Comprehensive Guide to Integers: Concepts, Operations, and Properties

Foundations of Numbers: Natural, Whole, and Integers

• Natural Numbers: These are the counting numbers beginning from 1. Their set is represented as $1, 2, 3, 4, \dots$
• Whole Numbers: This set includes all natural numbers plus zero. The set is represented as $0, 1, 2, 3, 4, \dots$
  • Property of Predecessor: While 1 is the smallest natural number and has no natural number predecessor, it has 0 as its predecessor in the set of whole numbers.
  • Additive Property: There is no natural number which, when added to another natural number, results in the same number. However, whole numbers possess this property through the number 0 ($a + 0 = a$).
• Negative Numbers: Numbers represented with a negative sign (e.g., $-1, -2, -3$) which indicate values less than zero.
• Integers: A combined collection of whole numbers and negative numbers. The set is theoretically infinite in both directions: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
  • Determination of Limits: The greatest and the smallest integers cannot be determined.

The Need for Integers in Real-World Scenarios

• Limitations of Whole Numbers: Natural and whole numbers cannot describe opposite situations such as profit and loss, price increases and decreases, or temperature changes effectively.
• Temperature: Temperatures are measured relative to $0^\circ C$.
  • Example: A temperature of $20^\circ C$ below $0^\circ C$ is written as $-20^\circ C$.
  • Coolest Place Analysis: In a list including Jammu ($9^\circ C$ below $0^\circ C$), Ladakh ($14^\circ C$ below $0^\circ C$), and Shimla ($6^\circ C$ below $0^\circ C$), Ladakh ($-14^\circ C$) is the coolest.
  • High Temperatures: Delhi ($25^\circ C$), Kanpur ($28^\circ C$), Bhopal ($32^\circ C$), and Jaipur ($35^\circ C$) all represent temperatures above $0^\circ C$.
• Altitude and Depth:
  • Positive Altitude: Mount Everest has the highest altitude at $8,850\,m$ above mean sea level.
  • Depth (Negative Altitude): Mauna Kea stands at $4,207\,m$ above sea level, but its base is approximately $6,000\,m$ below sea level. The total height from foot to summit is approximately $10,000\,m$.
  • Submarines: A submarine floating at a depth of $640\,m$ below sea level is represented as $-640\,m$. If it ascends $230\,m$, its new position is $-640 + 230 = -410\,m$ (or $410\,m$ below sea level).

Integers on a Number Line

• Structure:
  • Zero (0): The central point, which is an integer that is neither positive nor negative.
  • Negative Integers: Located to the left of 0 ($-1, -2, -3, \dots$). The term $-1$ is read as "negative of 1" or "minus 1".
  • Positive Integers: Located to the right of 0 ($1, 2, 3, \dots$). These can be written with a "+" sign (e.g., $+5$) or without any sign (e.g., $7$ represents $+7$).

Comparing and Ordering Integers

• Comparison Principles:
  • Smaller integers are always to the left on a number line; larger integers are always to the right.
  • Example: Comparing $-4$ and $6$. Since $-4$ is to the left of $6$, $-4 < 6$. Conversely, $6 > -4$.
  • Universal Rules:
    • Every positive integer is greater than every negative integer.
    • Zero is less than every positive integer but greater than every negative integer.
• Ordering:
  • Ascending Order: Arranging from smallest to greatest (e.g., $-6 < -3 < 0 < 2 < 4 < 7$).
  • Descending Order: Arranging from greatest to smallest (e.g., $7 > 4 > 2 > 0 > -3 > -6$).

Absolute Value of an Integer

• Definition: The numerical value of an integer regardless of its sign.
• Notation: Symbolized as $|a|$.
• Examples:
  • $|2| = 2$
  • $|-3| = 3$
  • $|0| = 0$

Addition and Subtraction of Integers

• Addition with Like Signs: Add the absolute values of the integers and affix the common sign.
  • Example: $-3 + (-12) = -(|-3| + |-12|) = -(3 + 12) = -15$.
• Addition with Unlike Signs: Subtract the smaller absolute value from the larger absolute value and append the sign of the integer with the larger absolute value.
  • Example: $(-23) + 15 = -(|-23| - |15|) = -(23 - 15) = -8$.
• Additive Inverse: For any integer "a", there exists an opposite "-a" such that $a + (-a) = 0$. Integers $a$ and $-a$ are additive inverses of each other.
• Subtraction: Subtraction is the inverse operation of addition.
  • To subtract an integer, add its additive inverse to the other integer.
  • $a - b = a + (-b)$
  • $a - (-b) = a + (+b)$
  • Example: $7 - (-8) = 7 + 8 = 15$.

Multiplication of Integers

• One Positive and One Negative Integer: Multiply them as whole numbers and place a minus sign before the product. The product is always negative.
  • Formula: $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$.
  • Example: $4 \times (-5) = -20$.
• Two Negative Integers: Multiply their absolute values and place a positive sign before the product. The product of two negative integers is always positive.
  • Formula: $(-) \times (-) = (+)$.
  • Example: $(-3) \times (-5) = 15$.
• More than Two Integers:
  • Find the product of absolute values.
  • If the count of negative factors is even, the product is positive.
  • If the count of negative factors is odd, the product is negative.
  • Example: $(-16) \times (-5) \times (-3) = -240$ (3 negative factors).
  • Example: $(-2) \times (-3) \times (-11) \times (-4) = 264$ (4 negative factors).

Division of Integers

• Like Signs: The quotient is always positive.
  • Formula: $(-a) \div (-b) = a \div b$.
  • Example: $(-36) \div (-4) = 9$.
• Unlike Signs: The quotient is always negative.
  • Formula: $(-a) \div b = -(a \div b)$.
  • Example: $56 \div (-7) = -8$.
• Division by Zero: Any integer divided by 0 is meaningless and not defined ($a \div 0 = \text{undefined}$).
• Division of Zero: Zero divided by any non-zero integer resulted in zero ($0 \div a = 0$).

Properties of Operations on Integers

• Closure Property:
  • Addition: Integers are closed under addition. For any integers $a$ and $b$, $a + b$ is an integer.
  • Subtraction: Integers are closed under subtraction. For any integers $a$ and $b$, $a - b$ is an integer.
  • Multiplication: Integers are closed under multiplication ($a \times b$ is an integer).
  • Division: Integers are not closed under division (e.g., $(-8) \div (-10)$ is not an integer).
• Commutative Property:
  • Addition: $a + b = b + a$.
  • Multiplication: $a \times b = b \times a$.
  • Note: Subtraction and division are not commutative.
• Associative Property:
  • Addition: $(a + b) + c = a + (b + c)$.
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$.
  • Note: Subtraction and division are not associative.
• Identity Properties:
  • Additive Identity: 0 is the additive identity because $a + 0 = 0 + a = a$.
  • Multiplicative Identity: 1 is the multiplicative identity because $a \times 1 = 1 \times a = a$. Note that $-1$ is not the multiplicative identity.
• Distributive Property:
  • Over Addition: $a \times (b + c) = (a \times b) + (a \times c)$.
  • Over Subtraction: $a \times (b - c) = (a \times b) - (a \times c)$.
• Zero Factor Property: For any integer $a$, $a \times 0 = 0 \times a = 0$.

Practical Problems and Case Studies

• Temperature Change: A city temperature drops by $15^\circ C$ from a starting point to reaching $-4^\circ C$. The original temperature was $11^\circ C$ ($-4 + 15 = 11$).
• Freezing Process: A room's temperature drops from $40^\circ C$ at a rate of $5^\circ C$ per hour. After 10 hours, the temperature is $40 + (10 \times -5) = -10^\circ C$.
• Commercial Gains/Losses:
  • Raghav earns a profit of $\text{ }8$ per bag of white cement and a loss of $\text{ }5$ per bag of grey cement. Selling 3,000 white and 5,000 grey bags results in: $(3,000 \times 8) + (5,000 \times -5) = 24,000 - 25,000 = -1,000$ (a loss of $\text{ }1,000$).
• Competitive Exams:
  • In a 40-question test, students get $+2$ for correct answers and $-1$ for wrong answers. If a student attempts all but has 12 wrong answers, their score is $(28 \times 2) + (12 \times -1) = 56 - 12 = 44$.
  • Saumya scored 30 marks with 20 correct answers. Calculations: $(20 \times 2) = 40$. Difference needed: $30 - 40 = -10$. Total incorrect answers = 10.
• Scientific Data (NASA Case Study):
  • Global Temperature Anomalies show recorded changes from 1900 to 2010.
  • 1900: $-0.20^\circ C$; 1910: $-0.35^\circ C$; 1930: $-0.28^\circ C$; 1970: $+0.00^\circ C$; 2010: $+0.63^\circ C$.
  • A reading of $-0.28^\circ C$ in 1930 indicates the temperature was $0.28^\circ C$ below the reference mean.

Logic and Challenges

• Ant on a Pipe: An ant climbs a $10\,m$ pipe, moving $2\,m$ up and slipping $1\,m$ down per move. It will reach the top in 9 moves (at move 8 it is at $8\,m$, on move 9 it reaches $10\,m$ and does not slip).
• Stone in Water: A stone thrown from a $23\,m$ high bridge at $1\,m/s$ for 25 seconds. Position = $23 - 25 = -2\,m$ (2 meters below the water surface).
• Mathematical Operations Defined on Integers:
  • Example: $a * b = (a + b) - (a \times b)$. For $2 * 3$, the value is $(2 + 3) - (2 \times 3) = 5 - 6 = -1$.

Questions & Discussion

• Question: What is the sign of the product if 180 negative integers and 11 positive integers are multiplied?
  • Response: Since 180 is an even number, the product of the negative integers will be positive. Multiplying this by positive integers retains the positive sign. The final product is positive.
• Question: Is there a natural number that when added to another natural number gives the same number as the sum?
  • Response: No. This property (Additive Identity) only applies to whole numbers and integers through the number 0.
• Question: Can two cyclists starting from the same point be separated if one travels $1,200\,m$ downward and another travels $350\,m$ upward?
  • Response: Yes, they are in opposite directions. The distance between them is $|1,200| + |350| = 1,550\,m$.