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Math Bootcamp > II] Numbers and arithmetics > Flashcards

II] Numbers and arithmetics Flashcards

1
Q

What is the sign for “positive integers”?

A

IN

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2
Q

What is the set of “positive integers”?

A

IN = {0, 1, 2, 3 … }

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3
Q

What is the symbol for “set of integers”?

A

Z

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4
Q

What is the set of “integers”?

A

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

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5
Q

What is the symbol for the “set or rationals”?

A

ℚ

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6
Q

What is the set of “rationals”?

A

ℚ = {x⎥∃ p, q ∈Z, x=p/q

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7
Q

What is the symbol for the “set or reals”?

A

ℝ

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8
Q

What is the set of “reals”?

A

ℝ = {all the numbers}

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9
Q

What is “transitivity of equalities”?

A

(a=b, b=c) => a=c

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10
Q

What is “closure property” and which operations does it concern?

A

Addition and product

∀x, y ∈ℝ, x+y ∈ℝ, x×y ∈ℝ

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11
Q

To which operations does “commutativity” apply?

A

Addition and product in ℝ

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12
Q

To which operations does “associative property” apply? And what is it?

A

Addition and product

a+(b+c) = (a+b)+c

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13
Q

What are the “neutral elements” for addition and product?

A

+ : 0

× : 1

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14
Q

How do you determine the “neutral element” for an operation?

A

y is the inverse of x for ✻

x✻y = neutral element for ✻

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15
Q

What is a subtraction?

A

∀a, b ∈ℝ

a-b = a+(-b)

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16
Q

What is a division?

A

∀a, b ∈ℝ

a/b = a×b⁻¹

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17
Q

What is “distributive property”?

A

∀a, b, c ∈ℝ a(b+c) = ab+ac

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18
Q

How to simplify a fraction?

A

a/b = a/b × c/c = ac/bc

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19
Q

What is an exponent?

A

Repetition of a certain operation a certain number of time

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20
Q

What are the results:
a⁻ⁿ
a⁰

A 
a⁻ⁿ = 1/aⁿ
a⁰ = 1
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21
Q

Find the radical of y=rⁿ

A

r = ⁿ√y

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22
Q

∀y∈ℝ⁺, ⁿ√y =?

A

ⁿ√y = y^1/n

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23
Q

Which sign do you choose for the radical when possible?

A

Positive

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24
Q

xᵐ×xⁿ=?

A

xᵐ×xⁿ = xᵐ⁺ⁿ

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25
Q

xᵐ∖xⁿ=?

A

xᵐ∖xⁿ = xᵐ⁻ⁿ

26
Q

(xᵐ)ⁿ =?

A

(xᵐ)ⁿ = xᵐⁿ

27
Q

x^(-1/n) =?

A

x^(-1/n) = 1/ⁿ√x

28
Q

ⁿ√x×ⁿ√y = ?

A

ⁿ√x×ⁿ√y =ⁿ√xy

29
Q

ⁿ√ᵐ√x = ?

A

ⁿ√ᵐ√x =ᵐⁿ√x

30
Q

x^m/n = ?

A

x^m/n = ⁿ√xᵐ = (ⁿ√x)ᵐ

31
Q

(ⁿ√x)/(ⁿ√y) = ?

A

(ⁿ√x)/(ⁿ√y) = ⁿ√(x/y)

32
Q

¬(A∧B) = ?

A

¬(A∧B) = (¬A∨¬B)

Same for ∨

33
Q

What is an “algebraic expression”?

A

Any expression where “unknowns” are combined using addition, subtraction, multiplication, division, exponents, fractions, radical & possibly with some numbers or constants.

34
Q 
How is called an algebraic expression
with 1 unknown?
with 2 unknowns?
with 3 unknowns?
with n unknowns?
A

1: monomial
2 binomial
3: trinomial
n: multinomial ≠ polynomial

35
Q

What is a polynomial?

A

Algebraic expression that can be written:
anxⁿ + an-1xⁿ⁻¹….. a2x²+a1x+a0
n: degree of the polynomial

36
Q

(x+a)(x+b) = ?

A

(x+a)(x+b) = x²+x(a+b)+ab

37
Q

(x+a)² = ?

A

(x+a)² = x²+2ax+a²

38
Q

(x-a)² = ?

A

(x-a)² = x²-2ax+a²

39
Q

(x+a)(x-a) = ?

A

(x+a)(x-a) = x²-a²

40
Q

(x+a)³ = ?

A

(x+a)³ = x³+3ax²+3a²x+3a³

41
Q

(x-a)³ = ?

A

(x-a)³ = x³-3ax²+3a²x-a³

42
Q

x³+a³ = ?

A

x³+a³ = (x+a)(x²-ax+a²)

43
Q

x³-a³ = ?

A

x³-a³ = (x-a)(x²-ax+a²)

44
Q

What is an equation?

A

Equality of 2 algebraic expressions.

45
Q

What’s solving an equation?

A

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.

46
Q

What is a literal equation?

A

When parameters are directly involved in the equation.

Discussion is depending on the value of these parameters.

47
Q

What is a quadratic equation?

A

ax²+bx+c=0

48
Q

For a quadratic equation, what is “∆”?

A

The discriminent.
If ∆=0, 1 solution for the equation
If ∆0, 2 solutions for the equation.

49
Q

How do you compute ∆?

A

∆ = b²-4ac

50
Q

What are the 2 solutions when ∆>0?

A 
x1 = (-b+√∆)/2a
x2 = (-b-√∆)/2a
51
Q

ax²+bx+c = ?

A

ax²+bx+c = (x-x1)(x-x2)

Where x1 and x2 are the 2 solutions of the equation.

52
Q

What is an inequality?

A

Statement where an algebraic expression is or ≥ some other algebraic expression.

53
Q

a<b>< = > a+c ? b+c</b>

∀c∈ℝ

A

a+c<b+c

54
Q

a<b>< = > ac ? bc</b>

∀c∈ℝ*⁺

A

ac<bc

55
Q

a<b>< = > ac ? bc</b>

∀c∈ℝ*⁻

A

ac>bc

56
Q

If a>0 & b>0,
a>b < = > aⁿ ? bⁿ
a>b < = > ⁿ√a ? ⁿ√b

A

aⁿ>bⁿ

ⁿ√a>ⁿ√b

57
Q

What is an “absolute value”?

A

⎟x⎟is given by:
x if x≥0
-x if x<0

58
Q

⎟x+y⎟= ?

A

⎟x+y⎟= ⎟-x-y⎟

59
Q

⎟x-y⎟= ?

A

⎟x-y⎟= ⎟y-x⎟

60
Q

⎟x⎟= ?

A

⎟x⎟=⎟-x⎟

Math Bootcamp (3 decks)

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