Properties of Real Numbers

Questions and Answers

What is the Commutative Property of real numbers, and how does it relate to addition and multiplication?

The Commutative Property states that the order of numbers does not change the result, i.e., a + b = b + a and a × b = b × a.

What is the Distributive Property of real numbers, and how does it relate to multiplication and addition?

The Distributive Property states that multiplication distributes over addition, i.e., a × (b + c) = a × b + a × c.

What is the difference between rational and irrational numbers, and provide examples of each?

Rational numbers can be expressed as a fraction (p/q) where p and q are integers and q ≠ 0, while irrational numbers cannot be expressed as a fraction. Examples of rational numbers include 3/4, 22/7, and 1/2, while examples of irrational numbers include π, e, and √2.

What is the Fundamental Theorem of Arithmetic, and how does it relate to prime numbers?

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a product of prime numbers in a unique way. Signup and view all the answers

What is the method of contradiction used to prove irrationality, and provide an example?

The method of contradiction assumes that the number is rational and shows that it leads to a contradiction. For example, to prove that √2 is irrational, assume √2 is rational, so √2 = p/q where p and q are integers, and then show that it leads to a contradiction. Signup and view all the answers

Prove that √3 is irrational using the method of contradiction.

Assume √3 is rational, so √3 = p/q where p and q are integers. Square both sides to get 3 = p^2/q^2. Then, p^2 = 3q^2, which implies that p^2 is a multiple of 3. Since p^2 is a multiple of 3, p must be a multiple of 3. Let p = 3k, then p^2 = 9k^2. Substituting this into the equation, we get 9k^2 = 3q^2, which implies that q^2 is a multiple of 3. This implies that q is also a multiple of 3, which contradicts the assumption that p and q are integers and have no common factors. Signup and view all the answers

Study Notes

Properties of Real Numbers

Commutative Property: The order of numbers does not change the result.
- a + b = b + a
- a × b = b × a
Associative Property: The order in which numbers are grouped does not change the result.
- (a + b) + c = a + (b + c)
- (a × b) × c = a × (b × c)
Distributive Property: Multiplication distributes over addition.
- a × (b + c) = a × b + a × c
Additive Identity: The number 0 does not change the result when added.
- a + 0 = a
Multiplicative Identity: The number 1 does not change the result when multiplied.
- a × 1 = a
Additive Inverse: Each number has an opposite that, when added, results in 0.
- a + (-a) = 0
Multiplicative Inverse: Each non-zero number has a reciprocal that, when multiplied, results in 1.
- a × (1/a) = 1

Rational and Irrational Numbers

Rational Numbers: Can be expressed as a fraction (p/q) where p and q are integers and q ≠ 0.
- Examples: 3/4, 22/7, 1/2
Irrational Numbers: Cannot be expressed as a fraction.
- Examples: π, e, √2

Fundamental Theorem of Arithmetic

Every positive integer can be expressed as a product of prime numbers in a unique way.
- Example: 60 = 2 × 2 × 3 × 5

Proving Irrationality

Method of Contradiction: Assume the number is rational and show that it leads to a contradiction.
Example: Proving √2 is Irrational
- Assume √2 is rational, so √2 = p/q where p and q are integers.
- Square both sides: 2 = (p/q)²
- Simplify: 2 = p²/q²
- Multiply both sides by q²: 2q² = p²
- Since the left side is even, the right side must also be even, so p is even.
- Let p = 2k, then 2q² = (2k)²
- Simplify: 2q² = 4k²
- Divide both sides by 2: q² = 2k²
- Since the left side is odd, the right side must also be odd, so q is odd.
- This is a contradiction, since we assumed p and q are integers, and p is even while q is odd.
- Therefore, √2 is irrational.

Properties of Real Numbers

The Commutative Property states that the order of numbers does not change the result, e.g. a + b = b + a and a × b = b × a.
The Associative Property states that the order in which numbers are grouped does not change the result, e.g. (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
The Distributive Property states that multiplication distributes over addition, e.g. a × (b + c) = a × b + a × c.
The Additive Identity is 0, which does not change the result when added, e.g. a + 0 = a.
The Multiplicative Identity is 1, which does not change the result when multiplied, e.g. a × 1 = a.
The Additive Inverse is the opposite of each number, which when added, results in 0, e.g. a + (-a) = 0.
The Multiplicative Inverse is the reciprocal of each non-zero number, which when multiplied, results in 1, e.g. a × (1/a) = 1.

Rational and Irrational Numbers

Rational Numbers are numbers that can be expressed as a fraction (p/q) where p and q are integers and q ≠ 0, e.g. 3/4, 22/7, 1/2.
Irrational Numbers are numbers that cannot be expressed as a fraction, e.g. π, e, √2.

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a product of prime numbers in a unique way, e.g. 60 = 2 × 2 × 3 × 5.

Proving Irrationality

The Method of Contradiction is used to prove that a number is irrational, by assuming it is rational and showing that it leads to a contradiction.
The Method of Contradiction is used to prove that √2 is irrational, by assuming √2 is rational, and then showing that it leads to a contradiction, thus proving that √2 is irrational.