II] Numbers and arithmetics Flashcards
What is the sign for “positive integers”?
IN
What is the set of “positive integers”?
IN = {0, 1, 2, 3 … }
What is the symbol for “set of integers”?
Z
What is the set of “integers”?
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
What is the symbol for the “set or rationals”?
ℚ
What is the set of “rationals”?
ℚ = {x⎥∃ p, q ∈Z, x=p/q
What is the symbol for the “set or reals”?
ℝ
What is the set of “reals”?
ℝ = {all the numbers}
What is “transitivity of equalities”?
(a=b, b=c) => a=c
What is “closure property” and which operations does it concern?
Addition and product
∀x, y ∈ℝ, x+y ∈ℝ, x×y ∈ℝ
To which operations does “commutativity” apply?
Addition and product in ℝ
To which operations does “associative property” apply? And what is it?
Addition and product
a+(b+c) = (a+b)+c
What are the “neutral elements” for addition and product?
+ : 0
× : 1
How do you determine the “neutral element” for an operation?
y is the inverse of x for ✻
x✻y = neutral element for ✻
What is a subtraction?
∀a, b ∈ℝ
a-b = a+(-b)
What is a division?
∀a, b ∈ℝ
a/b = a×b⁻¹
What is “distributive property”?
∀a, b, c ∈ℝ a(b+c) = ab+ac
How to simplify a fraction?
a/b = a/b × c/c = ac/bc
What is an exponent?
Repetition of a certain operation a certain number of time
What are the results:
a⁻ⁿ
a⁰
a⁻ⁿ = 1/aⁿ a⁰ = 1
Find the radical of y=rⁿ
r = ⁿ√y
∀y∈ℝ⁺, ⁿ√y =?
ⁿ√y = y^1/n
Which sign do you choose for the radical when possible?
Positive
xᵐ×xⁿ=?
xᵐ×xⁿ = xᵐ⁺ⁿ
xᵐ∖xⁿ=?
xᵐ∖xⁿ = xᵐ⁻ⁿ
(xᵐ)ⁿ =?
(xᵐ)ⁿ = xᵐⁿ
x^(-1/n) =?
x^(-1/n) = 1/ⁿ√x
ⁿ√x×ⁿ√y = ?
ⁿ√x×ⁿ√y =ⁿ√xy
ⁿ√ᵐ√x = ?
ⁿ√ᵐ√x =ᵐⁿ√x
x^m/n = ?
x^m/n = ⁿ√xᵐ = (ⁿ√x)ᵐ
(ⁿ√x)/(ⁿ√y) = ?
(ⁿ√x)/(ⁿ√y) = ⁿ√(x/y)
¬(A∧B) = ?
¬(A∧B) = (¬A∨¬B)
Same for ∨
What is an “algebraic expression”?
Any expression where “unknowns” are combined using addition, subtraction, multiplication, division, exponents, fractions, radical & possibly with some numbers or constants.
How is called an algebraic expression with 1 unknown? with 2 unknowns? with 3 unknowns? with n unknowns?
1: monomial
2 binomial
3: trinomial
n: multinomial ≠ polynomial
What is a polynomial?
Algebraic expression that can be written:
anxⁿ + an-1xⁿ⁻¹….. a2x²+a1x+a0
n: degree of the polynomial
(x+a)(x+b) = ?
(x+a)(x+b) = x²+x(a+b)+ab
(x+a)² = ?
(x+a)² = x²+2ax+a²
(x-a)² = ?
(x-a)² = x²-2ax+a²
(x+a)(x-a) = ?
(x+a)(x-a) = x²-a²
(x+a)³ = ?
(x+a)³ = x³+3ax²+3a²x+3a³
(x-a)³ = ?
(x-a)³ = x³-3ax²+3a²x-a³
x³+a³ = ?
x³+a³ = (x+a)(x²-ax+a²)
x³-a³ = ?
x³-a³ = (x-a)(x²-ax+a²)
What is an equation?
Equality of 2 algebraic expressions.
What’s solving an equation?
Find the set of solutions of x, y, z… such that the equation is True.
Sometimes, the solution will depend on 1, or more, parameter.
What is a literal equation?
When parameters are directly involved in the equation.
Discussion is depending on the value of these parameters.
What is a quadratic equation?
ax²+bx+c=0
For a quadratic equation, what is “∆”?
The discriminent.
If ∆=0, 1 solution for the equation
If ∆0, 2 solutions for the equation.
How do you compute ∆?
∆ = b²-4ac
What are the 2 solutions when ∆>0?
x1 = (-b+√∆)/2a x2 = (-b-√∆)/2a
ax²+bx+c = ?
ax²+bx+c = (x-x1)(x-x2)
Where x1 and x2 are the 2 solutions of the equation.
What is an inequality?
Statement where an algebraic expression is or ≥ some other algebraic expression.
a<b>< = > a+c ? b+c</b>
∀c∈ℝ
a+c<b+c
a<b>< = > ac ? bc</b>
∀c∈ℝ*⁺
ac<bc
a<b>< = > ac ? bc</b>
∀c∈ℝ*⁻
ac>bc
If a>0 & b>0,
a>b < = > aⁿ ? bⁿ
a>b < = > ⁿ√a ? ⁿ√b
aⁿ>bⁿ
ⁿ√a>ⁿ√b
What is an “absolute value”?
⎟x⎟is given by:
x if x≥0
-x if x<0
⎟x+y⎟= ?
⎟x+y⎟= ⎟-x-y⎟
⎟x-y⎟= ?
⎟x-y⎟= ⎟y-x⎟
⎟x⎟= ?
⎟x⎟=⎟-x⎟
