II] Numbers and arithmetics

Question 1

What is the sign for “positive integers”?

Answer 1

IN

Question 2

What is the set of “positive integers”?

Answer 2

IN = {0, 1, 2, 3 … }

Question 3

What is the symbol for “set of integers”?

Answer 3

Z

Question 4

What is the set of “integers”?

Answer 4

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

Question 5

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

Answer 5

ℚ

Question 6

What is the set of “rationals”?

Answer 6

ℚ = {x⎥∃ p, q ∈Z, x=p/q

Question 7

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

Answer 7

ℝ

Question 8

What is the set of “reals”?

Answer 8

ℝ = {all the numbers}

Question 9

What is “transitivity of equalities”?

Answer 9

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

Question 10

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

Answer 10

Addition and product

∀x, y ∈ℝ, x+y ∈ℝ, x×y ∈ℝ

Question 11

To which operations does “commutativity” apply?

Answer 11

Addition and product in ℝ

Question 12

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

Answer 12

Addition and product

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

Question 13

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

Answer 13

+ : 0

× : 1

Question 14

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

Answer 14

y is the inverse of x for ✻

x✻y = neutral element for ✻

Question 15

What is a subtraction?

Answer 15

∀a, b ∈ℝ

a-b = a+(-b)

Question 16

What is a division?

Answer 16

∀a, b ∈ℝ

a/b = a×b⁻¹

Question 17

What is “distributive property”?

Answer 17

∀a, b, c ∈ℝ a(b+c) = ab+ac

Question 18

How to simplify a fraction?

Answer 18

a/b = a/b × c/c = ac/bc

Question 19

What is an exponent?

Answer 19

Repetition of a certain operation a certain number of time

Question 20

What are the results:
a⁻ⁿ
a⁰

Answer 20

a⁻ⁿ = 1/aⁿ
a⁰ = 1

Question 21

Find the radical of y=rⁿ

Answer 21

r = ⁿ√y

Question 22

∀y∈ℝ⁺, ⁿ√y =?

Answer 22

ⁿ√y = y^1/n

Question 23

Which sign do you choose for the radical when possible?

Answer 23

Positive

Question 24

xᵐ×xⁿ=?

Answer 24

xᵐ×xⁿ = xᵐ⁺ⁿ

Question 25

xᵐ∖xⁿ=?

Answer 25

xᵐ∖xⁿ = xᵐ⁻ⁿ

Question 26

(xᵐ)ⁿ =?

Answer 26

(xᵐ)ⁿ = xᵐⁿ

Question 27

x^(-1/n) =?

Answer 27

x^(-1/n) = 1/ⁿ√x

Question 28

ⁿ√x×ⁿ√y = ?

Answer 28

ⁿ√x×ⁿ√y =ⁿ√xy

Question 29

ⁿ√ᵐ√x = ?

Answer 29

ⁿ√ᵐ√x =ᵐⁿ√x

Question 30

x^m/n = ?

Answer 30

x^m/n = ⁿ√xᵐ = (ⁿ√x)ᵐ

Question 31

(ⁿ√x)/(ⁿ√y) = ?

Answer 31

(ⁿ√x)/(ⁿ√y) = ⁿ√(x/y)

Question 32

¬(A∧B) = ?

Answer 32

¬(A∧B) = (¬A∨¬B) | Same for ∨

Question 33

What is an "algebraic expression"?

Answer 33

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

Question 34

``` How is called an algebraic expression with 1 unknown? with 2 unknowns? with 3 unknowns? with n unknowns? ```

Answer 34

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

Question 35

What is a polynomial?

Answer 35

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

Question 36

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

Answer 36

(x+a)(x+b) = x²+x(a+b)+ab

Question 37

(x+a)² = ?

Answer 37

(x+a)² = x²+2ax+a²

Question 38

(x-a)² = ?

Answer 38

(x-a)² = x²-2ax+a²

Question 39

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

Answer 39

(x+a)(x-a) = x²-a²

Question 40

(x+a)³ = ?

Answer 40

(x+a)³ = x³+3ax²+3a²x+3a³

Question 41

(x-a)³ = ?

Answer 41

(x-a)³ = x³-3ax²+3a²x-a³

Question 42

x³+a³ = ?

Answer 42

x³+a³ = (x+a)(x²-ax+a²)

Question 43

x³-a³ = ?

Answer 43

x³-a³ = (x-a)(x²-ax+a²)

Question 44

What is an equation?

Answer 44

Equality of 2 algebraic expressions.

Question 45

What's solving an equation?

Answer 45

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.

Question 46

What is a literal equation?

Answer 46

When parameters are directly involved in the equation. | Discussion is depending on the value of these parameters.

Question 47

What is a quadratic equation?

Answer 47

ax²+bx+c=0

Question 48

For a quadratic equation, what is "∆"?

Answer 48

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

Question 49

How do you compute ∆?

Answer 49

∆ = b²-4ac

Question 50

What are the 2 solutions when ∆>0?

Answer 50

``` x1 = (-b+√∆)/2a x2 = (-b-√∆)/2a ```

Question 51

ax²+bx+c = ?

Answer 51

ax²+bx+c = (x-x1)(x-x2) | Where x1 and x2 are the 2 solutions of the equation.

Question 52

What is an inequality?

Answer 52

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

Question 53

a< = > a+c ? b+c | ∀c∈ℝ

Answer 53

a+c

Question 54

a< = > ac ? bc | ∀c∈ℝ*⁺

Answer 54

ac

Question 55

a< = > ac ? bc | ∀c∈ℝ*⁻

Answer 55

ac>bc

Question 56

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

Answer 56

aⁿ>bⁿ | ⁿ√a>ⁿ√b

Question 57

What is an "absolute value"?

Answer 57

⎟x⎟is given by: x if x≥0 -x if x<0

Question 58

⎟x+y⎟= ?

Answer 58

⎟x+y⎟= ⎟-x-y⎟

Question 59

⎟x-y⎟= ?

Answer 59

⎟x-y⎟= ⎟y-x⎟

Question 60

⎟x⎟= ?

Answer 60

⎟x⎟=⎟-x⎟