What is a real number?
Posted: Fri Jan 10, 2020 8:57 am
Let's introduce some Mathematical Statements and the quantifiers before diving into math deeply.
∀ - "for each", "for all"
∃ - "there exists some"
⇒ - "result" or "conclusion"
⇔ - "equivalence"
def - "definition"
: - "such that"
Examples:
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∀ \: a>0 \: ∃ \: x>0 : x^2=a
Translation of this expression into human language looks like
for each a>0 there exists some x>0 such that x^2=a
Let's consider some real numbers a>0:
1)a=4 then x=\sqrt{4}=2
2)a=2 then x=\sqrt{2}
3)a=0.5 then x=\sqrt{0.5}
................
these statements will be true for any real number a>0 which you can pick up
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a>b>0 ⇒ a^2 > b^2
Let's consider some real numbers a>b>0:
1)a=3,b=2
3>2 ⇒ 9>4
2)a=5.2,b=4
5.2>4 ⇒ 5.2^2>16
................
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∀ - "for each", "for all"
∃ - "there exists some"
⇒ - "result" or "conclusion"
⇔ - "equivalence"
def - "definition"
: - "such that"
Examples:
-------------------------------------------------
∀ \: a>0 \: ∃ \: x>0 : x^2=a
Translation of this expression into human language looks like
for each a>0 there exists some x>0 such that x^2=a
Let's consider some real numbers a>0:
1)a=4 then x=\sqrt{4}=2
2)a=2 then x=\sqrt{2}
3)a=0.5 then x=\sqrt{0.5}
................
these statements will be true for any real number a>0 which you can pick up
------------------------------------------------
a>b>0 ⇒ a^2 > b^2
Let's consider some real numbers a>b>0:
1)a=3,b=2
3>2 ⇒ 9>4
2)a=5.2,b=4
5.2>4 ⇒ 5.2^2>16
................
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