Fractions and representation

Fractions: Basics and Representation

Fraction Components

• A fraction is represented as b/c.
• b is the numerator.
• c is the denominator.

Fraction as Portions of a Whole

• Fractions represent parts of a whole.
• Example: A pie cut into six equal slices; eating one slice means you've consumed 1/6 of the pie.

Visual Representation of Fractions

• Example: If there are eight slices and three are shaded, the fraction representing the shaded portion is 3/8.
• To represent 2/5, divide a shape into five equal parts and shade two of them.

Fractions as Division

• Fractions also represent division relationships.
• The relationship can be checked using multiplication.
• If $a/b = n$, then $b * n = a$.
• Example 1: $20/5 = 4$, and $5 * 4 = 20$.
• Example 2: $63/63 = 1$, and $63 * 1 = 63$.

Using a Calculator

• $20/5$ is calculated as 20 divided by 5, resulting in 4.
• $63/63$ is calculated as 63 divided by 63, resulting in 1.

Zero in Fractions

Zero as the Numerator

• If the numerator is zero, such as $0/12$, the fraction equals zero.
• $0/12 = 0$, because $12 * 0 = 0$.

Zero as the Denominator

• If the denominator is zero, such as $12/0$, the fraction is undefined.
• Testing zero as a solution: If $12/0 = 0$, then $0 * 0$ should equal 12, which is false.
• Testing 12 as a solution: If $12/0 = 12$, then $0 * 12$ should equal 12, which is also false.

Undefined Expressions

• A fraction with zero in the denominator has no numerical solution; it's an undefined expression.
• There is no number that, when multiplied by zero, yields a non-zero numerator.

Calculator Behavior

• $0/12$ (zero divided by 12) yields zero.
• $12/0$ (12 divided by zero) results in an error because division by zero is undefined.