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Ch. 10: What is a number?

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MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space) that leaves the object apparently unchanged.

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What is a rigid motion of the plane? Of space? This doesn’t make sense until we decide:

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MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space) that leaves the object apparently unchanged. What is a rigid motion of the plane? Of space? This doesn’t make sense until we decide: What is the plane? What is space? This doesn’t make sense until we decide:

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MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space) that leaves the object apparently unchanged. What is a rigid motion of the plane? Of space? This doesn’t make sense until we decide: What is the plane? What is space? This doesn’t make sense until we decide: Since the plane is made of pairs (x,y) of numbers and space is made of triples (x,y,z) of numbers, this doesn’t make sense until we decide: What is a number?

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MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space) that leaves the object apparently unchanged. What is a rigid motion of the plane? Of space? This doesn’t make sense until we decide: What is the plane? What is space? This doesn’t make sense until we decide: What is a number? Since the plane is made of pairs (x,y) of numbers and space is made of triples (x,y,z) of numbers, this doesn’t make sense until we decide: Let’s start our backtrack here!

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What is a number?

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers”

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Good for counting sheep

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…}

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5. …or like this: 300 = 10×30

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5. …or like this: 300 = 10×30 = 2×5×3×10

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. For example, 20 is not prime because 20 = 5×4. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5. …or like this: 300 = 10×30 = 2×5×3×10 = 2×5×3×2×5.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Including the primes {2, 3, 5, 7, 11, 13, 17, 19…} DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers. THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes. For example, the prime factorization of 300 is: 300 = 2×2×3×5×5. We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down… …like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5. …or like this: 300 = 10×30 = 2×5×3×10 = 2×5×3×2×5. Same answer after re-ordering! Same answer after re-ordering! This is the hard part to prove!

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000?

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000? 90000 = 300×300

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000? 90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000? 90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5 (reorder) = 2×2×2×2×3×3×5×5×5×5.

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000? 90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5 (reorder) = 2×2×2×2×3×3×5×5×5×5. NOTE: 300 2 has twice as many of each prime as 300 has.

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The prime factorization of a square The prime factorization of 300 is: 300 = 2×2×3×5×5. What is the prime factorization of 300 2 = 300×300 = 90000? 90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5 (reorder) = 2×2×2×2×3×3×5×5×5×5. NOTE: 300 2 has twice as many of each prime as 300 has. FACT: The square of any natural number greater than 1 has an even number of occurrences of each prime in its prime factorization.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Good for counting your sheep But has limits…

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Good for counting positive and negative sheep… But still has limits…

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} “the rational numbers” Examples: 1/2, -17/4, 7/1, -15/38

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} “the rational numbers” synonyms: quotient, fraction, ratio, rational number

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} “the rational numbers” synonyms: quotient, fraction, ratio, rational number Some fractions are the same as other, like 2/3 = 4/6.

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} “the rational numbers” synonyms: quotient, fraction, ratio, rational number Some fractions are the same as other, like 2/3 = 4/6. General rule: a/b = c/d whenever ad = bc. This rule is an addendum to the definition of Q

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What is a number? N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers”

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers” What is a number? Good for counting parts of sheep… But are the rational numbers enough?

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers” What is a number? “All numbers are rational.” Greek Mathematician, 500 BC

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers” What is a number? “All numbers are rational.” Greek Mathematician, 500 BC SOME REASONS FOR THIS BELIEF: (1)Finite divisibility of all matter (2)Rational numbers are gifts from the gods (3)They can measure arbitrarily small lengths.

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers” What is a number? “All numbers are rational.” Greek Mathematician, 500 BC

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers” What is a number? “All numbers are rational.” Greek Mathematician, 500 BC “Which rational number squares to 2?”

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“All numbers are rational.” Greek Mathematician, 500 BC “Which rational number squares to 2?” THEOREM: There is no rational number that squares to 2.

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“All numbers are rational.” Greek Mathematician, 500 BC “Which rational number squares to 2?” THEOREM: There is no rational number that squares to 2. “ὦ περίσσωμα”

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“All numbers are rational.” Greek Mathematician, 500 BC “Which rational number squares to 2?” THEOREM: There is no rational number that squares to 2. Greek for “Oh shit.” “ὦ περίσσωμα”

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. Today, we call this length “the square root of 2,” And we refer to it as an “irrational number.” But to the Greek mathematicians in 500 BC, “number” meant “rational number”.

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF:

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2.

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2 This has an even number of 2s in its prime factorization. (twice as many as p has)

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2 This has an even number of 2s in its prime factorization. This has an odd number of 2s in its prime factorization. (twice as many as p has) (one more than twice as many as q has)

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2 This has an even number of 2s in its prime factorization. This has an odd number of 2s in its prime factorization. But the prime factorization of a number is unique. If the left and right side of this equation really equaled each other, there would not be a difference between their prime factorizations.

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is: (p/q) 2 = 2 ↔ p 2 /q 2 = 2 ↔ p 2 = 2×q 2 This has an even number of 2s in its prime factorization. This has an odd number of 2s in its prime factorization. But the prime factorization of a number is unique. If the left and right side of this equation really equaled each other, there would not be a difference between their prime factorizations. SUMMARY: We assumed that there is a rational number that squares to 2, but this assumption lead to a contradiction; therefore, this assumption must be wrong!

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Greek Mathematician, 500 BC THEOREM: There is no rational number that squares to 2. “I need a new kind of number.”

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237…

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… indicates a continuation of the pattern (an unending string of 6s) indicates some unending string of digits, which might not have a pattern that you understand.

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R.

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2?

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = ???

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = ??? The answer is less than 12.75 – 12.74 = 0.01

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = ??? The answer is less than 12.75 – 12.74 = 0.01 and less than 12.750 – 12.749 = 0.001

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = ??? The answer is less than 12.75 – 12.74 = 0.01 and less than 12.750 – 12.749 = 0.001 and less than 12.7500 – 12.7499 = 0.0001

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = ??? The answer is less than 12.75 – 12.74 = 0.01 and less than 12.750 – 12.749 = 0.001 and less than 12.7500 – 12.7499 = 0.0001 and less than 12.75000 – 12.74999 = 0.00001 and so on…

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = 0 The answer is less than 12.75 – 12.74 = 0.01 and less than 12.750 – 12.749 = 0.001 and less than 12.7500 – 12.7499 = 0.0001 and less than 12.75000 – 12.74999 = 0.00001 and so on… THE ANSWER MUST BE ZERO!

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2? GUESS: 12.75000000000 … – 12.74999999999 … = 0 CONCLUSION: 12.75000000000 … = 12.74999999999 … If they differ by zero, they must be the same real number! (This is similar to how 2/3 and 2/6 are two different ways of writing the same rational number)

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Today, we put fractions like 2/3 and lengths like √2 on equal footing by them as decimal expressions. 2/3 = 0.66666666… and √2 = 1.41421356237… DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R. CONCLUSION: 12.75000000000 … = 12.74999999999 … REAL REDUNDANCY RULE: A digit (other than 9) followed by an unending string of 9s can be replaced by the next larger digit followed by an unending string of 0s. There are no other redundancies among real numbers.

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N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rational numbers” with the understanding that a/b = c/d whenever ad = bc. R = {all decimal expressions} “the real numbers” with the understanding that the real redundancy rule determines which are the same. What is a number?

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Picture each real number as a point on the real number line. 0123456-2-3-4-5-6

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Picture each real number as a point on the real number line. 0123456-2-3-4-5-6 -13/5π13/3

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Picture each real number as a point on the real number line. 0123456-2-3-4-5-6 -13/5π13/3 The number line is like a yard stick, which matches the ancient Greek idea that numbers should represent all possible lengths… …like this one:

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. 0123456-2-3-4-5-6

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. 0123456-2-3-4-5-6 √2 = 1.41421356237 … How do these digits locate the number?

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. 0123456-2-3-4-5-6 √2 = 1.41421356237 …

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. 12 √2 = 1.41421356237 …

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 12

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.02.0

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.02.0 0123456789

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.02.0 0123456789

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 4

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 4

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.41.5

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.41.5 0123456789

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.41.5 0123456789

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Picture each real number as a point on the real number line. The digits of a decimal expression locate the number on the real number line. √2 = 1.41421356237 … 1.41.5 0123456789 … and so on. Each next digit provides a ten fold increase in the accuracy with which we know the number’s location on the real number line.

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Which real numbers are rational?

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Q: Q: Use long division to convert the fraction 3/7 into a decimal expression.

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Which real numbers are rational? Q: Q: Use long division to convert the fraction 3/7 into a decimal expression. A: A: 3/7 = 0.428571428571428517….

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Which real numbers are rational? Q: Q: Use long division to convert the fraction 3/7 into a decimal expression. A: A: 3/7 = 0.428571428571428517…. With long division, as soon as the remainder repeats, the digits always begin repeating. How are you guaranteed that the remainder will repeat?

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Which real numbers are rational? Q: Q: Use long division to convert the fraction 3/7 into a decimal expression. A: A: 3/7 = 0.428571428571428517…. With long division, as soon as the remainder repeats, the digits always begin repeating. How are you guaranteed that the remainder will repeat? FACT: Long division converts any fraction into an eventually repeating decimal expression. FACT: Long division converts any fraction into an eventually repeating decimal expression.

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Which real numbers are rational? Q: Q: Use long division to convert the fraction 3/7 into a decimal expression. A: A: 3/7 = 0.428571428571428517…. With long division, as soon as the remainder repeats, the digits always begin repeating. How are you guaranteed that the remainder will repeat? FACT: Long division converts any fraction into an eventually repeating decimal expression. FACT: Long division converts any fraction into an eventually repeating decimal expression. Like this: 103.45873065306530653065… or like this: 4/5 = 0.80000000000…

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Which real numbers are rational? Q: Q: Convert N = 0.428571428571428517…. into a fraction.

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Which real numbers are rational? Q: Q: Convert N = 0.428571428571428517…. into a fraction. TRICK: Since N has a 6-digit repeating string, we multiply it by 1,000,000 (which has 6 zeros), and then subtract: 1000000×N = 428571. 428571428571 … N = 0. 428571428571 … subtract 999999×N = 428571.000000000000 … We learn that N = 428571/999999 (which reduces to N = 3/7).

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Which real numbers are rational? Q: Q: Convert N = 0.428571428571428517…. into a fraction. TRICK: Since N has a 6-digit repeating string, we multiply it by 1,000,000 (which has 6 zeros), and then subtract: 1000000×N = 428571. 428571428571 … N = 0. 428571428571 … subtract 999999×N = 428571.000000000000 … We learn that N = 428571/999999 (which reduces to N = 3/7). FACT: This trick works to convert any eventually repeating decimal expression into a fraction.

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Which real numbers are rational? FRACTION Eventually repeating decimal expression Long division Subtraction trick

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Which real numbers are rational? FRACTION Eventually repeating decimal expression Long division Subtraction trick THEOREM: A real number is rational precisely when its decimal expression is eventually repeating.

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Which real numbers are rational? FRACTION Eventually repeating decimal expression Long division Subtraction trick THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating.

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Which real numbers are rational? FRACTION Eventually repeating decimal expression Long division Subtraction trick THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating. COROLLARY: √2 = 1.41421356237 … is NOT eventually repeating! (It’s surprising that we could prove this without understanding the digits)

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Which real numbers are rational? THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating. COROLLARY: √2 = 1.41421356237 … is NOT eventually repeating! (It’s surprising that we could prove this without understanding the digits) Q: Q: Is this rational or irrational: 0.01001000100001000001… (pattern continues)

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Which real numbers are rational? THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating. COROLLARY: √2 = 1.41421356237 … is NOT eventually repeating! (It’s surprising that we could prove this without understanding the digits) Q: Q: Is this rational or irrational: 0.01001000100001000001… (pattern continues) A: A: Irrational. The pattern is more complicated than “eventually repeating”.

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Which real numbers are rational? THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating. COROLLARY: √2 = 1.41421356237 … is NOT eventually repeating! (It’s surprising that we could prove this without understanding the digits) Q: Q: Is this rational or irrational: 0.01001000100001000001… (pattern continues) A: A: Irrational. The pattern is more complicated than “eventually repeating”. Make up your own irrational numbers.

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Which real numbers are rational? THEOREM: A real number is rational precisely when its decimal expression is eventually repeating. That is: Every rational number has a decimal expression that’s eventually repeating. Every irrational number has a decimal expression that’s not eventually repeating. COROLLARY: √2 = 1.41421356237 … is NOT eventually repeating! (It’s surprising that we could prove this without understanding the digits) Q: Q: Is this rational or irrational: 0.01001000100001000001… (pattern continues) A: A: Irrational. The pattern is more complicated than “eventually repeating”. The famous numbers π and e are irrational, but this is hard to prove. Make up your own irrational numbers.

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How many primes are there?

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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … How long until we reach the largest prime? Is there a largest prime?

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers.

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. We know: (1) There are infinitely many natural numbers. (2) Each natural number is a product of primes. But this does not prove Euclid’s Theorem. After all, infinitely many things can be built from unlimited supplies of finitely many different building blocks.

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes.

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. In Andy’s list, the next prime, 17, is missing. But “choosing the next prime” is not a good general procedure, since it begs the question of whether there always is a next prime. In Andy’s list, the next prime, 17, is missing. But “choosing the next prime” is not a good general procedure, since it begs the question of whether there always is a next prime.

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031 This is bigger than anything on Andy’s list It would be nice if numbers formed this way were always prime. This is bigger than anything on Andy’s list It would be nice if numbers formed this way were always prime.

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031 = 59×509 But it’s not prime. Here is its prime factorization

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031 = 59×509 But here are two primes that are missing from Andy’s list!

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13. Andy How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes. First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031 = 59×509 But here are two primes that are missing from Andy’s list! 59 and 509 divide evenly into 30031. Andy’s primes all leave a remainder 1 when divided into 30031. Thus, 59 and 509 are different from Andy’s primes!

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. PROOF: Given any finite list of primes p 1, p 2, p 3, p 4, …, p n we can find a prime that is missing from the list as follows:

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. PROOF: Given any finite list of primes p 1, p 2, p 3, p 4, …, p n we can find a prime that is missing from the list as follows: First multiply them and add 1: N = p 1 ×p 2 ×p 3 ×p 4 ×…×p n + 1

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. PROOF: Given any finite list of primes p 1, p 2, p 3, p 4, …, p n we can find a prime that is missing from the list as follows: First multiply them and add 1: N = p 1 ×p 2 ×p 3 ×p 4 ×…×p n + 1 Each prime factor of N is a prime that’s missing from the list!

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. PROOF: Given any finite list of primes p 1, p 2, p 3, p 4, …, p n we can find a prime that is missing from the list as follows: First multiply them and add 1: N = p 1 ×p 2 ×p 3 ×p 4 ×…×p n + 1 Each prime factor of N is a prime that’s missing from the list! (because they divide evenly into N, while everything on the list leaves a remainder 1 when divided into N.)

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. PROOF: Given any finite list of primes p 1, p 2, p 3, p 4, …, p n we can find a prime that is missing from the list as follows: First multiply them and add 1: N = p 1 ×p 2 ×p 3 ×p 4 ×…×p n + 1 Each prime factor of N is a prime that’s missing from the list! (because they divide evenly into N, while everything on the list leaves a remainder 1 when divided into N.) Thus, no finite list of primes could ever be complete!

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? Are primes in abundance, or are they a rare breed? Do most United States citizens have a prime social security number, or very few?

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 10 100 1,000 10,000 100,000 1,000,000

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 100 1,000 10,000 100,000 1,000,000 Up to N=10:1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 4/10 = 0.40 (40% are prime)

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 100 0.25 1,000 10,000 100,000 1,000,000 Up to N=100: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100. 25/100 = 0.25 (25% are prime)

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 1000.25 1,0000.168 10,0000.1229 100,0000.09592 1,000,0000.078498

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 1000.25 1,0000.168 10,0000.1229 100,0000.09592 1,000,0000.078498 What is the pattern? How does the pattern continue for larger and larger N? What is the pattern? How does the pattern continue for larger and larger N?

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 1000.25 1,0000.168 10,0000.1229 100,0000.09592 1,000,0000.078498

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How many primes are there? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … EUCLID’S THEOREM: There are infinitely many prime numbers. After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers? NFraction of numbers up to N that are prime 100.40 1000.25 1,0000.168 10,0000.1229 100,0000.09592 1,000,0000.078498 1/(twice 7) = 1/14 = 0.0714… 7 digits

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Famous unsolved questions about primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Q: Q: Are there infinitely many “twin primes”? consecutive odd numbers that are both prime, like: 11 & 13, 29 & 31, 41 & 43 …

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Famous unsolved questions about primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Q: Q: Are there infinitely many “twin primes”? consecutive odd numbers that are both prime, like: 11 & 13, 29 & 31, 41 & 43 Q: Q: (Goldbach) Can every positive even number larger than 2 be written as a sum of two primes? 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 10 = 5 + 5 12 = 7 + 5 14 = 7 + 7 …

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