11 keys to learn oral/Spoken English

How to read / present math in English?

How to read / present math, equation, formula in English? 

1. Logic

∃ there exist
∀ for all
pp implies q / if p, then q
pp if and only if q /p is equivalent to q / p and q are equivalent


xx belongs to A / x is an element (or a member) of A
xx does not belong to A / x is not an element (or a member) of A
AA is contained in B / A is a subset of B
AA contains B / B is a subset of A
AA cap B / A meet B / A intersection B
AA cup B / A join B / A union B
A\B A minus B / the difference between A and B
A×B A cross B / the Cartesian product of A and B

3. Real numbers

x+1 x plus one
x-1 x minus one
x±1 x plus or minus one
xy xy / x multiplied by y
(x-y)(x+y) x minus y, x plus y
the equals sign
x=5 x equals 5 / x is equal to 5
x≠5 x (is) not equal to 5
xx is equivalent to (or identical with) y
x>y x is greater than y
x≥y x is greater than or equal to y
x<y x is less than y
x≤y x is less than or equal to y
0<x<1 zero is less than x is less than 1
0≤x≤1 zero is less than or equal to x is less than or equal to 1
|x| mod x / modulus x
x2  x squared / x (raised) to the power 2
x3  x cubed
x4  x to the fourth / x to the power 4
xn  x to the nth / x to the power n
x (−n) x to the (power) minus n
x^2 (square) root x / the square root of x
x^3 cube root (of) x
x^4 fourth root (of) x
x^n nth root (of) x
(x+y)2 x plus y all squared
n! n factorial
x^x hat
x¯ x bar
x˜ x tilde
xi xi / x subscript i / x suffix i / x sub i
∑(i=1~n) ai the sum from i equals one to n ai / the sum as i runs from 1 to n of the ai
xthe norm (or modulus) of x
OAOA / vector OA
OA¯ OA / the length of the segment OA
AT A transpose / the transpose of A
A−1 A inverse / the inverse of A

5. Functions

f(x) fx / f of x / the function f of x
f:Sa function f from S to T
xx maps to y / x is sent (or mapped) to y
f’(x) f prime x / f dash x / the (first) derivative of f with respect to x
f”(x) f double-prime x / f double-dash x / the second derivative of f with respect to x
f”’(x) triple-prime x / f triple-dash x / the third derivative of f with respect to x
f (4) (x) f four x / the fourth derivative of f with respect to x
∂f/∂x1 the partial (derivative) of f with respect to x1
2f/∂x12 the second partial (derivative) of f with respect to x1
0 the integral from zero to infinity
limx0the limit as x approaches zero
limx0+the limit as x approaches zero from above
limx0−the limit as x approaches zero from below
logelog y to the base e / log to the base e of y / natural log (of) y
lnlog y to the base e / log to the base e of y / natural log (of) y
how can i read math set in English?
how to read math formulas in English? 

Useful template sentences for English writing test of essay

Useful template sentences of English writing test of essay


1. with the booming of economy

2. with the remarkable improvement of people's living standard

3. advanced science and technology

4. add much spice/ flavor to our daily life.

5. It is commonly believed that

6. I give my vote to the former / latter opinion

7. Sth. has aroused wide public concern./ Sth has drawn great public attention.

8. It is undeniable that ...

9. a heated discussion /debate

10. a controversial issue

11. As far as I am concerned, / Personally

12. be supported by sound reasons

13. argument on both sides

14. play an increasingly important role in ...

15. be indispensable to

16. As the proverb goes:

17. exert positive / negative effects on ...

18. lead to / give rise to /contribute to / result in

19. The advantages far outweight the disadvantages.

20. a complicated social phenomenon

21. sense of responsibility/ achivenment

22. sense of competition and cooperation

23. widen one's horizon / broaden one's vision

24. acquire knowledge and skills

25. financial burden / psychological burden

26. take many factors into consideration

27. from another perspective

28. make joint efforts.

29. be beneficial to / be conducive to ...

30. make contributions to the society

31. lay a solid foundation for ...

32. comprehensive quality

33. be committed / devoted to ...

34. Admittedly

35. unshakable duty

36. satisfy / meet the needs of ...

37. a reliable source of information

38. valuable natural resources

39. the Internet (note to have 'the', and 'i' should be captilized)

40. convenient and efficient

41. in all aspects of human life

42. environmentally friendly materials

43. a symbol of society progress

44. Sth has greatly facilitated people's lives.

45. hold different attitudes towards this issue

46. to some extent

47. integrate theory with practice

48. an irresistible trend of ...

49. the increasingly keen social competition

50. immediate interest / short-term interest

51. long-term interest

52. has its own merits and demerits / pros and cons

53. do harm to / be harmful to / be detrimental to

54. exchange ideas / emotions / information

55. keep pace with /keep abreast with the latest development of ...

56. the healthy development of ...

57. attach great importance to ...

58. social status

59. focus one's time and energy on ...

60. expand one's scope of knowledge

61. both physically and mentally

62. be directly / indirectly related to ...

63. give rise to / lead to / spell various problems

64. the words to replace 'think' : believe, claim, maintain, argue, insist, hold the opinion / belief / view that

65. relieve stress / burden

66. give (top) priority to sth

67. compared with .../ in comparison with

68. degradable / decomposable material

69. replace / substitute / take the place of

70. offer job opportunities

71. mirror the social progress / advance

72. enhance / promote mutual understanding

73. make full use of / take advantage of

74. suffer from heavier work pressure

75. guarantee the stability and prosperity of our society

76. realize one's dream

77. adapt oneself to the social development

78. put more emphasis on ...

79. the main / leading reasons are list as follows:

80. we still have a long way to go

6 ways to make your day more than 24-hour

1. Check  email  less

2.  Plan the next day the night before

3. Don't fight against your body clock.
    if you get used to stay up late and get up late, just do it. It's good as long as it is regular.
    You can be efficient either way.

4.  eat less eat well, but eat more frequently

5. stay hydrated

6. Do the hard work up front 

Cool English

How to book flight ticket

Frequently used sentences on booking a flight ticket.

1. I want a package deal including airfare and hotel.
2. I'd like to change this ticket to the first class.
3. I'd like to reserve a sleeper to Chicago.
4. I'd like to sit in the front of the plane.
5. I'd like two seats on today's Northwest Flight 8 to Toronto, please
6. I'd like to buy an excursion pass instead.
7. I'd like a refund on this ticket.
8. I'd like to have a seat by the window.
9. we have only one a day for New York.
10. The flight number is CA1024 on Sep 5th.
11. There's a ten thirty flight in the morning.
12. I'd like to reconfirm my flight from London to Tokyo.
13. My reservation number is 3425.
14. Do you have any tickets available for that date?
15. Would you please make my reservation to Chicago for tomorrow?
16. What is the fare to New York, Economy Class?
17. Where do I pick up the ticket?
18. Can I have a second - class one way ticket to Chicago, please?
19. Can I stop over on the way?
20. What time does the plane take off ?

How to face the past, the present and future

Face the past with the least regrets, face the present with the least waste and face the future with the most dreams.