Numbers should be well distinguished by high school sophomores. Thus, this second grade math course on recall of middle school concepts must be acquired.
In this lesson, the objectives will be:
- Know how to distinguish the exact value of a number from its approximate values.
- Know how to develop with distributivity and remarkable identities.
- Know how to use in both directions.
- Demonstrate the nature of a quadrilateral.
- Know how to distinguish necessary and sufficient conditions for quadrilaterals.
- Know how to use the theorems of Pythagoras, Thales and the properties of angles.
I. Numbers and sets of numbers
1. Sets of numbers
- A number is said to be decimal if it can be written as a quotient of an integer by a power of 10.
- A rational is a number that can be written as a quotient of two integers.
- An irrational is a number that is not rational.
- In a relative number, we distinguish the sign (+ or -) and the absolute value.
-3 has sign – and absolute value 3. We note |-3| = 3.
- ℕ : Set of positive integers, or natural numbers.
- ℤ : Set of relative integers.
- Set of decimal numbers.
- ℚ : Set of rational numbers.
- ℝ : Set of real numbers.
So we have the following inclusions:
Integer notations :
We often note n a natural number. The next number is therefore n + 1.
The previous n – 1.
The even integers are the 2k for k ∈ ℕ and the odd ones are the 2k + 1 for k∈ ℕ.
In the same way the multiples of 3 can be noted 3k , those of 4 are noted 4k.
2. rational and irrational numbers
- For all these numbers, we do not have an exact decimal writing. Therefore, we cannot use an equality sign between and 3.141 592 653 for example. We note ≈ 3,141,592,653 .
It is very important to distinguish the exact value of a number from an approximate value (by excess or by default).
Ex: 3.14 < < 3.15 is a frame of of magnitude 10-2.
3. Calculate with square roots
We call square root of a positive number a, the unique positive number, noted , whose square is a .
i.e. for .
- For .
- For and : .
- For and : .
This equality is generally false.
4. solving equations
The equation has two solutions when : and .
5. remarkable identities
For all real numbers a and b we have the following equalities:
6. quadrilaterals :
For each type of quadrilateral, each property is both necessary and sufficient: it is a characteristic property.
A quadrilateral ABCD is a parallelogram if and only if :
- its opposite sides are of equal length two by two;
- its opposite sides are parallel two by two;
- its diagonals intersect in the middle;
- its opposite angles are equal;
- it is uncrossed and two of its opposite sides are parallel and of the same length.
A quadrilateral ABCD is a rhombus if and only if :
- its four sides are of equal length;
- it is a parallelogram which has two consecutive sides of the same length;
- it is a parallelogram which has its diagonals perpendicular.
A quadrilateral ABCD is a rectangle if and only if :
- it has 3 right angles
- it is a parallelogram which has a right angle
- it is a parallelogram which has its diagonals of the same length
A quadrilateral ABCD is a square if and only if :
- its four sides are of equal length and it has a right angle;
- it is a parallelogram that has two consecutive sides of equal length and perpendicular;
- it is a parallelogram which has its diagonals perpendicular and of the same length.
A square is a parallelogram, a rectangle and a rhombus at the same time.
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