representation of integers 

Introduction to Integers

• Importance of mathematics in daily life involves both positive and negative numbers.

  • Example:

    • Jose has $3 (positive amount)

    • Maria owes $5 (negative amount)

• Learning to manipulate both positive and negative numbers is essential for everyday applications.

• This lesson introduces integers and methods used for their representation.

Definition of Integers

• Integers are defined as whole numbers that do not contain fractions or decimals.

• Set of integers includes:

  • Positive numbers (greater than zero)

  • Negative numbers (less than zero)

  • Zero

• Integers can be represented in various ways:

  • Number line model

  • Signed numbers

  • Red and black chip model

Representation of Integers

Number Line Model

• A number line is a linear representation showing the relationship between integers.

• Key features of a number line:

  • Zero is positioned at the center.

  • Positive integers are located to the right of zero; they increase as they move away from zero.

  • Negative integers are located to the left; they decrease as they move away from zero.

• Indication of signs:

  • Signed numbers have a symbol indicating positive or negative value.

  • A minus sign precedes negative numbers.

  • A plus sign may precede positive numbers, but is often omitted since numbers are presumed to be positive by default.

Red and Black Chip Model

• The red and black chip model is a visual tool for representing integers.

  • One black chip = one positive unit.

  • One red chip = one negative unit.

• Examples of representation with chips:

  • Positive three is represented by three black chips.

  • Negative four is represented by four red chips.

• Question: What color would be used to represent negative three?

Mathematical Operations with Integers

Absolute Value

• The absolute value of a number is its distance from zero on a number line without regard to direction.

• Concept of absolute value is crucial to understanding integers and their calculations.

Operations with Integers

• Basic operations involving integers include:

  • Addition

  • Subtraction

  • Multiplication

  • Division

• Concept of zero in chip model:

  • One black chip and one red chip together equal zero.

  • Adding zero to a number does not change its value.

  • You can add any number of pairs of chips (one black and one red) without affecting the total.

• Example of representing the number negative two:

  • Two red chips represent the same value.

  • Alternatively, three red chips and one black chip represent negative two, since the pair cancels each other out (equaling zero).

Practice Question

• Explore the challenge of representing positive five using the red and black chip method.

  • Determine two different representations for positive five.