[Mathematic Form 4] Lower or Upper?

When quantities are measured, their data can be grouped into several classes. The range of each class is called the class interval.

For a class interval of 70 - 79, the lower limit is 70 and the upper limit is 79.

For a class interval of 80 - 89, the lower limit is 80 and the upper limit is 89.

The lower boundary of each class refers to the middle value between the lower limit of the class and the upper limit of the previous class.

The upper boundary of each class refers to the middle value between the upper limit of the class and the lower limit of the following class.

The size of a class interval is the difference between the upper boundary and the lower boundary of the class.

[Mathematic Form 4] Classified

A SET is a well-defined group of objects, or a collection of objects that have a common property.

Objects in a set are called members or elements of the set.

Sets can be defined either by description, or set notation with braces { } .

Eg:
By description, P is a set of consonants in the word "EXCELLENT".
By set notation with braces, P = { consonants in the word "EXCELLENT"} or
P = {x:x is a consonant in the word "EXCELLENT"} or
P = {X, C, L, N, T}

(Note the use of commas, and non-repetition of the same elements in the set, i.e. the letter "L")

[Mathematic Form 4] Cartesian Coordinates

Cartesian coordinates are rectilinear two-dimensional or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The three axes of three-dimensional Cartesian coordinates, conventionally denoted the x-, y-, and z-axes (a notation due to Descartes) are chosen to be linear and mutually perpendicular. In three dimensions, the coordinates x, y, and z may lie anywhere in the interval

In René Descartes' original treatise (1637), which introduced the use of coordinates for describing plane curves, the axes were omitted, and only positive values of the x- and the y-coordinates were considered, since they were defined as distances between points. For an ellipse this meant that, instead of the full picture which we would plot nowadays (left figure), Descartes drew only the upper half (right figure).

The inversion of three-dimensional Cartesian coordinates is called 6-sphere coordinates.

The scale factors of Cartesian coordinates are all unity, h_i=1. The line element is given by



and the volume element by



The gradient has a particularly simple form,



as does the Laplacian




The vector Laplacian, , is

The divergence is




and the curl, , is






The gradient of the divergence, , is

[Mathematic Form 4] True Or False?

A statement is a sentence that is either true or false, but not both. For example:

a) A hexagon has six sides. (A true statement)

b) 8 + 5 > 14 (A false statement: 8 + 5 = 13; 13 < 14)

c) What is the square root of 25? (A question; not a statement)

d) Stand up. (An instruction; not a statement)

e) Wow! (An exclamation; not a statement)


True or false statements can also be constructed from numbers and mathematical symbols.

Eg:

True statements:
a) 9 + 21 > 16
b) 16 - 10 < 7
c) 16 = 9 + 7

False statements:
a) 21 + 9 < 16 + 7
b) 7 + 9 = 21 - 16
c) 7 > 16 + 21

[Mathematic Form 4] Lines of Argument

An argument involves a given set of statements called premises. From the statements, a conclusion can be made.

The following are three simple forms of arguments that can be used to make a conclusion.

Argument (Form I)
Premise 1 : All A are B
Premise 2 : C is A
Conclusion: C is B
Eg:
Premise 1 : All heptagons have 7 sides.
Premise 2 : Object Y is a heptagon.
Conclusion: Object Y has 7 sides

Argument (Form II)
Premise 1 : If p, the q.
Premise 2 : p is true.
Conclusion: q is true
Eg:
Premise 1 : If k = 5, then 3k - 1 = 14
Premise 2 : k = 5
Conclusion: 3k - 1 = 14

Argument (Form III)
Premise 1 : If p, then q.
Premise 2 : Not q is true.
Conclusion: Not p is true.
Eg:
Premise 1 : If an integer, x, is a factor of 6, it is also a factor of 12.
Premise 2 : x is not a factor of 12
Conclusion: x is not a factor of 6

More Arguments...

1. Premise 1 : If x is an even number, it is divisible by 2.
Premise 2 : x is an even number.
Conclusion: x is divisible by 2.

2. Premise 1 : If p > q, then p > r.
Premise 2 : p > q.
Conclusion: p > r.

3. Premise 1 : All hexagons have six sides.
Premise 2 : J is a hexagon.
Conclusion: J has six sides.

4. Premise 1 : If cos x = 0.5, then x = 60^o or 120^o
Premise 2 : cos x = 0.5
Conclusion: x = 60^o or 120^o

5. Premise 1 : All prime numbers have only two factors.
Premise 2 : 9 is not a prime number.
Conclusion: 9 has more than two factors.

[Mathematic Form 4] To Round Off Numbers

To round off number to an appropriate number of significant figures, you may use the following steps:

1. Identify digit x that is to be rounded off.
2. Is the digit after x greater than, or equal to, 5?
3. If it is either greater than, or equal to, 5, add 1 to x.
4. If it isn't, then x remains unchanged.
5. Do the digits after x lie before or after the decimal point?
6. If before, replace each digit with a zero.
7. If after, drop the digits.

Let's try the steps on a few examples.

Suppose we have to round off the following numbers:

a) 34,782 to 1 significant figure
Solution:
34,782
The digit to be rounded off is 3. The digit after 3 is 4. 4 is less than 5.
Therefore, 3 remains unchanged and each digit to the right of 3 (4, 7, 8, 2) is to be replaced with zero.

34,782 = 30,000 (1 significant figure)

b) 54.78 to 3 significant figures
Solution:
54.78
The digit to be rounded off is 7. The digit after 7 is 8. 8 > 5.
Therefore, add 1 to 7 and drop the digit 8 because it lies after the decimal point.

54.78 = 54.8 (3 significant figure)

c) 0.0050327 to 2 significant figures
Solution:
0.0050327
The digit to be rounded off is 0. The digit after 0 is 3. 3 is less than 5, so leave 0 unchanged. Digit 2 and 7 are after the decimal point, so drop the digits.

0.0050327 = 0.0050 (2 significant figure)

*********************

• 7245.9 to be rounded off to 2 significant figure (sf) is 7,200.
• 0.0011056 to be rounded off to 3 significant figure (sf) is 0.00111.
• 986,468 to be rounded off to 1 significant figure (sf) is 1,000,000.
• 5.00402 to be rounded off to 5 significant figure (sf) is 5.0040.
• 67.9081 to be rounded off to 4 significant figure (sf) is 67.91.

[Mathematic Form 4] Get The Right Figure

In scientific or technical studies, very large or very small numbers are used sometimes.

Eg:
The speed of light is approximately 300,000,000 m/s.

The mass of one oxygen atom is approximately 0.000000000000000000027 g.

That's quite a number of zeros, isn't it? Imagine the amount of space they would take up in your exercise books.

To manage these extremely large or small numbers more easily, we use significant figures and the standard form.

The accuracy level of a measurement in scientific work is indicated by the number of significant figures it has.

Here are some rules to follow when dealing with significant figures;

Rule 1: All non-zero digits are significant.
Eg:
6.78 has 3 significant figures
97.122 has 5 significant figures

Rule 2: Zeros between non-zero digits are significant figures.
Eg:
1,007 (4 significant figure)
3.0002 (5 significant figure)

Rule 3: A zero after the decimal point of a decimal number is a significant figure.
Eg:
6.0 (2 significant figure)
18.00 (4 significant figure)

Rule 4: In a decimal, zeros before the first non-zero digit are not significant.
Eg:
0.00865 (3 significant figure) [the first three zeros are not significant]
0.06 (1 significant figure) [the first two zeros are not significant]

Rule 5: In a whole number, zeros after the last non-zero digit may or may not be significant, depending on the level of accuracy specified.
Eg:
74,000 has 2 significant figure when rounded off to the nearest thousand.
74,000 has 3 significant figure when rounded off to the nearest hundred.

[Mathematic Form 4] A Quadratic Equation In General Form

A QUADRATIC equation with one unknown has an equal sign and the highest power of the unknown is 2.

A quadratic equation in general form is presented thus: aX² + bx + c = 0

The following are quadratic equations that are not in general form:
i) x(x + 7) = 17
ii) (x - 5) (x + 3) = 8
iii) 7/x = 9 + x

To rewrite them in general form, you may have to follow some or all of the following steps.

i)
x(x + 7) = 17
X² + 7x = 17
X² + 7x - 17 = 0 (general form)

ii)

(x - 5) (x + 3) = 8

X² + 3x - 5x - 15 = 8
X² - 2x - 15 - 8 = 0
X² - 2x - 23 = 0 (general form)

iii)
7/x = 9 + x
7 = x(9 + x)
7 = 9x + X²
X² + 9x - 7 = 0 (general form)