Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero. They include both positive and negative integers, fractions, and terminating or repeating decimals.
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Rational numbers can be represented on the number line and are dense, meaning there are infinitely many rational numbers between any two given rational numbers.
All integers are rational numbers, as they can be expressed as a ratio with a denominator of 1.
The set of rational numbers is closed under the four basic arithmetic operations: addition, subtraction, multiplication, and division (except division by zero).
Rational numbers can be used to simplify and perform operations with square roots, as they can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers.
The distributive property can be applied to rational numbers, allowing for the simplification of expressions involving rational numbers and variables.
Review Questions
Explain how the properties of rational numbers can be used to simplify and perform operations with square roots.
Rational numbers can be used to simplify and perform operations with square roots because they can be expressed as a ratio of two integers. This allows for the representation of square roots in the form $\frac{a}{b}$, where $a$ and $b$ are integers. By using the properties of rational numbers, such as the ability to add, subtract, multiply, and divide (except division by zero), square root expressions can be simplified and manipulated more easily.
Describe how the distributive property can be applied to expressions involving rational numbers and variables.
The distributive property can be used to simplify expressions involving rational numbers and variables. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the results. For example, $\frac{1}{2}(3x + 4y) = \frac{1}{2}(3x) + \frac{1}{2}(4y)$. By applying the distributive property, the expression can be broken down into simpler, more manageable terms that involve rational numbers and variables.
Analyze the significance of the set of rational numbers being closed under the four basic arithmetic operations in the context of simplifying and manipulating algebraic expressions.
The fact that the set of rational numbers is closed under the four basic arithmetic operations (addition, subtraction, multiplication, and division, except division by zero) is crucial for simplifying and manipulating algebraic expressions. This property ensures that the result of any arithmetic operation performed on rational numbers will also be a rational number. This allows for the simplification and transformation of expressions involving rational numbers without the need to transition to a different number system, making the process more efficient and straightforward. The closure property of rational numbers under these operations is a fundamental characteristic that enables the effective manipulation of algebraic expressions.