# rational number

Real Numbers and the Real Line

Calculus depends on properties of the real number system.

Real numbers are numbers that can be expressed as decimals.

5 = 5.000…
-¾ = -0.75000…
1/3 = 0.333…
√2 = 1.414…
π = 3.141…

In each example above, the three dots “…” indicate that the sequence of decimal digits goes on forever.

For the first three examples above, the pattern of the digits are obvious, where the subsequent digits are easily known. However, for √2 and π, there is no obvious pattern of their decimal digits.

Real Line
The real numbers can be represented geometrically as points on a number line called the real line.

The symbol ℝ is used to denote either the real number system or the real number line.

Properties of Real Numbers
The properties of real numbers fall into three categories: algebraic properties, order properties, and completeness.

The algebraic properties assert that the real numbers can be added, subtracted, multiplied, and divided (except zero) to produce another real number, and so the rules of arithmetic are valid.

Order Properties
The order properties of the real numbers refer to the order the real numbers appears on the number line.

If x lies to the left of y on the real line, then x is less than y, or y is greater than x, written as x < y, y > x, respectively.

The inequality x ≤ y means either x = y or x < y.

The following are order properties of the real numbers:

The symbol => means “implies” while <=> (seen later) means “equivalent to.”

For Rules 1-4 and 6 (for 0 < a), they also hold if < and > are replaced with ≤ and ≥, respectively.

Note that, for Rules 3 and 4, the rules for multiplying or dividing an inequality by a positive number c preserves the inequality. If the number c is negative, the inequality is reversed.

Completeness Property
If A is any set of real numbers having at least one number in the set, and if there exists a real number y with the property x ≤ y for every x in A, then there exists a smallest number y with this same property.

Therefore, there are no holes on the real line, where every point corresponds to a real number.

The study of infinite sequences will use the property of completeness.

Subsets of the Real Numbers
The set of real numbers ℝ has some special subsets.

1. The set of natural numbers ℕ = {1, 2, 3, 4, …}.
2. The set of integers ℤ = {…, -2, -1, 0, 1, 2, …}
3. The set of rational numbers ℚ, which is the set of numbers that can be expressed as a fraction m/n, where m, n ∈ ℤ and n ≠ 0.

Rational Numbers
Rational Numbers are real numbers with decimal expansion that does either of the following:

1. Terminate: ending with an infinite string of zeros, such as ¾ = 0.75000…, or
2. Repeat: ending with a string of digits that repeat infinitely, such as 23/11 = 2.090909… = 2.̄0̄9̄ . The bar over the digits indicate the pattern of repeating digits.

Real numbers that are not rational are called irrational numbers.

The set of all rational numbers possesses all algebraic and order properties of the real numbers, but it does not possess the completeness property. For example, √2 is irrational, so there is a “hole” on the rational line where √2 should be.

Because the real line has no such “holes,” it is the appropriate setting for the study of calculus.

Intervals
A subset of the real line is called an interval if it contains at least two numbers and all real numbers between any two of its elements.

For example, the set of real numbers x such that x > 6 is an interval, but the set of real numbers such that y ≠ 0 is not an interval, because it consists of two intervals.

If a and b are real numbers and a < b:

1. The open interval from a to b, denoted as (a, b), consists of all real numbers x satisfying a < x < b.

2. The closed interval from a to b, denoted as [a, b], consists of all real numbers x satisfying a ≤ x ≤ b.

3. The half-open interval from a to b, denoted as [a, b), consists of all real numbers x satisfying a ≤ x < b.

4. The half-open interval from a to b, denoted as (a, b], consists of all real numbers x satisfying a < x ≤ b.

Note that, hollow dots are used to indicate endpoints that are not included in the interval, and the solid dots are used to indicate endpoints of the interval that are included. The endpoints are also called boundary points.

The above type of intervals are called finite intervals, such that each interval has a finite length of b – a.

Intervals with infinite length are called infinite intervals.

The following are the infinite intervals (a, ∞) and (-∞, a]:

The whole real line ℝ is an infinite interval denoted as (-∞, ∞).

Infinity ∞ does not denote a real number, and so it is never allowed to belong to an interval.

Union and Intersection of Intervals
The symbol ⋃ is used to denote the union of intervals.

A real number is in the union of intervals if it is in at least one of the intervals.

For example, [1, 3) ⋃ [2, 4] = [1, 4]. Even though 3 is not included in the first interval, it is included in the second interval, and so the union of these two intervals is simply from 1 to 4, inclusive.

The symbol ⋂ is used to denote the intersection of intervals.

A real number is in the intersection of intervals if it is in every one of those intervals.

For example, [1, 3) ⋂ [2, 4] = [2, 3). Even though 4 is included in the second interval, and 1 is included in the first interval, they are not included in both intervals.

Whenever “and” is mentioned in conditions for intervals, it will be one interval. Whenever “or” is mentioned in conditions for intervals, it will be a union of intervals.

The Absolute Value
The absolute value, or magnitude, of a number x, denoted as |x|, is defined as the following:

The vertical lines in the symbol |x| are called absolute value bars.

For example, |3| = 3, |0| = 0, and |-5| = 5.

Note that, √a always denotes the non-negative square root of a, and so the alternative definition for |x| is |x| = √x². It is important to remember that √a² = |a| and not just a, unless it is known that a ≥ 0.

Geometrically, |x| represents the non-negative distance from x to 0 on the real line.

In general, |x – y| represents the non-negative distance between x and y on the real line, since this distance is the same as the distance between x – y and 0.

The following demonstrates |x – y| = distance from x to y:

Properties of Absolute Value
The absolute value function has the following properties:

Equations and Inequalities Involving Absolute Values
The equation |x| = D, where D > 0, has two solutions: x = -D and x = D, which are two points on the real line that lie at distance D from the origin 0.

Equations and inequalities involving absolute values can be solved algebraically by using cases according to the definition of absolute value. Another way to solve them is geometrically interpreting absolute values as distances.

For example, the inequality |x – a| < D means the distance from x to a is less than D, and so x must lie between a – D and a + D, or a must lie between x – D and x + D.

If D is a positive number, then:

1. |x| = D <=> either x = -D or x = D
|x – a| = D <=> either x = a – D or x = a + D

2. |x| < D <=> -D < x < D
|x – a| < D <=> a – D < x < a + D

3. |x| ≤ D <=> -D ≤ x ≤ D
|x – a| ≤ D <=> a – D ≤ x ≤ a + D

4. |x| > D <=> either x < -D or x > D
|x – a| > D <=> either x < a – D or x > a + D

PDF reference: 27/1

Objectives

1. Writing repeating decimals.
2. Converting fractions into repeating decimals.
3. Converting repeating decimals into fractions.
4. Solving linear inequalities and graphing the solution set.
6. Solving absolute value equations and inequalities.
7. Using the Triangle Inequality to prove other inequalities.

Show that each of the following numbers is a rational number by expressing it as a quotient of two integers:

a) 1.323232… = 1.3̄2̄
b) 0.3405405405… = 0.34̄0̄5̄

a)

Let x = 1.323232…

Then the non-decimal digit is subtracted from both sides.

x – 1 = 0.323232…

Then 100 is multiplied on both sides of the original equation to get one repeating pattern on the non-decimal side.

100x = 132.323232… = 132 + 0.323232… = 132 + x – 1

Using algebraic techniques forms the decimal into a fraction and hence prove the number is rational.

100x = 132 + x – 1
99x = 131
x = 131/99

b)

Let x = 0.3405405405…

Since the non-repeating digit is in the decimal side, a multiple of 10 must be multiplied to make it into a non-decimal digit.

10x = 3.405405405…

Then the non-decimal digit can be subtracted from both sides.

10x – 3 = 0.405405405…

Then 10000 is multiplied on both sides of the original equation to get one repeating pattern on the non-decimal side.

10000x = 3405.405405405… = 3405 + 0.405405405… = 3405 + 10x - 3

Using algebraic techniques forms the decimal into a fraction and hence prove the number is rational.

10000x = 3405 + 10x – 3
9990x = 3402
x = 3402/9990 = 63/185

Express the following rational numbers as a repeating decimal. Use a bar to indicate the repeating digits:

a) 2/9
b) 1/11

a)

Using long division, it is found that 2/9 = 0.222… = 0.2̄ .

b)

Using long division, it is found that 1/11 = 0.090909… = 0.0̄9̄ .

Express the following repeating decimal as a quotient of integers in lowest terms:

a) 0.1̄2̄
b) 3.27̄

a)

Let x = 0.121212…

Since there are no non-repeating digits, many steps are skipped and now 100 is multiplied on both sides to get one repeating pattern on the non-decimal side.

100x = 12.121212… = 12 + 0.121212 = 12 + x

Using algebraic techniques forms the decimal into a fraction and then simplified into its lowest terms.

100x = 12 + x
99x = 12
x = 12/99 = 4/33

b)

Let x = 3.2777…

Since there is a non-repeating digit on the decimal side, 10 is multiplied on both sides.

10x = 32.777..

The non-repeating digits are subtracted from both sides.

10x – 32 = 0.777…

Now the original equation is multiplied by 100 to get one repeating pattern on the non-decimal side.

100x = 327.777… = 327 + 0.777… = 327 + 10x – 32

Using algebraic techniques forms the decimal into a fraction and then simplified into its lowest terms.

100x = 327 + 10x – 32
90x = 295
x = 295/90 = 59/18

Solve the following inequalities and express the solution sets in terms of intervals. Graph their intervals.

a) 2x – 1 > x + 3
b) -x/3 ≥ 2x - 1
c) 2/(x – 1) ≥ 5

a)

2x – 1 > x + 3
x – 1 > 3
x > 4

Solution: (4, ∞)

b)

-x/3 ≥ 2x – 1
x ≤ -6x + 3
7x ≤ 3
x ≤ 3/7

Solution: (-∞, 3/7]

c)

This inequality has x in the denominator, and so it is possible for x to be undefined at a certain point on the real line.

The inequality will be written such that all terms are on one side and then simplified into one fraction.

2/(x – 1) ≥ 5
2/(x – 1) – 5 ≥ 0
(2 – 5x + 5)/(x – 1) ≥ 0
(7 – 5x)/(x – 1) ≥ 0

The numerator is examined to determine when it equals 0.

7 – 5x = 0
x = 7/5

The denominator is examined to determine when it equals 0.

x – 1 = 0
x = 1

Since the fraction is undefined when x = 1, the interval will have a round bracket on 1 and a square bracket on 7/5.

Solution: (1, 7/5]

Solve the following systems of inequalities:

a) 3 ≤ 2x + 1 ≤ 5
b) 3x – 1 < 5x + 3 ≤ 2x + 15

a)

First, the left inequality is solved:

3 ≤ 2x + 1
2 ≤ 2x
x ≥ 1

Then the right inequality is solved:

2x + 1 ≤ 5
2x ≤ 4
x ≤ 2

Therefore, since x ≥ 1 and x ≤ 2, the interval is [1, 2].

b)

First, the left inequality is solved:

3x – 1 < 5x + 3
-4 < 2x
x > -2

Then the right inequality is solved:

5x + 3 ≤ 2x + 15
3x ≤ 12
x ≤ 4

Since x > -2 and x ≤ 4, the interval is (-2, 4].

a) x² – 5x + 6 < 0
b) 2x² + 1 > 4x

a)

x² – 5x + 6 < 0
(x – 2)(x – 3) < 0
x = 2, 3

Drawing a number line with labeled solutions, points between -∞ and 2 are positive, points between 2 and 3 are negative, and points between 3 and ∞ are positive.

Therefore, (2, 3).

b)

2x² + 1 > 4x
2x² – 4x + 1 > 0

The quadratic formula is used to determine the solutions.

x = 1 ± √2/2

(x - 1 - √2/2)(x - 1 + √2/2) > 0

Drawing a number line with labeled solutions, points between -∞ and 1 - √2/2 are positive, points between 1 - √2/2 and 1 + √2/2 are negative, and points between 1 + √2/2 and ∞ are positive.

Therefore, the solution is the union of the intervals (-∞, 1 - √2/2) ⋃ (1 + √2/2, ∞).

Express the set of all real numbers x satisfying the following conditions as an interval or a union of intervals.

a) x ≥ 0 and x ≤ 5
b) x < 2 and x ≥ -3
c) x > -5 or x < -6
d) x ≤ -1
e) x > -2
f) x < 4 or x ≥ 2

a)

Since “and” is used, the interval must satisfy when x ≥ 0 and when x ≤ 5.

[0, 5]

b)

Since “and” is used, the interval must satisfy when x < 2 and when x ≥ -3.

[-3, 2)

c)

Since “or” is used, the interval must satisfy when x > -5 or x < -6.

(-∞, -6) ⋃ (-5, ∞)

d)

x ≤ -1 means that the interval contains all real numbers such that x is less than or equal to -1.

(-∞, -1]

e)

x > -2 means that the interval contains all real numbers such that x is greater than -2.

(-2, ∞)

f)

Since “or” is used, the interval must satisfy when x < 4 or x ≥ 2, which is just the real line.

(-∞, 4) ⋃ [2, ∞) = (-∞, ∞)

Solve the following inequality and graph the solution set.

3/(x – 1) < -2/x

Bring all terms to one side.

3/(x – 1) + 2/x < 0
(3x + 2x – 2)/x(x – 1) = (5x – 2)/x(x – 1) < 0

Examine the numerator and determine when it is equal to zero.

5x – 2 = 0
x = 2/5

Examine the denominator and determine when it is equal to zero.

x(x – 1) = 0
x = 0, 1

The denominator is undefined when either x = 0 or x = 1 or both.

Testing points between 0, 2/5, and 1 to determine whether it is negative or positive.

Points to the left of zero are negative, points between 0 and 2/5 are positive, points between 2/5 and 1 are negative, and points to the right of one are positive.

Since it is less than zero and not less than or equal to zero, all three endpoints use a round bracket.

(-∞, 0) ⋃ (2/5, 1)

Solve the following inequalities, giving the solution set as an interval or union of intervals.

a) -2x > 4
b) 3x + 5 ≤ 8
c) 5x – 3 ≤ 7 - 3x
d) (6 – x)/4 ≥ (3x – 4)/2
e) 3(2 – x) < 2(3 + x)
f) x² < 9
g) 1/(2 – x) < 3
h) (x + 1)/x ≥ 2
i) x² – 2x ≤ 0
j) 6x² – 5x ≤ -1
k) x³ > 4x
l) x² – x ≤ 2
m) x/2 ≥ 1 + 4/x
n) 3/(x – 1) < 2/(x + 1)

a)

-2x > 4 <=> x < -2 <=> (-∞, -2)

b)

3x + 5 ≤ 8 <=> 3x ≤ 3 <=> x ≤ 1 <=> (-∞, 1]

c)

5x – 3 ≤ 7 – 3x <=> 8x – 3 ≤ 7 <=> 8x ≤ 10 <=> x ≤ 5/4 <=> (-∞, 5/4]

d)

(6 – x)/4 ≥ (3x – 4)/2 <=> 12 – 2x ≥ 12x – 16 <=> -14x ≥ -28 <=> x ≤ 2 <=> (-∞, 2]

e)

3(2 – x) < 2(3 + x) <=> 6 – 3x < 6 + 2x <=> -5x < 0 <=> x > 0 <=> (0, ∞)

f)

x² < 9 <=> x² – 9 < 0 <=> (x – 3)(x + 3) < 0 => x = -3, 3

Drawing a number line with labeled solutions, points between -∞ and -3 are positive, points between -3 and 3 are negative, and points between 3 and ∞ are positive.

Therefore, (-3,3).

g)

1/(2 – x) < 3 <=> 1/(2 – x) – 3 < 0 <=> (1 – 6 + 3x)/(2 – x) <=> (-5 + 3x)/(2 – x) => x = 5/3, 2

Drawing a number line with labeled solutions, points between -∞ and 5/3 are negative, points between 5/3 and 2 are positive, and points between 2 and ∞ are negative.

Since the fraction is undefined when x = 2, 2 is not allowed to be included in the interval.

Therefore, (-∞, 5/3) ⋃ (2, ∞).

h)

(x + 1)/x ≥ 2 <=> (x + 1)/x – 2 ≥ 0 <=> (x + 1 – 2x)/x ≥ 0 <=> (-x + 1)/x ≥ 0 => x = 0, 1

Drawing a number line with labeled solutions, points between -∞ and 0 are negative, points between 0 and 1 are positive, and points between 1 and ∞ are negative.

Since the fraction is undefined when x = 0, 0 is not allowed to be included in the interval.

Therefore, (0, 1].

i)

x² – 2x ≤ 0 <=> x(x – 2) ≤ 0 => x = 0, 2

Drawing a number line with labeled solutions, points between -∞ and 0 are positive, points between 0 and 2 are negative, and points between 2 and ∞ are positive.

Therefore, [0, 2].

j)

6x² – 5x ≤ -1 <=> 6x² – 5x + 1 ≤ 0 <=> 6x² – 3x – 2x + 1 ≤ 0 <=> 3x(2x – 1) – (2x – 1) ≤ 0 <=> (3x – 1)(2x – 1) ≤ 0 => x = 1/3, ½

Drawing a number line with labeled solutions, points between -∞ and 1/3 are positive, points between 1/3 and ½ are negative, and points between ½ and ∞ are positive.

Therefore, [1/3, ½].

k)

x³ > 4x <=> x³ – 4x > 0 <=> x(x² – 4) > 0 <=> x(x – 2)(x + 2) > 0 => x = -2, 0, 2

Drawing a number line with labeled solutions, points between -∞ and -2 are negative, points between -2 and 0 are positive, point between 0 and 2 are negative, and points between 2 and ∞ are positive.

Therefore, (-2, 0) ⋃ (2, ∞).

l)

x² – x ≤ 2 <=> x² – x – 2 ≤ 0 <=> (x – 2)(x + 1) ≤ 0 => x = -1, 2

Drawing a number line with labeled solutions, points between -∞ and -1 are positive, points between -1 and 2 are negative, and points between 2 and ∞ are positive.

Therefore, [-1, 2].

m)

x/2 ≥ 1 + 4/x <=> 0 ≥ 1 – x/2 + 4/x <=> 0 ≥ 0 ≥ (x – x²/2 + 4)/x <=>   0 ≥ (-2/-2)[(x – x²/2 + 4)/x] <=> 0 ≥ (x² – 2x – 8)/(-2x) <=> 0 ≥ (x – 4)(x + 2)/(-2x) <=> (x – 4)(x + 2)/(-2x) ≤ 0 => x = -2, 0, 4

Drawing a number line with labeled solutions, points between -∞ and -2 are positive, points between -2 and 0 are negative, point between 0 and 4 are positive, and points between 4 and ∞ are negative.

Since the fraction is undefined when x = 0, 0 is not allowed to be included in the interval.

Therefore, [-2, 0) ⋃ [4, ∞).

n)

3/(x – 1) < 2/(x + 1) <=> 3/(x – 1) – 2/(x + 1) < 0 <=> (3x + 3 – 2x + 2)/(x - 1)(x + 1) < 0 <=> (x + 5)/(x – 1)(x + 1) < 0 => x = -5, -1, 1

Drawing a number line with labeled solutions, points between -∞ and -5 are negative, points between -5 and -1 are positive, point between -1 and 1 are negative, and points between 1 and ∞ are positive.

Since the fraction is undefined when x = -1 or 1, -1 and 1 are not allowed to be included in the interval.

Therefore, (-∞, -5) ⋃ (-1, 1).

Solve the following:

a) |2x + 5| = 3
b) |3x – 2| ≤ 1

a)

Case 1: x > 0
|2x + 5| = 3 <=> 2x + 5 = 3 <=> x = -1

Case 2: x < 0
|2x + 5| = 3 <=> -(2x + 5) = 3 <=> 2x + 5 = -3 <=> x = -4

b)

Since the question deals with “less than or equal to,” the equivalent form is used.

|3x – 2| ≤ 1 <=> -1 ≤ 3x – 2 ≤ 1

The left inequality is solved first.

-1 ≤ 3x – 2
1 ≤ 3x
x ≥ 1/3

The right inequality is solved last.

3x – 2 ≤ 1
3x ≤ 3
x ≤ 1

Therefore, [1/3,1].

Solve the equation |x + 1| = |x – 3|.

Case 1: x < 0
|x + 1| = |x – 3| <=> -(x + 1) = x – 3 <=> -x – 1 = x – 3 <=> -2x = -2 <=> x = 1

Case 2: x > 0
|x + 1| = |x – 3| <=> x + 1 = x – 3 <=> no solution

What values of x satisfy the following inequality?

Since the inequality uses “less than,” the following equivalent form is used.

-3 < 5 – 2/x < 3 <=> -8 < -2/x < -2 <=> 4 > 1/x > 1

Solve the left inequality first.

4 > 1/x <=> 4x > 1 <=> x > ¼

Solve the right inequality last.

1/x > 1 <=> 1 > x <=> x < 1

Therefore, the values of x that satisfy the above inequality is in the interval (¼, 1).

Solve the following equations:

a) |x – 3| = 7
b) |2t + 5| = 4
c) |1 – t| = 1
d) |s/2 – 1| = 1

a)

|x – 3| = 7

Case 1: x < 0
-(x – 3) = 7 <=> x – 3 = -7 <=> x = -4

Case 2: x > 0
x – 3 = 7 <=> x = 10

Therefore, x = -4, 10.

b)

|2t + 5| = 4

Case 1: x < 0
-(2t + 5) = 4 <=> 2t + 5 = -4 <=> 2t = -9 <=> t = -9/2

Case 2: x > 0
2t + 5 = 4 <=> t = -½

Therefore, t = -½, -9/2.

c)

|1 – t| = 1

Case 1: x < 0
-(1 – t) = 1 <=> 1 – t = -1 <=> t = 2

Case 2: x > 0
1 – t = 1 <=> t = 0

Therefore, t = 0, 2.

d)

|s/2 – 1| = 1

Case 1: x < 0
-(s/2 – 1) = 1 <=> s/2 – 1 = -1 <=> s = 0

Case 2: x > 0
s/2 – 1 = 1 <=> s/2 = 2 <=> s = 4

Therefore, s = 0, 4.

Write the interval defined by the following inequalities:

a) |x| < 2
b) |x| ≤ 2
c) |t + 2| < 1
d) |3x – 7| < 2
e) |x/2 – 1| ≤ 1
f) |2 – x/2| < ½

a)

|x| < 2 <=> -2 < x < 2 <=> (-2, 2)

b)

|x| ≤ 2 <=> -2 ≤ x ≤ 2 <=> [-2, 2]

c)

|t + 2| < 1 <=> -1 < t + 2 < 1 <=> -3 < t < -1 <=> (-3, -1)

d)

|3x – 7| < 2 <=> -2 < 3x – 7 < 2 <=> 5 < 3x < 9 <=> 5/3 < x < 3 <=> (5/3, 3)

e)

|x/2 – 1| ≤ 1 <=> -1 ≤ x/2 – 1 ≤ 1 <=> 0 ≤ x/2 ≤ 2 <=> 0 ≤ x ≤ 4 <=> [0, 4]

f)

|2 – x/2| < ½ <=> -0.5 < 2 – 0.5x < 0.5 <=> -2.5 < -0.5x < 2.5 <=> 5 > x > 3 <=> (3, 5)

Solve the equation |x – 1| = 1 – x.

Case 1: x < 0
-(x – 1) = 1 – x <=> -x + 1 = 1 – x <=> no solution

Case 2: x > 0
x – 1 = 1 – x <=> 2x = 2 <=> x = 1

Additionally, the equation holds if |x – 1| = -(1 – x), since this is when they are always equal to each other. For this to be true, x – 1 < 0, or x < 1.

Therefore, x < 1.

Show that the following inequality holds for all real numbers a and b:

|a – b| ≥ ||a| - |b||

Begin with the Triangle Inequality.

|x + y| ≤ |x| + |y|
<=> |x| + |y| ≥ |x + y|
<=> |x| ≥ |x + y| - |y|

Let x = a – b and y = b.

|x| ≥ |x + y| - |y|
<=> |a – b| ≥ |a – b + b| - |b| = |a| - |b|

Similarly, |a – b| = |b – a| ≥ |b| - |a|.

Then ||a| - |b|| is equal to either |a| - |b| or |b| - |a|, and since |a – b| ≥ |b| - |a| and |a – b| ≥ |a| - |b|:

|a – b| ≥ ||a| - |b||

the beautiful thing about hermann gottlieb is that he’s terribly wounded at his core (”politics, poetry, promises, those are lies–”) because he’s someone who has been beaten down by life over and over and over and he buried himself in this defence mechanism in which he pretends like he has abandoned faith and only trusts in rationality and tangible rules (”–numbers are the handwriting of g-d”) because their inevitability is so comforting to him

but of course he’s lying. not only to other people but also to himself. the very language he uses to talk about this betrays him already.

and that’s what makes his story work: that this man who by all accounts presents himself to everyone as someone who scoffs at irrationality and avoids the company of human beings as a protective measure is also the one person who would, against all logic, be ready to sacrifice his life and all rights to privacy to protect the people he loves and the world in general

it’s ironic because drifting is a promise in a sense, and the type he and newt attempted is probably the biggest one of its kind, and he doesn’t even hesitate when it comes to it. he’s ready to tear down every single wall he’s built up over the course of his life because he believes so deeply in the cause and in people in general and i love him so much i could cry

Somali Astronomical/Astrological terms

Astronomy/Astrology;
Space = Fagaagga
Space and Time = Fagaagga iyo Semen
Solar System = Iskujoogga qorraxda
The Solar System = Isku-joog qorraxeed
Galaxies = Diillimo-caanoodyo

Planets = Meerayaasha
Planet = Meere
Planet = Malluug

The Sun = Qorraxda
Moon = Dayax
Mercury = Dusaa
Venus = Waxaraxirta(Sugra) ama Bakool
Earth = Dhulka
Mars = Farraare
Jupiter = Cirjeex
Saturn = Raage
Uranus = Uraano
Neptune = Docay

Saturn rings = Garaangaraha Raage ama Saxal
Dwarf planet = meere-cillin ah
Super Nova = Xiddig dhimanaaya

Comets = Dibdheeryo
Meteorider = Burburka jibinta ama xiddigaha ridma
Meteor = Jibin

Open Universe = Koonka Furan
Closed Universe = Koonka Xiran
Expanding Universe = isbaahidda Koonka
Light = ifka
Speed of Light = Xawliga ifka
Cosmos = Koonka
Color Spectrum = shucaaca ilayska-ifka

Astronomy = Xiddiga'aqoonta
Satellite = Dayaxgacmeed
Space Exploration = Sahminta fagaagga sare adduunka
Earth-based observatories = Rugaha kuurgalka fagaagga dhulkaku Saldhigan
Research Centres = Rugo cilmibaaris
Spaceships = Maraakiibta cir-maaxidda
Telescope = Doorbin
Zoom = Waxweyneynta
Observatories = Rugo-kuurgal

Spherical = Ugxan
Rotation = meerwareeg
Rotate = udub-wareeg
North Pole = Qudbiga waqooyi
South Pole = Qudbiga konfur
Northern Lights = Wirwirka Qudbiga woqoyi
Southern Lights = Wirwirka Qudbiga koofureede
Equator = dhulbaraha

Core = Bu’
Inner core = Bu’da gudo-xigeenka
Greenhouse effect = kulaylkaydinta

Astronomical unit = Halbeegga xiddigaha
Diameter = Dhexroorka
Formula = Hab-xisaabeed

Particles = bitaanbitooyin
Anti-Particles = lidka-bitaanbitooyinka
Particles = iniino
Nuclear reaction = iskushubmidda bu'aha
Magnetic Field = Birlabta
Charge electricity = Cabbeynta Shixnadaha Jacda

Circuit: Mareegta
Current= Qulqulka
Forces = Xoogag

Inertia Law = Xogta qaynuunka nuuxsi wax-negaadsan
Acceleration = Xowli
Response Act = Qaynuunka falka iyo falcelinta
Theory of Relativity = Fikriga isudhiganka
Mass = Jir
Energy = Tabarta
Electromagnetic wavelengths = Dhererka hirarka ku danabeysan birlabta

Aries = laxo
Scorpion =daba-alleele ama dib-qallooc
Cancer = naaf
Gemeni = mataanaha
Virgo = afaggaal
Sagittarius = dameerajoogeen
Leo = Libaax

Resistor = Caabiye
Diode = Laba Qotinle
Transformer = Dooriye
Socod Karaarsan = Accelerated Motion
Samaan = Time
Barobax = Displacement
Fogaan = Distance
Karaar = Acceleration
Keynaan = Velocity
Xawaare = Speed
Celcelis = Average
Socod Winiin = Circular Motion
Barobax –xagleed = Angular Displacement
Karaar –xagleed = Angular Acceleration
Keynaan –xagleed = Angular Velocity
Xoog = Force
Culeys = Weight
Xoog –cuf –isjiidad = Gravitational Force
Xoog –dilaac = Shear Force
Xoog –giigsan = Tensile Force (tension)
Xoog –islis = Frictional Force
Xoog –ligan = Normal Force
Xoog –taab = Tangential Force
Xoog –urrur = Compressive Force (compression)
Xoog –xudumeed = Centripetal Force
Weheliyaha isliska = Coefficient of Friction
Gujo = Impulse
Gujo –xagleed = Angular impulse
Kalka = Period
Maroojin = Moment
Maroojinta Wahsiga = Moment of inertia (Angular mass)
Tamar = Energy
Hawl = Work
Awood = Power
Tamar-keyd = Potential Energy
Tamar-socod = Kinetic Energy

Speed = Xawaare
Velocity = Kaynaan
Acceleration = Karaar (I think this is more appropriate than xowli and this is how i remember it from Fiisigiska)
Deceleration = Karaar-jab

Average Velocity： Kaynaan Celcelis
Diffusion: Saydhin
Infrared: Casaan Dhiimeed
Decay: Qudhmis
Decay series: Qudhmis isdaba Joog
Resistance: Iska caabin
Concave Mirror: Bikaaco Golxeed
Convex Mirror: Bikaaco Tuureed
Virtual Image: Humaag Beeneed
Real Image: Humaag runeed
Displace Ment: Baro bixin
Vector: Leeb
Hir = Wave
Daryan = Echo
Itaal = Intensity
Tooxda maqalka = Range of audibility
Danan = Pitch
Isku duba-dhac = Resonance
Ileys = Light
Noqod = Reflection
Daahfurran/gudbiye = Transparent
Falaar abaareed = Incident ray
Golxo = Concave
Tuur = Convex
Xagasha qiirqiirka = Critical angle

Xisaab - Maths

Tiro = Number
Tiro tirsiimo = Natural number
Tiro idil = Whole number
Abyoone = Integer
Jajab/Tiro lakab = Rational number
Mutaxan = Prime
Farac = Composite
Tiro maangal ah = Real number
Tiro maangad ah = Imaginary number
Tiro kakan = Complex number
Faraq = difference
Taran = Product
Qeyb = Quotient
Urur = Set
Hormo-urur = Subset
Hormo-urur quman = Proper subset
Dhextaal = Intersection
Isutag = Union
Duleedin = Complement
Aljebro = Algebra
Doorsoome = Variable
Tibix = Term
Hawraar = Expression
Tibxaale = Polynomial
Heer = Degree
Hawraar lakab = Rational expression
Isirayn = Factorization
Isir = Factor
Isle’eg = Equation
Isle’eg toosan = Linear equation
Sunsun = Sequence
Dareerimo = Series
Faansaar = Function
Wanqar = Symmetry
Tikraar = Intercept
Weydaar = Inverse
Qiime sugan = Absolute value

Joomitiri - Geometry

Bar = Point
Xariiq = Line
Xagal = Angle
Fool = Bearing
Barbaro = Parallel
Qoton = Perpendicular
Sallax = Plane
Goobo = Circle
Dhexroor/Dhexfur = Diameter
Qaanso = Arc
Fatuuq = Sector
Labojibaarane = Square
Dhinac = Side
Laydi = Rectangle
Joog = Length
Balac = Width
Sal = base
Dherer = Height
Gundho = Centroid
Barbaroole = Parallelogram
Koor = Trapezoid
Qardhaas = Rhombus
Gunbur = Pyramid
Toobin = Cone
Dhululubo = Cylinder
Gabal Toobineed = Conic section
Goobo = Circle
Saab = Parabola
Gees = Vertex
Jeedshe = Directrix
Kulmis = Focus
Qabaal = Ellipse
Kulmisyo = Foci
Dhidib weyne = Major axis
Dhidib yare = Minor axis
Labosaab = Hyperbola
Dhidib-xisti = Conjugate axis
Taxaneyaal = Matrices
Taxane = Matrix
Jiiftax = Row
Joogtax = Colunm
Suge = Determinant
Leeb = Vector
Itimaal = Probability
Abnaqan = Factorial
Racayn = Combination
Waqdhac = Event
Xigidda = Differentiation
Xigsin = Derivative
Lid-xigsin = antiderivative
Abyan = Integration
Abyane huban = Definite integral

a little surprise

Lukas sits in the diner down the street from his Computer Studies class and eats his burger, drumming his fingers on the table. He hates Tuesdays now because his and Philip’s schedules are exactly opposite, and they don’t get to see each other until late because Philip’s History of Photography class runs until seven.

He texts Philip. Thinking about me in class?

Always Philip replies, a moment later. You eating dinner?

Lukas picks at his fries, dipping one into the ketchup. He types out a response with one finger. Sadly. Lonely. We only get three hours together tonight before we have to go to sleep.

Lukas sighs, wishing it was Friday already. On Friday they both only have two classes early in the morning, and then they have the whole weekend off.

We’ve got date night Thursday. Then Friday and the weekend are all ours :)

Bartho, help me! I have trouble creating aliens but I feel like a failure of an author for asking for help, yet I just have tried and feel like the aliens I make are dumb or lame. Can you please help me?

anon, you’re very brave. people should ask for help when they need it. that’s what helps you improve.

my first tip is to look around, especially when there’s a lot of artists who love drawing aliens. pick and choose at things you like. changing colors? awesome, keep it. wings? sure, why not? nine eyes? yes, please. feet for hands and hands for feet? love it

tip number two, rationalize it. why does your species have the ability to breathe in the water and above it? “why that’s easy, bartho, they need that ability, because when their waters started getting polluted and to survive, they had to adapt outside of their natural element” awesome, good idea

tip number three, think of the origins and their planet. assuming you did tip number two, and you thought about the type of habitat your creatures live in, you’ve already partially thought about the kind of planet you want them to live on. how did they evolve?

tip number three, what do they eat? think about their teeth and how many stomachs they have. how do they catch their food? are they hunters, gatherers, scavengers?

tip number four, biology. organs in their body, are they like humans? what about their sexes and genders? how to they raise their young? how many young can they have, at once, and in total?

some other things you can look up are languages, different dialects, their intelligence, their economy, and how advanced their tech is. i recommend checking articles to see what pet peeves readers or other writers have when it comes to sci fi worlds. most times, i see people getting mad at people for making the feminine aliens like feminine humans. like, why would a lizard have boobs if they lay eggs?

i hope this helps, anon

Is there a theorem or conjecture that the square root of any non-square integer is irrational?

Keep reading for the proof. [EDIT: I tried to hide this but apparently it didn’t show up properly on mobile. So sorry for the wall of text if you didn’t want the proof]

Your question could be rephrased as “if you square a rational number that is not an integer, you can’t get an integer”. So consider a rational number- that is, a fraction x=p/q where p and q have no common factors. Assume that q is not 1 so that x is not a whole number.

Now if you square x, you get p^2/ q^2. If p and q have no common factors, then p^2 and q^2 have no common factors- it suffices to consider whether they have a prime factor in common, and the prime factors of p^2 are the same as the prime factors of p (and similarly for q). So p^2/q^2 is a fraction in its lowest terms. The denominator q^2 still isn’t 1, so x^2 is not a whole number. Which is what we wanted to prove.

Ever since I’ve taken Algebra 2, I internally panic if my answer produces a solid clean rational number instead of something like 0.23037087905…

Could you explain this tfw no ZF joke? I really dont get it... :D

Get ready for a long explanation! For everyone’s reference, the joke (supplied by @awesomepus​) was:

Q: What did the mathematician say when he encountered the paradoxes of naive set theory?
A: tfw no ZF

You probably already know the ‘tfw no gf’ (that feel when no girlfriend) meme, which dates to 2010. I’m assuming you’re asking about the ZF part.

Mathematically, ZF is a reference to Zermelo-Fraenkel set theory, which is a set of axioms commonly accepted by mathematicians as the foundation of modern mathematics. As you probably know if you’ve taken geometry, axioms are super important: they are basic assumptions we make about the world we’re working in, and they have serious implications for what we can and can’t do in that world.

For example, if you don’t assume the Parallel Postulate (that consecutive interior angle measures between two parallel lines and a transversal sum to 180°, or twice the size of a right angle), you can’t prove the Triangle Angle Sum Theorem (that the sum of the angle measures in any triangle is also 180°). It’s not that the Triangle Angle Sum Theorem theorem is not true without the Parallel Postulate — simply that it is unprovable, or put differently, neither true nor false, without that Postulate. Asking whether the Triangle Angle Sum Theorem is true without the Parallel Postulate is really a meaningless question, mathematically. But we understand that, in Euclidean geometry (not in curved geometries), both the postulate and the theorem are “true” in the sense that we have good reason to believe them (e.g., measuring lots of angles in physical parallel lines and triangles). Clearly, the axioms we choose are important.

Now, in the late 19th and early 20th century, mathematicians and logicians were interested in understanding the underpinnings of the basic structures we use in math — sets, or “collections,” being one of them, and arithmetic being another. In short, they were trying to come up with an axiomatic set theory. Cantor and Frege were doing a lot of this work, and made good progress using everyday language. They said that a set is any definable collection of elements, where “definable” means to provide a comprehension (a term you’re familiar with if you program in Python), or rule by which the set is constructed.

But along came Bertrand Russell. He pointed out a big problem in Cantor and Frege’s work, which is now called Russell’s paradox. Essentially, he made the following argument:

Y’all are saying any definable collection is a set. Well, how about this set: R, the set of all sets not contained within themselves. This is, according to you, a valid set, because I gave that comprehension. Now, R is not contained within itself, naturally: if it is contained within itself, then it being an element is a violation of my construction of R in the first place. But R must be contained within itself: if it’s not an element of itself, then it is a set that does not contain itself, and therefore it is an element of itself. So we have that R ∈ R and also R ∉ R. This is a contradiction! Obviously, your theory is seriously messed up.

This paradox is inherently a part of Cantor and Frege’s set theory — it shows that their system was inconsistent (with itself). As Qiaochu Yuan explains over at Quora, the problem is exactly what Russell pointed out: unrestricted comprehension — the idea that you can get away with defining any set you like simply by giving a comprehension. Zermelo and Fraenkel then came along and offered up a system of axioms that formalizes Cantor and Frege’s work logically, and restricts comprehension. This is called Zermelo-Fraenkel set theory (or ZF), and it is consistent (with itself). Cantor and Frege’s work was then retroactively called naive set theory, because it was, of course, pretty childish:

There are two more things worth knowing about axiomatic systems in mathematics. First, some people combine Zermelo-Fraenkel set theory with the Axiom of Choice¹, resulting in a set theory called ZFC. This is widely used as a standard by mathematicians today. Second, Gödel proved in 1931 that no system of axioms for arithmetic can be both consistent and complete — in every consistent axiomatization, there are “true” statements that are unprovable. Or put another way: in every consistent axiomatic system, there are statements which you can neither prove nor disprove. For example, in ZF, the Axiom of Choice is unprovable — you can’t prove it from the axioms in ZF. And in both ZF and ZFC, the continuum hypothesis² is unprovable.³ Gödel’s result is called the incompleteness theorem, and it’s a little depressing, because it means you can’t have any good logical basis for all of mathematics (but don’t tell anyone that, or we might all be out of a job). Luckily, ZF or ZFC has been good enough for virtually all of the mathematics we as a species have done so far!

The joke is that, when confronted with Russell’s paradox in naive set theory, the mathematician despairs, and wishes he could use Zermelo-Fraenkel set theory instead — ‘that feel when no ZF.’

I thought the joke was incredibly funny, specifically because of the reference to ‘tfw no gf’ and the implication that mathematicians romanticize ZF (which we totally do). I’ve definitely borrowed the joke to impress friends and faculty in the math department…a sort of fringe benefit of having a math blog.

– CJH

squaring my face is a problem. it is the challenge of constructing a square with the same area as my given face by using only a finite number of steps with compassion and straightedge. in 2017, the task was proven to be impossible, yet there’s work to be done. approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of persons, since there are rational numbers arbitrarily far from the axis of symmetry. the expression “squaring my face” is never allowed to be used for it is not a metaphor. it is reality as you do the impossible. the term quadrature of my face is sometimes used to mean the same thing as squaring my face, but it may also refer to approximate or verbose methods for finding the area of my terrified face to cover with the complacent hands

The number of rational hypotheses that can explain any given phenomenon is infinite
—  Robert Pirsig
flickr

rational circles by stephen schiller
Via Flickr:
This image consists only of circles rendered at about 1 pixel thick, of varying intensity. As in the “line thicket” image earlier in the photo stream, the ellipsoidal dark areas emerge only due to a lack of arcs drawn in those regions. Because the circles are restricted to the set of “rational” circles they tend to avoid certain areas and also tend to intersect at certain other locations, giving rise to the background pattern of light and dark. (Also in this image the circles are drawn white on a black background as opposed to dark on white.) Mathematical information: These are rational circles in the following way. A circle can be represented as a 4-tupple of numbers, say {a, b, c, d}, where a point (x,y) is on the circle if a*(x^2 + y^2) + b*x + c*y + d equals 0. For the rational circles the numbers a,b,c,d are integers. The subset of rational circles drawn above have a^2 + b^2 + c^2 <= 9, and d equal to any integer. The area shown in the image is the unit square. (I call them “rational” because a rational number is a ratio of integers and the density of these rational circles in the space of all circles resembles that of the density of rational numbers in the space of all (real) numbers.)

Dirichlet Function.

Notice the graph looks like 2 solid horizontal lines, but it is still a function and will still pass the vertical line test.

How is this possible?

Remember a graph is merely a model or representation of the true function. In this case since rational and irrational numbers are both dense in the real numbers, the “lines” y=0 and y=1 both appear solid, but in actuality have gaps. The “line” y=1 has gaps at each irrational x value and similarly, y=0 has gaps at each rational x. For example, a vertical line x=½ would hit the top “line” but miss the bottom one.

…Wait, what? How can you miss a line when you are headed straight for it?

A line (or curve) is defined as a continuum of points. So the 2 “lines” in the graph of the Dirichlet function above are not truly lines as they do not represent a continuum of points. If one could represent the Dirichelt function accurately there would be small gaps in between the points that make up these “lines” but the gaps would be so, so, very small and perhaps impossible to locate. So the vertical line x=½ hits the top “line” but slips through a gap in the bottom one.

Another way to think about it is to recall a relation is a function if each input has one and only one output. Since every real number input is either rational or irrational and never both, every input in the formula above gives exactly one output and thus the Dirichlet function is indeed a function.

This is an example of a function that is defined everywhere but continuous nowhere, and is rarely seen in mathematics classrooms.

A brief history lesson....

Oh so the only thing Arabs and Muslims bring is terrorism? I guess everyone forgot that….

1. Surgery
Around the year 1,000, the celebrated doctor Al Zahrawi published a 1,500 page illustrated encyclopedia of surgery that was used in Europe as a medical reference for the next 500 years. Among his many inventions, Zahrawi discovered the use of dissolving cat gut to stitch wounds – beforehand a second surgery had to be performed to remove sutures. He also reportedly performed the first caesarean operation and created the first pair of forceps.

2. Coffee
Now the Western world’s drink du jour, coffee was first brewed in Yemen around the 9th century. In its earliest days, coffee helped Sufis stay up during late nights of devotion. Later brought to Cairo by a group of students, the coffee buzz soon caught on around the empire. By the 13th century it reached Turkey, but not until the 16th century did the beans start boiling in Europe, brought to Italy by a Venetian trader.

3. Flying machine
“Abbas ibn Firnas was the first person to make a real attempt to construct a flying machine and fly,” said Hassani. In the 9th century he designed a winged apparatus, roughly resembling a bird costume. In his most famous trial near Cordoba in Spain, Firnas flew upward for a few moments, before falling to the ground and partially breaking his back. His designs would undoubtedly have been an inspiration for famed Italian artist and inventor Leonardo da Vinci’s hundreds of years later, said Hassani.

4. University In 859 a young princess named Fatima al-Firhi founded the first degree-granting university in Fez, Morocco. Her sister Miriam founded an adjacent mosque and together the complex became the al-Qarawiyyin Mosque and University. Still operating almost 1,200 years later, Hassani says he hopes the center will remind people that learning is at the core of the Islamic tradition and that the story of the al-Firhi sisters will inspire young Muslim women around the world today.

5. Algebra
The word algebra comes from the title of a Persian mathematician’s famous 9th century treatise “Kitab al-Jabr Wa l-Mugabala” which translates roughly as “The Book of Reasoning and Balancing.” Built on the roots of Greek and Hindu systems, the new algebraic order was a unifying system for rational numbers, irrational numbers and geometrical magnitudes. The same mathematician, Al-Khwarizmi, was also the first to introduce the concept of raising a number to a power.
6. Optics/Magnifying Glass
Not only did the Arab world revolutionize mathematics – it also revolutionized optics. The scholar Alhazen (Abu al-Hasan) from Basra was the first person to describe how the eye works.He carried out experiments with reflective materials and proved that the eye does not sense the environment with “sight rays,” as scientists had believed up until then. He also discovered that curved glass surfaces can be used for magnification.His glass “reading stones” were the first magnifying glasses. It was from these that glasses were later developed. Furthermore, Alhazen wrote important scholarly texts on astronomy and meteorology.“ Many of the most important advances in the study of optics come from the Muslim world,” says Hassani. Around the year 1000 Ibn al-Haitham proved that humans see objects by light reflecting off of them and entering the eye, dismissing Euclid and Ptolemy’s theories that light was emitted from the eye itself. This great Muslim physicist also discovered the camera obscura phenomenon, which explains how the eye sees images upright due to the connection between the optic nerve and the brain.
7. Music
Muslim musicians have had a profound impact on Europe, dating back to Charlemagne tried to compete with the music of Baghdad and Cordoba, according to Hassani. Among many instruments that arrived in Europe through the Middle East are the lute and the rahab, an ancestor of the violin. Modern musical scales are also said to derive from the Arabic alphabet. The guitar, as we know it today, has its origins in the Arabic oud – a lute with a bent neck. During the Middle Ages, it found its way to Muslim Spain, where it was referred to as “qitara” in the Arabic of Andalusia. It is said that a music teacher brought one to the court of the Umayyad ruler Abdel Rahman II in the ninth century. The modern guitar developed as a result of many influences, but the Arabic lute was an important predecessor.
8. Toothbrush
According to Hassani, the Prophet Mohammed popularized the use of the first toothbrush in around 600. Using a twig from the Meswak tree, he cleaned his teeth and freshened his breath. Substances similar to Meswak are used in modern toothpaste.
9. The crank
Many of the basics of modern automatics were first put to use in the Muslim world, including the revolutionary crank-connecting rod system. By converting rotary motion to linear motion, the crank enables the lifting of heavy objects with relative ease. This technology, discovered by Al-Jazari in the 12th century, exploded across the globe, leading to everything from the bicycle to the internal combustion engine.
10. Hospitals
“Hospitals as we know them today, with wards and teaching centers, come from 9th century Egypt,” explained Hassani. The first such medical center was the Ahmad ibn Tulun Hospital, founded in 872 in Cairo. Tulun hospital provided free care for anyone who needed it – a policy based on the Muslim tradition of caring for all who are sick. From Cairo, such hospitals spread around the Muslim world.

11. Marching bands Military marching bands date back to the Ottoman Mehterhane. These were bands which played during the entire battle and only ceased their music-making when the army retreated or the battle was over.During the wars with the Ottoman Empire, the bands are thought to have made a considerable impression on European soldiers – after which they adapted the principle for their own use.

12. Parachute
A thousand years before the Wright brothers a Muslim poet, astronomer, musician and engineer named Abbas ibn Firnas made several attempts to construct a flying machine. In 852 he jumped from the minaret of the Grand Mosque in Cordoba using a loose cloak stiffened with wooden struts. He hoped to glide like a bird. He didn’t. But the cloak slowed his fall, creating what is thought to be the first parachute, and leaving him with only minor injuries. In 875, aged 70, having perfected a machine of silk and eagles’ feathers he tried again, jumping from a mountain. He flew to a significant height and stayed aloft for ten minutes but crashed on landing - concluding, correctly, that it was because he had not given his device a tail so it would stall on landing.Baghdad international airport and a crater on the Moon are named after him.

13. Shampoo
Washing and bathing are religious requirements for Muslims, which is perhaps why they perfected the recipe for soap which we still use today. The ancient Egyptians had soap of a kind, as did the Romans who used it more as a pomade. But it was the Arabs who combined vegetable oils with sodium hydroxide and aromatics such as thyme oil. One of the Crusaders’ most striking characteristics, to Arab nostrils, was that they did not wash. Shampoo was introduced to England by a Muslim who opened Mahomed’s Indian Vapour Baths on Brighton seafront in 1759 and was appointed Shampooing Surgeon to Kings George IV and William IV.

14. Vaccination
The technique of inoculation was not invented by Jenner and Pasteur but was devised in the Muslim world and brought to Europe from Turkey by the wife of the English ambassador to Istanbul in 1724. Children in Turkey were vaccinated with cowpox to fight the deadly smallpox at least 50 years before the West discovered it.

15.  Pay Cheques The modern cheque comes from the Arabic saqq, a written vow to pay for goods when they were delivered, to avoid money having to be transported across dangerous terrain. In the 9th century, a Muslim businessman could cash a cheque in China drawn on his bank in Baghdad.

16. Earth’s Shape?
By the 9th century, many Muslim scholars took it for granted that the Earth was a sphere. The proof, said astronomer Ibn Hazm, “is that the Sun is always vertical to a particular spot on Earth”. It was 500 years before that realisation dawned on Galileo. The calculations of Muslim astronomers were so accurate that in the 9th century they reckoned the Earth’s circumference to be 40, 253.4km - less than 200km out. The scholar al-Idrisi took a globe depicting the world to the court of King Roger of Sicily in 1139.

17. Gardens
Medieval Europe had kitchen and herb gardens, but it was the Arabs who developed the idea of the garden as a place of beauty and meditation. The first royal pleasure gardens in Europe were opened in 11th-century Muslim Spain. Flowers which originated in Muslim gardens include the carnation and the tulip.

18) Refinement
Distillation, the means of separating liquids through differences in their boiling points, was invented around the year 800 by Islam’s foremost scientist, Jabir ibn Hayyan, who transformed alchemy into chemistry, inventing many of the basic processes and apparatus still in use today - liquefaction, crystallisation, distillation, purification, oxidisation, evaporation and filtration. As well as discovering sulphuric and nitric acid, he invented the alembic still, giving the world intense rosewater and other perfumes and alcoholic spirits (although drinking them is haram, or forbidden, in Islam). Ibn Hayyan emphasised systematic experimentation and was the founder of modern chemistry.

So……..

We literally SHAPED THE MODERN WORLD kiss my entire ass.

Home, Chapter 3

TITLE: Home
CHAPTER NUMBER: 3/?
AUTHOR: Losille2000
WHICH TOM/CHARACTER: Actor!Tom
GENRE: Romance/Drama
FIC SUMMARY: Tom returns home grouchy and exhausted from a cramped flight after four months away for work. Unfortunately, there’s already someone sleeping in his bed.
RATING: M (sex, language)
WARNINGS:  None.

Previous Chapter

Myshka (Bard the Bowman x Reader one shot)

Title: Myshka

Rating: PG

Word Count: 6537 ( I know, I  know, it’s a long one!)

Notes/Summary:  So I had a request over on deviantart to write a one shot about Bard the Bowman rescuing the reader form the lake, and it turned into this.  I also realised it kind of fits in with this imagine over at imaginexhobbit and they are always looking for more Bard.  So here we go!

TBT ENTP #12

i skim when i read and make up the text based on the letters i see. I once thought i spotted a book called pirates of the carribbean, brought my friend to the library to show her the pirates section… turns out it was called planes of composition

Nine Hundred and Sixty-Two

This was an anon request for a jealous Hermione. You had asked for it to be about an Auror colleague of Ron’s and I tried but I couldn’t make it work. Sorry, anon, I hope you still like it and thanks for sending it in.

If Hermione was certain of anything in this life, it was that Ron Weasley loved her with all his heart.

But that didn’t mean she didn’t worry a little.

After the war, the trio had reached an odd, feverish level of celebrity in the wizarding world that brought with it a crush of attention and many ardent admirers. Harry was used to the scrutiny, Hermione brushed it off but Ron, Ron had been overlooked and underappreciated his whole life and some of those ardent admirers were busty and blonde…

“Ah, come on,” Ron said. “Why would I go for those stewed mushrooms when I already have a bacon sandwich?”

“How flattering,” Hermione sighed.

“Are you thinking about that girl from this afternoon?” he said. “She was getting a little touchy, yes, but I was just being polite. When dealing with difficult women, I have found it works best to be complimentary.” He paused. “Have I mentioned you look lovely tonight?”

She swatted at him half-heartedly.

“Hermione?” he said and there was no more joking in his voice. “Do you really not know?”

When she didn’t reply he took her face in his hands and gave her a firm kiss. And not just any kiss, but one of those ardent, slow, toe-curlingly wonderful kisses that he was so terribly good at.

“Nine hundred and sixty-two,” he said when he finally let her resurface.

“What?” she said dimly, still trying to regain her bearings.

“Nine hundred and sixty-two. I have kissed you nine hundred and sixty-two times.”

She laughed. “You’re counting?”

“Yes,” he said seriously. “I’ve been keeping track since the first one. Mouths only. No cheeks or, uh, other parts. It gets a little hard because they sometimes sort of run into each other but I think I’m pretty accurate. Plus or minus five, anyway.”

She looked at him, amused and flattered but also rationally skeptical. The number couldn’t possibly be that high. They hadn’t been together all that long, and she’d have to subtract all the time she spent with her parents and those first horrible days that had been filled with funerals and tears. But on the other hand, there had been some secret, wonderful days where it seemed like all they had done was kiss each other and if she averaged them all out…

“See?” Ron said smugly. “Coming up on a thousand. Cause for celebration, that, and I’ve planned accordingly. We might even get there tonight if you play your cards right.”

She stood up on her toes and kissed him. “I think we can do it.”

“And just wait until you see what I’ve got planned for ten thousand,” Ron said, pulling her closer. “And then one hundred thousand. I’ve already got them all sorted.” He gave a sad shake of his head. “You’re stuck with me, I’m afraid.”

But that was something that didn’t scare her at all.

hey! are you okay?

Hey! I’m ok I guess I just got back from school and it wasn’t that bad but I forgot what a rational number is and like.,,, wtf madison,,,, how do you forget something you learned 4 years ago???,, anyway in another class we talked about boobs, French kissing, butts, moms who date guys with greasy hair, guys sending nudes but the teacher couldn’t say nudes bc she’ll get in trouble so she said “he didn’t send her a picture of his big toe” and yeah I ate and felt sick but didn’t purge or cry in the school bathrooms and I’m so tired I didn’t sleep all night and we also talked about sex in books so :) it’s been eh…. also I’m reading Carry On by Rainbow Rowell which is really cool and this one kid I knew from 3-4 years ago is back and he knows a lot about me so he better keep his mouth closed or I will attack him when he walks by my house bc we live in the same neighborhood.