaps you were like, "Nah, that's bullshit". But now, as an adult, you've likely come to realize that your former teacher was, in fact, right. Math isn't just confined to textbooks; it's the underlying structure of the world we live in.
In this article (or rather, this series of articles), I'll dive into the world of mathematics, building from the ground up. Our goal is to demystify math and show you its practical relevance and maybe, make you like it a little more.
Here's what we'll cover to start:
is a multifaceted and deeply complex human endeavor that transcends mere calculation. Its nature has been a subject of philosophical inquiry for centuries, revealing layers that encompass it being a language, showing itself as a powerful tool. It provides a framework for describing, analyzing, and predicting patterns, relationships, and structures. Unlike empirical sciences that rely on observation and experimentation to validate theories.
The understanding and notation of these operations evolved over millennia. Early civilizations developed methods for counting and performing basic arithmetic for practical purposes like trade, construction, and astronomy. The development of place-value number systems, particularly the Hindu-Arabic numeral system with its symbol for zero, greatly simplified calculations. The symbols for the operations themselves also evolved, with the symbols we use today becoming standardized relatively late in mathematical history.
These four operations are not isolated concepts but are deeply interconnected. Addition and subtraction are inverse operations, as are multiplication and division. 1 Multiplication can be defined in terms of repeated addition, and division in terms of repeated subtraction. 2 The distributive property elegantly links multiplication and addition/subtraction. 3 This intricate web of relationships allows for the manipulation and simplification of mathematical expressions and the development of more advanced mathematical theories
The four fundamental operations of mathematics – addition, subtraction, multiplication, and division are the bedrock of arithmetic and the building blocks for virtually all higher mathematical concepts. While seemingly simple at a glance, they might surprise with its flexibility.
Conceptually, addition is the process of combining two or more quantities or sets of objects. It answers the question "how many in total?" or "what is the total amount?"
Properties:
- Commutative Property: The order of the numbers being added does not affect the sum (a+b=b+a).
- Associative Property: The way numbers are grouped when adding does not affect the sum ((a+b)+c=a+(b+c)).
- Identity Property: Adding zero to any number does not change the number (a+0=a). Zero is the additive identity.
Subtraction is the inverse operation of addition. It represents taking away a quantity from another, finding the difference between two quantities, or determining what needs to be added to one quantity to get another. It answers questions like "how many are left?" or "what is the difference?"
Properties:
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Inverse of Addition: a−b=c is equivalent to a=b+c.
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Identity Property: Subtracting zero from a number does not change the number (a−0=a). However, 0−a=a (unless a=0), highlighting the lack of a simple identity property like addition.
Multiplication can be understood as repeated addition of a number by itself a certain number of times. It also represents scaling a quantity or finding the area of a rectangle. It answers questions like "what is the total of several equal groups?" or "how much is something increased by a certain factor?"
Properties:
- Commutative Property: The order of the numbers being multiplied does not affect the product (a×b=b×a).
- Associative Property: The way numbers are grouped when multiplying does not affect the product ((a×b)×c=a×(b×c))
- Identity Property: Multiplying any number by one does not change the number (a×1=a). One is the multiplicative identity.
- Zero Property: Multiplying any number by zero results in zero (a×0=0).
- Closure Property: For a given set of numbers, the product of any two numbers in the set is also in the set.
- Distributive Property: Multiplication distributes over addition and subtraction (a×(b+c)=(a×b)+(a×c) and a×(b−c)=(a×b)−(a×c)). This property links addition and multiplication
Division is the inverse operation of multiplication. It represents splitting a quantity into equal parts or groups, or determining how many times one quantity is contained within another. It answers questions like "how many are in each group?" or "how many groups can be made?"
Properties:
- Inverse of Multiplication: a÷b=c (with b=0) is equivalent to a=b×c.
- Identity Property: Dividing any number by one does not change the number (a÷1=a).
As we discussed, a fraction represents a part of a whole or a ratio of a part to a whole. It's written as ba, where a is the numerator (the number of parts you have) and b is the denominator (the total number of equal parts in the whole).
Types
- Proper Fractions (ba where a<b): Represent values between 0 and 1. Think of a piece of a pie – you have a fraction of the whole pie.
- Improper Fractions (ba where a≥b): Represent values equal to or greater than 1. This could be more than one whole pie, or a whole pie plus a fraction of another.
- Mixed Numbers (Whole number and a proper fraction): A practical way to express improper fractions, clearly showing the number of whole units and the remaining fractional part (e.g., 121).
Equivalence and Simplification: The idea of equivalent fractions (21=42=84) shows that the same proportion can be represented in infinitely many ways. Simplifying fractions to their lowest terms by dividing the numerator and denominator by their greatest common divisor provides the most concise representation of that proportion.
Operations (Brief Recap): The rules for adding, subtracting, multiplying, and dividing fractions are built upon the concept of combining or dividing these parts of a whole, often requiring a common ground (like a common denominator for addition and subtraction) or leveraging the inverse relationship (like multiplying by the reciprocal for division).
Decimals are a way of expressing fractions where the denominator is a power of ten (10, 100, 1000, etc.). They extend the place value system we use for whole numbers to represent fractional parts.
Terminating vs. Repeating Decimals:
- Terminating Decimals: Result when the fractional part can be expressed with a denominator that is a power of ten (e.g., 41=10025=0.25).
- Repeating Decimals: Occur when the division of the numerator by the denominator results in a pattern of digits that repeats infinitely (e.g., 31=0.333...). These decimals represent rational numbers that cannot be expressed as terminating decimals.
Operations (Brief Recap): Operations with decimals are often more straightforward than with fractions because the place value system aligns naturally. Lining up decimal points for addition and subtraction ensures that you are combining or separating the same fractional units. Multiplication involves counting the total number of decimal places, and division can be transformed into a whole number division problem by shifting the decimal points
A percentage is a way of expressing a fraction with a denominator of 100. The word "percent" comes from the Latin phrase "per centum," meaning "by the hundred." Percentages are denoted by the symbol %