Lecture Notes: Introduction to Integers and Basic Mathematics

Introduction

• Lecturer: Jonathan Gardner
• Topic: Integers and Basic Mathematical Concepts

The Integers

Positive Integers

• Numbers like 1, 2, 3, 4, etc.
• Known as positive integers, forming an infinite series.

Zero

• Zero represents the concept of 'nothing'.
• Significant number with special properties, used frequently in mathematics.

Number Line

• Visual tool to represent numbers as distances from an origin point.
• Origin (0) is an arbitrary choice on the line.
• Positive integers move to the right of the origin.

Natural Numbers

• Combination of zero and positive integers.
• Used to measure real-world quantities like temperature.
• Thermometer examples:
  • Celsius: 0 degrees is freezing, 100 degrees is boiling.
  • Fahrenheit: 0 degrees is below freezing, 100 degrees is a hot day.

Negative Integers

• Numbers like -1, -2, -3, etc.
• Represent distances to the left of the origin on the number line.
• Expand the number line into the negative direction forming negative infinity.

Operations on Integers

Addition

• Basic operation learned at a young age.
• "5 + 7 = 12" example shows combining quantities.
• Symbols: '+' (add), '=' (equals).
• Reflexive Property: Equality can be swapped, e.g., 5 + 7 = 12 is the same as 12 = 5 + 7.

Zero in Addition

• Adding zero to any number doesn't change the value.
• "n + 0 = n" or "0 + n = n"; known as Equation N1.

Negative Numbers in Addition

• Adding a negative number moves to the left.
• Examples:
  • "10 - 5 = 5" shown as taking away 5 from 10.
  • "10 - 15 = -5" involves moving beyond zero to the left.
• "n - a" or "n + (-a)" results in moving left on the number line.

Important Properties

Additive Inverse

• Any number plus its negative equals zero.
• Examples:
  • "5 + (-5) = 0"
  • "-3 + 3 = 0"
• Known as Equation N2.
• Visualized as mirror images on the number line: a and -a.

Notation

• '-' sign can indicate subtraction or a negative sign.
• "-a" is read as minus a, not necessarily negative.
• Additive inverse examples: 3 and -3 are inverses of each other.

Conclusion

• Introduction to additive and multiplicative inverses.
• Upcoming topic: Rules for addition in the next section.

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