Podcast
Questions and Answers
What property is described by the statement: "For every real number r, we have (-r + r) = 0"?
What property is described by the statement: "For every real number r, we have (-r + r) = 0"?
Which of the following is an irrational number?
Which of the following is an irrational number?
Which property is demonstrated by the statement: "If r and s are any two real numbers, then r * (s - t) = r * s - r * t for all real numbers t"?
Which property is demonstrated by the statement: "If r and s are any two real numbers, then r * (s - t) = r * s - r * t for all real numbers t"?
What is the purpose of the real number line?
What is the purpose of the real number line?
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Which operation with real numbers follows the closure property and associativity?
Which operation with real numbers follows the closure property and associativity?
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Which of the following statements about rational numbers is true?
Which of the following statements about rational numbers is true?
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Which of the following properties allows us to combine like terms in an expression?
Which of the following properties allows us to combine like terms in an expression?
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If a and b are real numbers, which property guarantees that a * b is also a real number?
If a and b are real numbers, which property guarantees that a * b is also a real number?
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Which property allows us to factor out common terms in an expression?
Which property allows us to factor out common terms in an expression?
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Which of the following is an irrational number?
Which of the following is an irrational number?
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Which property allows us to change the order of operations in a multiplication expression?
Which property allows us to change the order of operations in a multiplication expression?
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Which property states that the sum of any real number and zero is that same real number?
Which property states that the sum of any real number and zero is that same real number?
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Study Notes
Real Numbers
Real numbers form the set of all numbers that can represent measurements and quantities in mathematics and physics. They include both rational and irrational numbers, which together span an infinite sequence of decimals.
Properties of Real Numbers
Properties of real numbers include:
- Closure under addition: If two real numbers
aandbexist, their sumc = a + bis also a real number. - Closure under multiplication: If two real numbers
aandbexist, their productd = a * bis also a real number. - Distributive property of multiplication over addition: For any three real numbers
a,b, andc, we have(a + b) * c = a * c + b * c. - Associative property of addition: For any four real numbers
a,b,c, andd, we have(a + b) + c = a + (b + c). - Associative property of multiplication: For any five real numbers
a,b,c,d, ande, we have(a * b) * c * d * e = a * (b * c * d * e). - Commutativity of addition: For any two real numbers
aandb, we havea + b = b + a. - Commutativity of multiplication: For any two real numbers
aandb, we havea * b = b * a. - Identity element for addition: There exists a unique real number
0such that for every real numberr, we haver + 0 = r. - Inverse elements for addition: For each real number
r, there exists another real number(-r)such thatr + (-r) = 0. - Identity element for multiplication: There exists a unique real number
1such that for every real numberr, we haver * 1 = r. - Multiplicative identity: For each real number
r, there exists another real number(1/r)such thatr * (1/r) = 1. - Distributivity of multiplication over subtraction: If
randsare any two real numbers, thenr * (s - t) = r * s - r * tfor all real numberst. - Inverse property of zero relative to addition: For every nonzero real number
r, we have(-r + r) = 0. - Associativity of taking a reciprocal: Given three real numbers
a,b, andcwitha ≠ 0,b ≠ 0, andc ≠ 0, ifd = a/bande = b/c, thena/b = c/d.
Operations with Real Numbers
Operations with real numbers include:
Addition
The sum of two real numbers is also a real number. The sum follows the closure property under addition, meaning it can be added with other real numbers without going out of the set.
Subtraction
Subtracting one real number from another results in another real number. This operation also follows the closure property.
Multiplication
Multiplying two real numbers together also results in another real number. Multiplication preserves the properties of closure and associativity.
Division by Non-Zero Real Numbers
Dividing a real number by another non-zero real number also results in another real number. Division follows the closure property and associativity.
Real Number Line
The real number line is a way of visualizing and organizing all real numbers on a single line. It represents that any arithmetic operation between two real numbers will always result in a unique real number as output.
Rational Numbers
Rational numbers are those which can be represented as the quotient or fraction p/q of two integers, with denominator q not equal to zero. They include whole numbers (e.g., 2, 3, 4, etc.) and fractions (e.g., 1/2, 2/3, 3/4, etc.). Every rational number has finite decimal representation and lies on the real number line.
Irrational Numbers
Irrational numbers cannot be expressed as a ratio of two integers. The most famous irrational number is pi (π), but there are many more. Pi is approximately 3.14159 and continues infinitely without repeating after the decimal point. Other examples of irrational numbers include square roots of positive numbers that are not perfect squares, such as √2 (approximately 1.41421). These numbers do not have a finite decimal representation and lie on the real number line.
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Description
Test your knowledge about real numbers, including rational and irrational numbers, and their properties such as closure, distributivity, and commutativity. Explore operations with real numbers like addition, subtraction, multiplication, and division. Learn about the real number line, rational numbers, and irrational numbers.
