Question
What is the definition of Natural Numbers?
Answer
They are the counting numbers: 1, 2, 3, 4, 5, and so on.
Question
What is the definition of Whole Numbers?
Answer
They are the natural numbers together with zero (0): 0, 1, 2, 3, 4, and so on.
Question
The set of _____ consists of the natural numbers, their negatives, and zero.
Answer
Integers
Question
What is the formal definition of a rational number?
Answer
A number that can be expressed in the form $\frac{p}{q}$, where p and q are integers and q is not equal to zero ($q \ne 0$).
Question
In the rational number $\frac{p}{q}$, what is the term for 'p'?
Answer
The numerator.
Question
In the rational number $\frac{p}{q}$, what is the term for 'q'?
Answer
The denominator.
Question
Why is zero (0) considered a rational number?
Answer
Because it can be written as a fraction with a non-zero denominator, such as $\frac{0}{1}$, $\frac{0}{5}$, or $\frac{0}{-10}$.
Question
Is every natural number a rational number? Explain why.
Answer
Yes, because any natural number 'n' can be written as $\frac{n}{1}$.
Question
Is every integer a rational number? Explain why.
Answer
Yes, because any integer 'z' can be written as $\frac{z}{1}$.
Question
Under what condition is a rational number considered positive?
Answer
When its numerator and denominator have the same sign (both positive or both negative).
Question
Under what condition is a rational number considered negative?
Answer
When its numerator and denominator have opposite signs.
Question
What are the two conditions for a rational number $\frac{p}{q}$ to be in standard form?
Answer
1. The integers p and q have no common divisor other than 1. 2. The denominator q is positive.
Question
What is the standard form of the rational number $\frac{-21}{36}$?
Answer
The standard form is $\frac{-7}{12}$.
Question
What does the Closure Property of Addition for rational numbers state?
Answer
If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers, their sum $(\frac{a}{b} + \frac{c}{d})$ is also a rational number.
Question
What does it mean to say that the addition of any two rational numbers is commutative?
Answer
It means that for any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, the equation $\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$ is always true.
Question
What does the Associative Property of Addition for rational numbers state?
Answer
For any three rational numbers $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{e}{f}$, the equation $(\frac{a}{b} + \frac{c}{d}) + \frac{e}{f} = \frac{a}{b} + (\frac{c}{d} + \frac{e}{f})$ is always true.
Question
What is the additive identity for rational numbers?
Answer
The number zero (0) is the additive identity for rational numbers.
Question
What property states that for any rational number $\frac{a}{b}$, $\frac{a}{b} + 0 = 0 + \frac{a}{b} = \frac{a}{b}$?
Answer
The property of the existence of an additive identity.
Question
What is the additive inverse of a rational number $\frac{a}{b}$?
Answer
The additive inverse is $-\frac{a}{b}$.
Question
The sum of a rational number and its additive inverse is always equal to what value?
Answer
The additive identity, which is 0.
Question
What is the additive inverse of $\frac{7}{15}$?
Answer
The additive inverse is $-\frac{7}{15}$.
Question
What is the additive inverse of $-\frac{8}{11}$?
Answer
The additive inverse is $\frac{8}{11}$.
Question
Does the set of rational numbers satisfy the closure property under subtraction?
Answer
Yes, the difference of any two rational numbers is always a rational number.
Question
Is the subtraction of rational numbers commutative? Why or why not?
Answer
No, because in general, $\frac{a}{b} - \frac{c}{d} \ne \frac{c}{d} - \frac{a}{b}$.
Question
Is the subtraction of rational numbers associative? Why or why not?
Answer
No, because in general, $(\frac{a}{b} - \frac{c}{d}) - \frac{e}{f} \ne \frac{a}{b} - (\frac{c}{d} - \frac{e}{f})$.
Question
Why is there no identity element for subtraction in the set of rational numbers?
Answer
Because while $\frac{a}{b} - 0 = \frac{a}{b}$, the reverse $0 - \frac{a}{b} = -\frac{a}{b}$, which is not equal to $\frac{a}{b}$ (unless $\frac{a}{b}=0$).
Question
Does an inverse element exist for subtraction in the set of rational numbers?
Answer
No, an inverse for subtraction does not exist.
Question
How is the product of two rational numbers, $\frac{a}{b}$ and $\frac{c}{d}$, calculated?
Answer
By multiplying the numerators and multiplying the denominators: $\frac{a \times c}{b \times d}$.
Question
What does the Closure Property of Multiplication for rational numbers state?
Answer
If two rational numbers are multiplied together, the result is always a rational number.
Question
What property of multiplication states that for any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, $\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}$?
Answer
The Commutative Property of Multiplication.
Question
What does the Associative Property of Multiplication for rational numbers state?
Answer
For any three rational numbers $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{e}{f}$, $(\frac{a}{b} \times \frac{c}{d}) \times \frac{e}{f} = \frac{a}{b} \times (\frac{c}{d} \times \frac{e}{f})$.
Question
What is the multiplicative identity for rational numbers?
Answer
The number one (1) is the multiplicative identity.
Question
What property states that for any rational number $\frac{a}{b}$, $\frac{a}{b} \times 1 = 1 \times \frac{a}{b} = \frac{a}{b}$?
Answer
The property of the existence of a multiplicative identity.
Question
What is the multiplicative inverse of a rational number also known as?
Answer
Its reciprocal.
Question
What is the multiplicative inverse of a non-zero rational number $\frac{a}{b}$?
Answer
The multiplicative inverse is $\frac{b}{a}$.
Question
The product of a non-zero rational number and its multiplicative inverse is always equal to what value?
Answer
The multiplicative identity, which is 1.
Question
Which rational number does not have a multiplicative inverse?
Answer
Zero (0).
Question
What is the multiplicative inverse (reciprocal) of $\frac{3}{5}$?
Answer
The reciprocal is $\frac{5}{3}$.
Question
What is the multiplicative inverse (reciprocal) of $-8$?
Answer
The reciprocal is $-\frac{1}{8}$.
Question
The property $\frac{a}{b} \times (\frac{c}{d} + \frac{e}{f}) = (\frac{a}{b} \times \frac{c}{d}) + (\frac{a}{b} \times \frac{e}{f})$ is known as the _____ property.
Answer
Distributive
Question
How is the division of one rational number $\frac{a}{b}$ by another non-zero rational number $\frac{c}{d}$ performed?
Answer
By multiplying $\frac{a}{b}$ by the reciprocal of $\frac{c}{d}$, which is $\frac{a}{b} \times \frac{d}{c}$.
Question
Why is division by the rational number 0 not defined?
Answer
Because the reciprocal of 0 (which would be $\frac{1}{0}$) is undefined, and division involves multiplying by the reciprocal.
Question
Is the set of rational numbers closed under division?
Answer
No, because division by zero is not a rational number (it is undefined). The set of non-zero rational numbers is closed under division.
Question
Is the division of rational numbers commutative?
Answer
No, because in general, $\frac{a}{b} \div \frac{c}{d} \ne \frac{c}{d} \div \frac{a}{b}$.
Question
Is the division of rational numbers associative?
Answer
No, because in general, $(\frac{a}{b} \div \frac{c}{d}) \div \frac{e}{f} \ne \frac{a}{b} \div (\frac{c}{d} \div \frac{e}{f})$.
Question
Does an identity element exist for division in the set of rational numbers?
Answer
No, an identity element for division does not exist.
Question
Does an inverse element exist for division in the set of rational numbers?
Answer
No, an inverse for division does not exist.
Question
Describe the first step in representing a rational number like $\frac{p}{q}$ on a number line.
Answer
Divide each unit segment (e.g., between 0 and 1, 1 and 2) into 'q' equal parts.
Question
To represent the rational number $\frac{5}{3}$ on a number line, how many parts would you divide each unit into?
Answer
You would divide each unit into 3 equal parts.
Question
What is one method to find a rational number between two given rational numbers 'a' and 'b'?
Answer
Calculate their average: $\frac{a+b}{2}$.
Question
How many rational numbers exist between any two distinct rational numbers?
Answer
An infinite number of rational numbers.
Question
What is the 'LCM method' for finding a large number of rational numbers between $\frac{a}{b}$ and $\frac{c}{d}$?
Answer
Find equivalent fractions with a common denominator (the LCM), then multiply the numerators and denominators by a large enough integer to create gaps between the numerators.
Question
To find 5 rational numbers between two fractions, what is the minimum number you should multiply their numerators and denominators by (after finding a common denominator)?
Answer
You should multiply by at least $5 + 1 = 6$.
Question
For any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, the number $\frac{a+c}{b+d}$ lies _____ them.
Answer
between
Question
Which number is its own additive inverse?
Answer
Zero (0).
Question
Which two numbers are their own multiplicative inverses (reciprocals)?
Answer
1 and -1.