02 - קבוצות של מספרים, הוכחות מתמטיות

Number Systems and Notation

The lecture introduces number systems, mathematical notation, and basic proof techniques as preparation for studying limits and other topics in calculus.

• Natural numbers ((\mathbb{N})): Presented as (1, 2, 3, \ldots), the numbers used for counting.
• Integers ((\mathbb{Z})): Include the natural numbers, zero, and negative whole numbers. Thus, (\mathbb{N} \subseteq \mathbb{Z}).
• Rational numbers ((\mathbb{Q})): Numbers that can be written as (m/n), where (m,n) are integers and (n \ne 0). Examples include (1/3), (-1/5), and (973/4). Every integer is rational, so (\mathbb{Z} \subseteq \mathbb{Q}).
• Real numbers ((\mathbb{R})): Informally, all numbers on the number line. The lecturer emphasizes that this is an intuitive description; a rigorous definition is more involved and will be developed later. The course works within the real numbers.
• Complex numbers ((\mathbb{C})): Mentioned as a broader number system, but not a focus of the course.

The lecturer stresses that mathematical symbols and notation form a language that takes time to learn. Each expansion of the number system allows operations that were not possible in the previous one. For example, (2-4) is not a natural number, but it is an integer; (3/2) is rational but not an integer.

Proofs and Logical Statements

Trying examples can help build intuition, but examples alone do not prove a statement for every number.

The lecture demonstrates that the square of an even number is even. Write an even number as (n=2k). Then

[ n^2=(2k)^2=4k^2=2(2k^2), ]

which is divisible by (2), and therefore even.

The lecturer introduces implication notation: (A \Rightarrow B) means that (A) implies (B). A statement and its converse are not generally equivalent. For example, “if (x>5), then (x>2)” is true, but its converse is not.

An implication is logically equivalent to its contrapositive:

[ A \Rightarrow B \quad\text{is equivalent to}\quad \text{not }B \Rightarrow \text{not }A. ]

For example, the claim that if (n^2) is even, then (n) is even can be proved by contrapositive. If (n) is odd, write (n=2k+1). Then

[ n^2=(2k+1)^2=4k^2+4k+1, ]

which is odd.

Proving That (\sqrt{2}) Is Irrational

The lecturer presents a proof by contradiction:

1. Assume, contrary to the claim, that (\sqrt{2}) is rational.
2. Write (\sqrt{2}=m/n) as a fraction in lowest terms, where (m,n) are integers and (n \ne 0).
3. Squaring gives (m^2=2n^2), so (m^2) is even. Therefore, (m) is even; write (m=2k).
4. Substituting gives (4k^2=2n^2), hence (n^2=2k^2). Thus, (n^2), and therefore (n), is also even.
5. Both (m) and (n) are divisible by (2), contradicting the assumption that the fraction was in lowest terms.
6. The assumption must be false, so (\sqrt{2}) is irrational. Consequently, (\mathbb{Q}) is a proper subset of (\mathbb{R}).

The lecturer also recounts a traditional anecdote about the Pythagoreans reacting strongly to the discovery of irrational numbers.

Density of Rational Numbers

Between any two distinct real numbers (x<y), there is a rational number. The lecturer notes that the analogous statement also holds for irrational numbers.

The proof uses the Archimedean principle, which says that natural numbers can be found larger than any given real number:

1. Choose a sufficiently large natural number (n) so that (n>1/(y-x)). Then (ny-nx>1).
2. There is an integer (m) between (nx) and (ny).
3. Dividing by (n) gives a rational number (m/n) between (x) and (y).

Thus, rational numbers occur between every pair of distinct real numbers, rather than being confined to separate intervals on the number line. The same density property holds for irrational numbers.

Further Note: Fields

The lecturer briefly describes (\mathbb{Q}) and (\mathbb{R}) as fields: number systems with familiar arithmetic operations and properties, such as addition, multiplication, additive inverses, and division by nonzero numbers. These properties are studied systematically in algebra.

Speakers and Sources

• Speaker: The course lecturer, identified in the video metadata as Dr. Aviv Tzancor.
• Historical figures and ideas referenced: Pythagoras and the Pythagorean school; the Archimedean principle.