Download Some Basic Notions and Notations 1

Some Basic Notions and Notations 1
1S1 - F. Jaëck
R is the set of all real numbers (all numbers, including negative numbers, rational
numbers etc.)
R∗ is the set of all real numbers except 0
R+ is the set of all non negative real numbers
R∗+ is the set of all (strictly) positive real numbers
Z is the set of all integers ( including negative ones )
Given two real numbers a and b, the intervall [a; b] is the set containing all numbers
between a and b (one can write [a; b] = {x ∈ R such that a ≤ x ≤ b}). Of course you
can replace one square bracket [ or ] by a ( or a ) in order to indicate that you wish to
consider all the numbers in the intervall BUT the concerned extremity. For example
[a; b) contains all the numbers between a and b but b.
Recall that X 2 = 16 is an equation which has two solutions: X = 4 and X = −4. This
comes from the easy trick: X 2 = 16 ⇔ X 2 − 16 = 0 ⇔ (X − 4)(X + 4) = 0 and by the
zero product property...
Just above we used: A2 − B 2 = (A − B)(A + B) and we knew that 16 was 42 .
√
√
Recall that 3 = ( 3)2 and X 2 − α = X 2 − ( α)2 = · · · for all non negative α.
√
Substitution: suppose that you consider an expression such as (2 X − 3)2 which is
obviously valid for any X ≥ 0. Then you are allowed to replace X by any expression
representing a non negative number. Think of Y 2 + 248 for example. You can put this
new number, with a strange form, in the place of X. This gives WITH BRACKETS
√
when necessary: (2 Y 2 + 248 − 3)2 .
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