Grade 8 Mathematics: Rational Numbers

Flashcards reviewing Grade 8 Mathematics concepts including natural numbers, whole numbers, integers, fractions, rational numbers, operations, equality, and equivalent forms.

What is the definition and notation for the set of natural numbers?

Natural numbers, also called counting numbers, are numbers that start with the number 1 and do not involve negatives, fractions, or decimals. The set of natural numbers is denoted by $N$ and is given by $N = \{1, 2, 3, 4, 5, \dots\}$.

What is the set of whole numbers $W$, and what is its relationship to the set of natural numbers $N$?

The set of whole numbers, denoted by $W$, is formed by combining the set of natural numbers and zero ($0$), represented as $W = \{0, 1, 2, 3, 4, \dots\}$. The relationship between them is $N \subset W$.

How is the set of integers $Z$ defined?

The set of integers, denoted by $Z$, is formed by combining whole numbers and negative numbers (like $-4, -3, -2, -1$). It is given by $Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

What is the definition of a fraction, including its parts?

A fraction is a number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are whole numbers and $b \neq 0$. In this form, $a$ is called the numerator and $b$ is called the denominator.

What is the formula for the sum of two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

The sum of fractions is given by $\frac{a}{b} + \frac{c}{d} = \frac{(a \times d) + (b \times c)}{b \times d}$, where $a, b, c, d$ are natural numbers and $b, d \neq 0$.

What is the formula for the difference of two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

The difference of fractions is given by $\frac{a}{b} - \frac{c}{d} = \frac{(a \times d) - (b \times c)}{b \times d}$, where $a, b, c, d$ are natural numbers and $b, d \neq 0$.

What is the formula for the product of two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

The product of fractions is given by $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$, where $a, b, c, d$ are natural numbers and $b, d \neq 0$.

What is the formula for the division of two fractions $\frac{a}{b}$ and $\frac{c}{d}$?

The division of fractions is given by $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$, where $a, b, c, d$ are natural numbers and $b, d, c \neq 0$.

According to Definition 1.1, how is a rational number defined?

A number that can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ is called a rational number. The set of rational numbers is denoted by $Q = \{\frac{a}{b} \mid a, b \in Z \text{ and } b \neq 0\}$.

How can $-1.5$ be shown to be a rational number according to Definition 1.1?

$-1.5$ is a rational number because it can be written in the form $\frac{a}{b}$ as $\frac{-15}{10}$, where $a = -15$ and $b = 10$ are both integers.

How can zero ($0$) be written to show that it is a rational number?

Zero can be written in the form $\frac{a}{b}$ as $\frac{0}{1}$, where $a = 0$ and $b = 1$ are both integers.

What is the full subset relationship between natural numbers, whole numbers, integers, and rational numbers?

The relationship between these number sets is $N \subset W \subset Z \subset Q$, meaning every natural number is a whole number, every whole number is an integer, and every integer is a rational number.

According to Definition 1.2, when are two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ equal?

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ (where $a, b, c, d$ are integers and $b \neq 0, d \neq 0$) are equal, written as $\frac{a}{b} = \frac{c}{d}$, if $a \times d = b \times c$.

How do you show that $\frac{3}{4}$ and $\frac{9}{12}$ are equal rational numbers?

By checking cross-multiplication: $3 \times 12 = 36$ and $4 \times 9 = 36$. Since $3 \times 12 = 4 \times 9$, the rational numbers $\frac{3}{4}$ and $\frac{9}{12}$ are equal.

How do you find the value of $a$ such that $\frac{a}{6} = \frac{3}{9}$?

If $\frac{a}{6} = \frac{3}{9}$, then cross-multiplication yields $a \times 9 = 3 \times 6$, which simplifies to $9a = 18$. Dividing by 9 gives $a = 2$.

What rule generates equivalent forms of a rational number $\frac{a}{b}$ using a nonzero integer $c$?

Given a rational number $\frac{a}{b}$, if $c$ is a nonzero integer, then $\frac{a}{b} = \frac{a \times c}{b \times c}$.

What are three different forms of the rational number $\frac{3}{4}$ as shown in Example 1.4?

Three different forms of $\frac{3}{4}$ are $\frac{6}{8}$ (using $c=2$), $\frac{9}{12}$ (using $c=3$), and $\frac{12}{16}$ (using $c=4$).

In a class where 47 students took a mathematics exam and 15 failed, what is the ratio of failing students to passing students?

The number of passing students is $47 - 15 = 32$. The ratio of the number of students who failed to the number of students who passed is $15$ to $32$ (or $\frac{15}{32}$).