Mathematics Dictionary
Dr. K. G. Shih
Question and Answer
Questions
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How to use ?
Q01 |
- Symbol Defintion
Q02 |
- Change x^n to mathematical symbols
Q03 |
- Change Abs(x) to mathemtical symbol
Q04 |
- Change sin(x)^2 to mathematical symbols
Q05 |
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Q06 |
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Q07 |
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Q08 |
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Q09 |
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Q10 |
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Answers
Q01. Symbol Definition
Characters
^ ................... It stands power
* ................... It stands multiplication
/ ................... It stands division
Numerical
2*3 ................. It menas 2 times 3 (2x3)
2/3 ................. It means 2 divide by 3
2^3 ................. It means 2 to power 3
Mathematical
abs(x) .............. It means absolute value of x. e.g. |-1|=1
atn(x) .............. It stands for arctan(x). e.g. atn(1)=pi/4
cis(x) .............. It is cos(x) + i*sin(x)
cos(x) .............. It is cosine function of x (x in deg or radians
C(n,r) .............. It stands for coefficients of binomial expansion
e^x ................. It means e to power x
exp(x) .............. It stands for exponent. e.g. exp(1)=2.718281829
log(x) .............. It is natural logarithm. e.g. ln(e) = 1, ln(1)=0
n! .................. It is n factorial. e.g. 5! = 1*2*3*4*5 = 120
pi .................. It is 3.141592...
sin(x) .............. It is sine function of x (x in deg or radians)
tan(x) .............. It is tangent function of x (x in deg or radians)
sqr(x) .............. It means square roots of x. e.g. sqr(16)=4
Sum[n^2] ............ It stands for summation of n^2 from 1 to n
Computer Language
EQ .................. It stands equal to
GE .................. It stands greater than and equal to
GT .................. It stands greater than
LE .................. It stands less than and equal to
LT .................. It stands less than
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Q02. Change x^n to mathematical symbol
Example 1 : y = a*(x^2) + b*x + c
Mathematics :
y = ax² + bx + c
Example 2 : y = a*(x^3) + b*(x^2) + c*x + d
Mathematics :
y = ax³ + bx² + c*x + c
Example 3 : Sum[n^2] with limit n from 1 to n
Mathematics :
Sum[n^2] = ∑(n²)
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Q03. Change Abs(x) to mathematical symbol
Example 1 Solve abs(x+1)=2
Mathematics : Solve | x + 1 | = 0
Solution
(x+1)=+1 and x= 0
(x+1)=-1 and x=-2
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Q04. Change sin(x)^2 to mathematical symbols
Example 1 : y = sin(x)^2
Mathematics :
y = sin²(x)
Example 2 : cos(x)^2 + sin(x)^2 = 1
Mathematics :
cos²(x) + sin²(x) = 1
Example 1 : tan(x)^2 + 1 = sec(x)^2
Mathematics :
tan²(x) + 1 = sec²(x)
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Q05. Answer
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Q06. Answer
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Q07. Answer
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Q08. Answer
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Q09. Answer
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Q10. Answer
Example 2 Find angle between y=m*x+n and x-axis
Answer 02
* Angle = atn(m) in radians
* Angle = 180*atn(m)/pi and pi = 3.1416
* Where m is slope and n is y-intercept
Example 3 Prove cos(pi/3) = 1/2
Answer 02
* Right triangle ABC and angle A = pi/6 = 30 degrees
* Then Opp = Hyp/2 and Adj = sqr(3)/2
* Sin(A) = Opp/Hyp = 1/2 and cos(A) = sqr(3)/2
* Sin(B) = sqr(3)/2 and cos(B) = 1/2
* Where A = 30 degrees and B = 60 degrees
Example 4 Prove C(10,2) = C(10,8)
Answer 02
* Definition : C(n,r) = n(n-1)....(n-r+1)/r!
* C(10,2) = (10*9)/(1*2) = 450 (n=10 and r=2)
* C(10,8) = (10*9*8*7*6*5*4*3)/(1*2*3*4*5*6*7*8) = 450
* Hnece C(10,2) = C(10,8)
* Hence C(n, r) = C(n,n-r)
* Note : 0! = 1 and C(n,0) = 1 and C(n,n) = 1
Example 5 How many trailor zeros in n! ?
Answer 05
* Trailor zeros = n/5 + n/25 + n/125 + n/625 + ....
* Each divistion takes only integer portion
* For 20! Trailor zeros = 20/5 + 20/25 = 4 + 0 = 4
Example 6 Composite function : e^(ln(x)) = x.
It can be expressed as Exp(Log(x)) = x.
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