Asked by Ezile
Prove that √2 not a rational number
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Assume that √2 is a rational number, that is, it can be expressed as a ratio of two integers a and b in the lowest terms:
√2 = a/b
Squaring both sides, we get:
2 = a^2 / b^2
Multiplying both sides by b^2, we get:
2b^2 = a^2
This means that a^2 is even. Therefore, a must be even, because the square of an odd number is always odd. Let a = 2c, where c is an integer:
2b^2 = (2c)^2
Simplifying, we get:
2b^2 = 4c^2
Dividing both sides by 2, we get:
b^2 = 2c^2
This means that b^2 is even, and therefore b must also be even. However, this contradicts our initial assumption that a and b are in the lowest terms, because they both have a common factor of 2. Therefore, our assumption that √2 is a rational number must be false.
Hence, we can conclude that √2 is an irrational number.
√2 = a/b
Squaring both sides, we get:
2 = a^2 / b^2
Multiplying both sides by b^2, we get:
2b^2 = a^2
This means that a^2 is even. Therefore, a must be even, because the square of an odd number is always odd. Let a = 2c, where c is an integer:
2b^2 = (2c)^2
Simplifying, we get:
2b^2 = 4c^2
Dividing both sides by 2, we get:
b^2 = 2c^2
This means that b^2 is even, and therefore b must also be even. However, this contradicts our initial assumption that a and b are in the lowest terms, because they both have a common factor of 2. Therefore, our assumption that √2 is a rational number must be false.
Hence, we can conclude that √2 is an irrational number.
Answered by
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There are many properties of real numbers, but some of the most basic ones are:
1. Commutative property of addition: a + b = b + a
This means that we can add two real numbers in any order and the result will be the same.
2. Associative property of addition: (a + b) + c = a + (b + c)
This means that we can group real numbers in any way we want when adding them and the result will be the same.
3. Identity property of addition: a + 0 = a
This means that if we add zero to any real number, the result is the same real number.
4. Inverse property of addition: a + (-a) = 0
This means that the sum of any real number and its opposite is zero.
5. Commutative property of multiplication: ab = ba
This means that we can multiply two real numbers in any order and the result will be the same.
6. Associative property of multiplication: (ab)c = a(bc)
This means that we can group real numbers in any way we want when multiplying them and the result will be the same.
7. Identity property of multiplication: a x 1 = a
This means that if we multiply any real number by one, the result is the same real number.
8. Inverse property of multiplication: a x 1/a = 1
This means that the product of any real number and its reciprocal is one, as long as the real number is not zero.
9. Distributive property: a(b + c) = ab + ac
This means that we can distribute the multiplication of a real number to the sum of two or more real numbers and the result will be the same.
1. Commutative property of addition: a + b = b + a
This means that we can add two real numbers in any order and the result will be the same.
2. Associative property of addition: (a + b) + c = a + (b + c)
This means that we can group real numbers in any way we want when adding them and the result will be the same.
3. Identity property of addition: a + 0 = a
This means that if we add zero to any real number, the result is the same real number.
4. Inverse property of addition: a + (-a) = 0
This means that the sum of any real number and its opposite is zero.
5. Commutative property of multiplication: ab = ba
This means that we can multiply two real numbers in any order and the result will be the same.
6. Associative property of multiplication: (ab)c = a(bc)
This means that we can group real numbers in any way we want when multiplying them and the result will be the same.
7. Identity property of multiplication: a x 1 = a
This means that if we multiply any real number by one, the result is the same real number.
8. Inverse property of multiplication: a x 1/a = 1
This means that the product of any real number and its reciprocal is one, as long as the real number is not zero.
9. Distributive property: a(b + c) = ab + ac
This means that we can distribute the multiplication of a real number to the sum of two or more real numbers and the result will be the same.
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