All Topics
mathematics-international-0607-advanced | cambridge-igcse
Your Flashcards are Ready!
15 Flashcards in this deck.
1.
Number
2.
Statistics
3.
Algebra
4.
Transformations and Vectors
5.
Geometry
6.
Functions
7.
Mensuration
8.
Coordinate Geometry
9.
Trigonometry
10.
Probability
Integers (positive, zero, and negative)
Topic 2/3
or
3
Still Learning
I know
12
Previous
Next
Integers (Positive, Zero, and Negative)
Introduction
Integers form a fundamental concept in mathematics, representing the set of whole numbers that extend infinitely in both positive and negative directions, including zero. In the Cambridge IGCSE Mathematics - International - 0607 - Advanced syllabus, understanding integers is crucial for developing numerical fluency and problem-solving skills. This article delves into the intricacies of integers, exploring their properties, operations, and applications within various mathematical contexts.
Key Concepts
Definition of Integers
Integers are a set of numbers that include all positive whole numbers, their negative counterparts, and zero. Formally, the set of integers is represented as:
$$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$$
Unlike natural numbers, integers extend infinitely in both the positive and negative directions, allowing for a comprehensive numerical framework essential in various mathematical operations and real-life applications.
Properties of Integers
Integers exhibit several key properties that facilitate their manipulation and application in mathematical problems:
- Closure Property: Integers are closed under addition, subtraction, and multiplication. This means that performing these operations on any two integers will always result in another integer.
- Associative Property: For addition and multiplication, integers satisfy the associative property. That is, for any integers a, b, and c: $$ (a + b) + c = a + (b + c) $$ $$ (a \times b) \times c = a \times (b \times c) $$
- Commutative Property: Addition and multiplication of integers are commutative. For any integers a and b: $$ a + b = b + a $$ $$ a \times b = b \times a $$
- Distributive Property: Multiplication distributes over addition: $$ a \times (b + c) = (a \times b) + (a \times c) $$
- Additive Identity: The number zero acts as the additive identity since adding zero to any integer leaves it unchanged: $$ a + 0 = a $$
- Multiplicative Identity: The number one serves as the multiplicative identity: $$ a \times 1 = a $$
- Additive Inverse: For every integer a, there exists an integer -a such that: $$ a + (-a) = 0 $$
Operations with Integers
Mastery of operations with integers is essential for solving a wide range of mathematical problems. The primary operations include addition, subtraction, multiplication, and division.
Addition of Integers
Adding integers involves combining their values while considering their signs. The rules are as follows:
- If both integers have the same sign, add their absolute values and assign the common sign to the result.
- If the integers have different signs, subtract the smaller absolute value from the larger absolute value and assign the sign of the integer with the larger absolute value to the result.
- $5 + 3 = 8$
- $-4 + (-6) = -10$
- $7 + (-2) = 5$
- $-5 + 9 = 4$
Subtraction of Integers
Subtracting integers is equivalent to adding the additive inverse of the subtracted number: $$ a - b = a + (-b) $$ Apply the addition rules accordingly. Examples:
- $10 - 4 = 6$
- $-3 - (-7) = -3 + 7 = 4$
- $5 - 8 = -3$
- $-6 - 2 = -8$
Multiplication of Integers
Multiplying integers follows specific sign rules:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
- $3 \times 4 = 12$
- $-5 \times (-2) = 10$
- $6 \times (-3) = -18$
- $-7 \times 5 = -35$
Division of Integers
Dividing integers adheres to the same sign rules as multiplication:
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative
- $12 \div 3 = 4$
- $-20 \div (-4) = 5$
- $15 \div (-3) = -5$
- $-16 \div 4 = -4$
Absolute Value
The absolute value of an integer is its distance from zero on the number line, irrespective of direction. It is denoted by two vertical bars:
$$ |a| $$
For any integer a:
- If $a \geq 0$, then $|a| = a$.
- If $a < 0$, then $|a| = -a$.
- $|5| = 5$
- $|-7| = 7$
- $|0| = 0$
Number Line Representation
A number line is a visual representation of integers arranged in order on a straight line. Positive integers are positioned to the right of zero, while negative integers are to the left. This graphical tool aids in understanding the relative sizes and operations involving integers.
Usage: The number line is instrumental in performing operations such as addition and subtraction by shifting left or right based on the operation's nature.
Usage: The number line is instrumental in performing operations such as addition and subtraction by shifting left or right based on the operation's nature.
Order of Operations
When dealing with integers in expressions, it's essential to follow the order of operations to ensure accurate results. The standard sequence is:
- Parentheses ()
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Properties of Even and Odd Integers
Integers can be classified as even or odd based on their divisibility by 2.
- Even Integers: Integers divisible by 2 without a remainder. $$ a = 2k, \text{ where } k \in \mathbb{Z} $$ Examples: ..., -4, -2, 0, 2, 4, ...
- Odd Integers: Integers that have a remainder of 1 when divided by 2. $$ a = 2k + 1, \text{ where } k \in \mathbb{Z} $$ Examples: ..., -3, -1, 1, 3, 5, ...
Applications of Integers
Integers are pervasive in various real-life scenarios and mathematical applications, including:
- Temperature Measurement: Representing temperatures above and below zero.
- Financial Transactions: Depicting profits and losses.
- Elevation Levels: Indicating heights above and below sea level.
- Game Scores: Tracking points gained or lost.
Advanced Concepts
Integer Properties in Algebraic Expressions
When dealing with algebraic expressions involving integers, it's essential to apply the properties of integers to simplify and solve equations.
Example:
Simplify the expression:
$$ 3(-2x + 4) - 2(x - 5) $$
Solution:
\begin{align*}
3(-2x) + 3(4) - 2(x) + 2(5) &= -6x + 12 - 2x + 10 \\
&= (-6x - 2x) + (12 + 10) \\
&= -8x + 22
\end{align*}
Integer Equations and Inequalities
Solving equations and inequalities involving integers requires a solid grasp of integer operations and properties.
Example:
Solve for x:
$$ -3x + 7 \leq 16 $$
Solution:
\begin{align*}
-3x + 7 &\leq 16 \\
-3x &\leq 16 - 7 \\
-3x &\leq 9 \\
x &\geq -3 \quad (\text{Note: Inequality sign reverses when dividing by a negative number})
\end{align*}
Integer Sequences and Patterns
Recognizing and analyzing integer sequences and patterns is a vital skill in advanced mathematics. Patterns can reveal properties such as arithmetic progression, geometric progression, and more.
Example:
Consider the sequence: -5, -3, -1, 1, 3, ...
Analysis:
This is an arithmetic sequence where each term increases by 2. The nth term can be expressed as:
$$ a_n = -5 + 2(n - 1) = 2n - 7 $$
Modulo Operation with Integers
The modulo operation finds the remainder after division of one integer by another. It is widely used in computer science, cryptography, and number theory.
Definition:
$$ a \mod n = r \quad \text{where } 0 \leq r < n $$
Examples:
- $10 \mod 3 = 1$
- $-7 \mod 4 = 1$
Diophantine Equations
Diophantine equations are polynomial equations that seek integer solutions. These equations are named after the ancient Greek mathematician Diophantus.
Example:
Solve for integers x and y:
$$ 2x + 3y = 7 $$
Solution:
By trial:
\begin{align*}
x &= 2 \quad \Rightarrow \quad 2(2) + 3y = 7 \quad \Rightarrow \quad 4 + 3y = 7 \quad \Rightarrow \quad y = 1 \\
x &= 5 \quad \Rightarrow \quad 2(5) + 3y = 7 \quad \Rightarrow \quad 10 + 3y = 7 \quad \Rightarrow \quad y = -1
\end{align*}
Thus, the integer solutions are $(x, y) = (2, 1)$ and $(5, -1)$.
Vector Operations Involving Integers
Vectors, which have both magnitude and direction, often involve integer components, especially in coordinate systems.
Example:
Consider vectors $\mathbf{A} = \langle 3, -2 \rangle$ and $\mathbf{B} = \langle -1, 4 \rangle$.
Operations:
- Addition: $\mathbf{A} + \mathbf{B} = \langle 3 + (-1), -2 + 4 \rangle = \langle 2, 2 \rangle$
- Scalar Multiplication: $2\mathbf{A} = \langle 2 \times 3, 2 \times (-2) \rangle = \langle 6, -4 \rangle$
Interdisciplinary Connections
Integers connect to various fields beyond pure mathematics, illustrating their broad applicability:
- Physics: Representing quantities like charge, direction, and energy levels.
- Economics: Modeling profits, losses, and net income.
- Computer Science: Managing memory addresses, data indexing, and algorithm optimization.
- Engineering: Designing systems where direction and magnitude are essential, such as electrical circuits and structural analysis.
Comparison Table
| Aspect | Positive Integers | Negative Integers |
|---|---|---|
| Definition | Whole numbers greater than zero | Whole numbers less than zero |
| Symbol Representation | e.g., +1, +2, +3 | e.g., -1, -2, -3 |
| Position on Number Line | Right of zero | Left of zero |
| Absolute Value | Same as the integer | Positive counterpart |
| Use Cases | Counting, measuring profit | Debts, temperatures below zero |
Summary and Key Takeaways
- Integers encompass positive numbers, negative numbers, and zero.
- Mastery of integer operations is essential for solving mathematical problems effectively.
- Advanced concepts include Diophantine equations, vector operations, and modulo arithmetic.
- Integers have wide-ranging applications across various disciplines such as physics, economics, and computer science.
- Understanding the properties and behaviors of integers forms a foundation for higher-level mathematical studies.
Coming Soon!
Examiner Tip
Tips
- Remember the Signs: Use the "rules of signs" for addition and multiplication to keep track of positive and negative numbers accurately.
- Use Number Line: Visualize integer operations on a number line to better understand the movement between positive and negative values.
- PEMDAS is Key: Always follow the order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to avoid calculation errors.
Did You Know
Did You Know
- The concept of negative integers was first used in ancient China to represent debts and losses, illustrating their practical importance in early commerce.
- Zero, an integer, was a later addition to the number system and is considered one of the most significant mathematical inventions, enabling the development of algebra and calculus.
- In computer science, integers are fundamental in programming, particularly in data types and memory addressing, making them essential for software development and algorithm efficiency.
Common Mistakes
Common Mistakes
- Ignoring the Sign: Students often forget to consider the sign of integers when performing operations. For example, calculating $-5 + 3$ incorrectly as $-2$ instead of the correct $-2$.
- Incorrectly Applying Operation Rules: When multiplying integers, students may misapply the sign rules, such as thinking a negative times a positive is positive.
- Division by Zero: Attempting to divide integers by zero, which is undefined, leading to incorrect solutions in equations.
FAQ
What are integers?
Integers are whole numbers that include positive numbers, negative numbers, and zero, without any fractional or decimal parts.
How do you add negative integers?
When adding negative integers, combine their absolute values and assign a negative sign to the result if both numbers are negative.
Why is division by zero undefined?
Division by zero is undefined because there is no number that you can multiply by zero to get a non-zero number, making the operation impossible.
What is the absolute value of an integer?
The absolute value of an integer is its distance from zero on the number line, regardless of direction, and is always a non-negative number.
How are integers used in real life?
Integers are used in various real-life situations, such as representing temperatures, financial transactions, elevation levels, and game scores.
1.
Number
2.
Statistics
3.
Algebra
4.
Transformations and Vectors
5.
Geometry
6.
Functions
7.
Mensuration
8.
Coordinate Geometry
9.
Trigonometry
10.
Probability
Get PDF
How would you like to practise?
