Comprehensive Guide to Integers, Rational Numbers, and Proportionality

Subtraction and Sign Rules

• Subtraction is defined as the process of adding the opposite of a number. This concept follows a specific mathematical formula:

• $p - q = p + (-q)$

• Integer rules are utilized in various real-world mathematical applications, including:

  • Calculating changes in temperature.

  • Managing and reconciling banking balances.

  • Determining shifts in elevation levels.

• When multiplying or dividing two numbers, the resulting sign of the product or quotient is determined by the relationship between the signs of the two original numbers:

  • Identical signs (both positive or both negative) always yield a positive result.

  • Opposite signs (one positive and one negative) always yield a negative result.

• Negative fractions can be expressed formally in three equivalent ways:

  • $-\frac{p}{q}$

  • $\frac{-p}{q}$

  • $\frac{p}{-q}$

The Number System and Rational Numbers

• Rational numbers are defined as any number that can be expressed as a fraction in the form $\frac{p}{q}$.

• A critical requirement for a rational number is that the denominator, represented by $q$, must not be zero ($q \neq 0$).

• Fractions can be converted into decimals through the use of long division. These resulting decimals fall into two categories:

  • Terminating decimals: These are decimals that come to an end, specifically when the long division process results in a remainder of $0$.

  • Repeating decimals: These are decimals that do not end; instead, the same digit or set of digits repeats infinitely.

• Additive inverses refer to opposites that always combine to make exactly zero. This follows the formula:

• $p + (-p) = 0$

• An example of an additive inverse is $-8 + 8 = 0$.

Number Lines and Adjustments

• Number lines provide a visual representation for tracking adjustments and changes to values:

  • Movement to the right indicates positive adjustments.

  • Movement to the left indicates negative changes.

• A standard number line range includes negative numbers on the left and positive numbers on the right of the zero point:

• $-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$

Proportional and Non-Proportional Relationships

• In a proportional relationship, the ratio between two variables always remains the same.

• To determine if a relationship is proportional, divide $y$ by $x$ (or vice versa); if the relationship is proportional, this value will be constant across all pairs.

• Example of a proportional relationship where the ratio between $y$ and $x$ stays constant:

  • For $x = 1$ and $y = 3$, $\frac{y}{x} = \frac{3}{1} = 3$

  • For $x = 2$ and $y = 6$, $\frac{y}{x} = \frac{6}{2} = 3$

  • For $x = 3$ and $y = 9$, $\frac{y}{x} = \frac{9}{3} = 3$

  • In this case, because the ratio stays the same ($3$), the relationship is proportional.

• In a non-proportional relationship, the ratio between the variables does not stay the same.

• Example of a non-proportional relationship involving variables $a$ and $b$:

  • For $a = 6$ and $b = 26$, $\frac{b}{a} = \frac{26}{6} = 4.33$

  • For $a = 10$ and $b = 35$, $\frac{b}{a} = \frac{35}{10} = 3.5$

  • Because $\frac{26}{6} \neq \frac{35}{10}$, these values do not stay the same and the relationship is not proportional.