Author Archives: Nalin Pithwa

I am a DSP Engineer and a mathematician working in closely related areas of DSP, Digital Control, Digital Comm and Error Control Coding. I have a passion for both Pure and Applied Mathematics.

Skill Check IV: IITJEE Foundation Maths

I. State whether the following statements are True or False:

(i) 0 is to the left of all negative numbers on the number line.

(ii) 3 is greater than -3333.

(iii) 1\frac{2}{5} will lie to the right of the mid-point between 1 and 2 on the number line.

(iv) 1\frac{2}{5} will lie to the left of the mid-point between 1 and 2 on the number line.

(v) If a decimal fraction is non-terminating and non-recurring, it is known as a rational number.

(vI) The rational number 3\frac{1}{5} lies between 3\frac{2}{11} and 3 \frac{3}{11}

II. How many natural numbers lie between 212 and 2120?

III. How many integers lie between -219 and +2190?

IV. Write the following numbers in descending order: (i) -213, +126, -212, +127, -127 (ii) -1\frac{7}{11}, -3\frac{7}{11}, -2\frac{7}{11}, -5\frac{7}{11}, -4\frac{7}{11} (iii) \frac{3}{5}, -\frac{2}{9}, \frac{5}{7}, -\frac{3}{10}, \frac{11}{21} (iv) -2.3838, -2.3388, -2.8838, -2.8833, -2.3883 (v) 3.\overline{8}, 3.\overline{6}, 3.88, 3.6\overline{8}, 3.8\overline{6}

V. Write the following numbers in ascending order: (i) +418, -481, -418, +481, -841 (II) -1\frac{3}{11}, -1\frac{4}{11}, -1\frac{5}{11}, -1\frac{6}{11}, -1\frac{7}{11} (iii) \frac{2}{5}, \frac{11}{23}, \frac{7}{15}, \frac{9}{20}, \frac{3}{7} (iv) 6.7134, 6.7431, 6.7341, 6.7413, 6.7143 (v)7.9\dot{8}, 7.\dot{9}, 7.\overline{98}, 7.8\dot{9}, 7.\dot{8}

VI. Insert a rational number between the following pairs of numbers: (i) -0.001 and +0.001 (ii) -8 and -3 (iii) 85 and 86 (iv) 5.5 and 6 (v) \frac{1}{4} and \frac{1}{5} (vi) 2\frac{2}{5}, 2\frac{3}{5} (vii) 3.0688 and 3.0699 (viii) 5.2168 and 5.2169 (ix) 1\frac{9}{15}, 1\frac{11}{15} (x) -8\frac{6}{7}, -8\frac{5}{7}

VII. Insert 2 rational numbers between the following numbers: (a) 3.18 and 3.19 (b) 2\frac{2}{5}, 2 \frac{3}{5}

VIII. Represent the following rational numbers on the number line:

(i) 2\frac{1}{3} (ii) -\frac{5}{7} (iii) 3.7 (iv) 4.85 (v) 6 \frac{7}{11}

IX. Which of the following rational numbers will have a terminating decimal value? (a) \frac{3}{5} (b) -\frac{5}{7} (c) \frac{1}{2} (d) -\frac{7}{10} (v) \frac{7}{15}

X. Convert each of the following decimal fractions in the form \frac{p}{q}, where p and q are both integers, but q is not zero: (a) 0.32 (b) 0.42 (c) 0.85 (d) 1.875 (e) 0.4375 (f) 3.\dot{7} (g) 1.6\dot{4} (h) 5.\overline{23} (i) 7.11\dot{3} (j) 8.9\overline{505}

We will continue this series later…

Regards,

Nalin Pithwa

Skill Check III: IITJEE Foundation Maths

State whether the following statements are true or false:

  1. If A = \{ x | x = 5n, 5 < n < 10, n \in N\}, then n(A)=4
  2. If n(A) = n(B), then set A \leftrightarrow set B
  3. If Set A= \{ x | x \in N, x<3\}, then A is a singleton set.
  4. The intelligent students of class VIII form a set.
  5. The students passing the half-yearly exams in Class VIII B of school is a set.
  6. A = \{ x | x = p^{3}, p<4, p \in \mathcal{N}\} and \{ x|x = m^{2}, m < 4, m \in \mathcal{N} \} are overlapping sets.
  7. A = \{ x | x \in \mathcal{N}\} is a subset of B = \{ x | x in \mathcal{Z}\}
  8. If we denote the universal set as \Omega = \{ p,q,r,s,t,u,v\} and A = \{ q,u,s,t\}, then \overline{A} = \{ p, r, v\}
  9. A = \{ x | x =2p, p \in \mathcal{N}\} and B = \{ x | x =3p, p \in \mathcal{N}\} are disjoint sets.
  10. If A = \{ 2,3,4\}, then P(A)= \{ \phi, A, \{ 2\}, \{ 3\}, \{ 4 \},        \{ 2,3\}, \{ 2,4\}, \{ 3,4\}\} where P(A) is the power set of set A.

II. If C is a letter in the word down all the subsets of C.

III. Write down the complements of all the 8 subsets of set C above.

IV. If Q = \{ x : x =a^{2}+1, 2 \leq a \leq 5\}, what is the power set of Q?

V. If x = \{ x | x<20, x \in \mathcal{N}\}, and if A = \{x | x = 2a, 3 < a < 8, a \in \mathcal{N} \}, and if B = \{ x | x = 3b, b < 5, b \in \mathcal{N}\}, and if C = \{x | x = c+1, 5 < c < 15, c \in \mathcal{N} \}, then find : (i) n(B) (ii) n(C) (iii) \overline{A} (iv) \overline{B} (v) P(B)

VI. If A = \{ x| x \in \mathcal{N}, 3 < x < 10\}, and if B = \{ x| x =4a-1, a<5, a \in \mathcal{N}\} and if C = \{ x | x = 3a+2, a<7, a \in \mathcal{N}\}, then confirm the following: (i) the commutative property of the unions of sets B and C (ii) the commutative property of intersection of two sets A and C (iii) the associative property of the union of the sets A, B and C (iv) the associative property of intersection of sets A, B and C.

VII. If A = \{ x | x \in \mathcal{N}, 4 \leq x \leq 12\}, and B = \{ x| x = a+1, a<8, a \in \mathcal{N}\}, and C= \{ x| x =2n, 1 < n <7, n \in |mathcal{N}\}, then find (i) A-B (ii) B-C (iii) B \bigcap C (iv) A - (B \bigcap C) (v) B - (A \bigcap C) (vi) A-C (vii) A- (B-C) (viii) A- (B \bigcup C)

VIII. If \xi = \{ x | x \hspace{0.1in}is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in}of \hspace{0.1in}the \hspace{0.1in} English \hspace{0.1in} alphabet \hspace{0.1in} between \hspace{0.1in} but \hspace{0.1in} not \hspace{0.1in} including \hspace{0.1in} d \hspace{0.1in} and \hspace{0.1in} o\}, and let A = \{ l, m , n\} and let B= \{ e,f,g,h,i,j,k,l\}, and let C = \{ j,k,l,m\}, find (i) \overline{A} \bigcup \overline{B} (ii) \overline{B} \bigcap \overline{C} (iii) A \bigcap C (iv) B - (A \bigcap C) (v) \overline{B-A} (vi) Is (B-C) \subset (B-A)? (vii) Is \overline{A} \bigcap \overline{B} = \phi?

IX. All 26 customers in a restaurant had either drinks, snacks, or dinner. 18 had snacks, out of which 6 had only snacks, 4 had snacks and drinks but not dinner, 2 had drinks and dinner but not snacks, and 3 had snacks and dinner but not drinks. If 14 customers had drinks, find (i) how many customers had all three — drinks, snacks as well as dinner. (ii) how many customers had dinner but neither snacks nor drinks (iii) how many customers had only drinks.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Near Dimple Arcade, Thakur Complex
Mumbai, Maharastra 400101
India

Maths will rock your world — a motivational article

keep dreaming the applications of math…try to count every thing….

Mathematics Hothouse

Math will rock your world

BusinessWeek

Jan 23 2006

A generation ago, quants turned finance upside down. Now they’re mapping out ad campaigns and building new businesses from mountains of personal data

 

Neal Goldman is a math entrepreneur. He works on Wall Street, where numbers rule. But he’s focusing his analytic tools on a different realm altogether: the world of words.

Goldman’s startup, Inform Technologies LLC, is a robotic librarian. Every day it combs through thousands of press articles and blog posts in English. It reads them and groups them with related pieces. Inform doesn’t do this work alphabetically or by keywords. It uses algorithms to analyze each article by its language and context. It then sends customized news feeds to its users, who also exist in Inform’s system as — you guessed it — math.

How do you convert written words into math? Goldman says it takes a…

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A motivation for Math and some Math competitive exams in India

what motivates you to keep going in math?

Mathematics Hothouse

Sometime back, there was a tremendous publicity in the Indian media to two Fields medallists of Indian origin. They also talked about what motivated them towards Math when they were young. One should  not do Math just lured by its glamorous applications in IT or other engineering disciplines. But, one can develop both aptitude and attitude  towards it if one works from a young age.

What you need is intrinsic motivation. In this context, I like to quote the following words of a famous mathematician:

“And, a final observation. We should not forget that the solution to any worthwhile problem very rarely comes to us easily and without hard work; it is rather the result of intellectual effort of days or weeks or months. Why should the young mind be willing to make this supreme effort? The explanation is probably the instinctive preference for certain values, that is, the attitude…

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Skill Check II: IITJEE foundation maths

Set Theory Primer/basics/fundamentals/preliminaries:

I. Represent the following sets in Venn Diagrams: (a) \Xi = \{ x : x = n, n <40, n \in \mathcal{N} \} (b) \mu = \{ x : x = 6n, n < 6. n \in \mathcal{N}\} (c) \alpha = \{ x: x = 3n, n<8, n \in \mathcal{N} \}

2. If x = \{ x: x<29 \hspace{0.1in}and \hspace{0.1in}prime\} and A = \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} prime \hspace{0.1in} factor \hspace{0.1in} of \hspace{0.1in} 210\}, represent A in Venn diagram and find \overline{A}.

3. 95 boys of a school appeared for a physical for selection in NCC and Boy Scouts. 21 boys got selected in both NCC and Boy Scouts, 44 boys were not selected in Boy Scouts and 20 boys were not selected only in boy scouts. Draw Venn diagram and find : (i) how many boys did not get selected in NCC and boy scouts. (ii) how many boys did not get selected only in NCC (iii) how many boys got selected in NCC (iv) how many boys got selected in boy Scouts (v) How many boys got selected in NCC not in boy Scouts?

Regards,

Nalin Pithwa.

Skill Check I: IITJEE Foundation Maths

  1. Simplify: (+1) \times (-1) + (+1) \div (-1) -()-1 +(-1) \div (-1) \times (-1) +(-1)
  2. Simplify: \{ 7 \hspace{0.1in} of \hspace{0.1in} 6  \div 2 - 4(8 \times 12 \div 3 + \overline{-3 \hspace{0.1in} of \hspace{0.1in}+6 -2 })\} \div (-3)
  3. Simplify: 24 \div (-8) + 3 \times (-3)
  4. Simplify: (+7) - (-3) + (+4) \times (-3) \div (+3) of (-2)
  5. Simplify: (-3) of (-5) \div (-3) \times (-2) + (-5) - (-2) \div (+2)
  6. Simplify: (-7) + (-8) - (-3) \hspace{0.1in} of \hspace{0.1in} (-6) \div (+2) - (-4) \times (-4) \div (+2)
  7. Simplify: (+24) \div (-3) \hspace{0.1in} of \hspace{0.1in} (+4)  - (-25) \times (-6) \div (-3) + (-15) \div (-3) \times (-10)
  8. Simplify: (-3) \hspace{0.1in} of \hspace{0.1in} (-8) \div (-6)  - (-8 +4-3)
  9. Simplify: (-5) [ (-6) - \{ -5 + (-2 + 1 - \overline{3-2})  \} ]
  10. Simplify: (-3) [ (-8) - \{ +7 - (4-5 - \overline{2-5-1})\}] \div (-11)
  11. Simplify: (+8) \times (-3) \times (+2) \div [ -1 - \{ -3 + 8 - (6 -2 - \overline{3+5-4}) \} ]
  12. Simplify: (+32) \div (+2) of (-4) \div [(-7) of 3 \div \{4 - 5(3 - \overline{4 of 2 - 2 of 5}) \}]
  13. Simplify: (-30) + (-8) \div (-4) \time 2
  14. Simplify: (-3) \times (-6) \div (-2) + (-1)
  15. Simplify: 56 \div (16 + \overline{4-6}) + (6-8)
  16. Simplify: (7+6) \times [19 + \{ (-15) + \overline{6-1}\}]

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

B.S. in Mathematics: IIT Bombay program:

http://www.math.iitb.ac.in/Academics/bs_programme.php

Note that the admission is through IITJEE Advanced only.

Nalin Pithwa.

Math Basics Division by Zero

the eleventh commandment of Moses: Though shall not divide by zero !!

Mathematics Hothouse

Let’s pause Geometry for a little time and start thinking of some basic rules of the game of Math. Have you ever asked “why is division by zero not allowed in Math?” Try to do 1/2 in a calculator and see what you get!!

This was also a question an immortal Indian math genius, Srinivasa Ramanujan had asked his school teacher when he was a tiny tot. Note the following two arguments against the dangers of division by zero:

(a) Suppose there are 4 apples and two persons want to divide them equally. So, it is 4/2 apples per person, that is, 2 apples per person. But, now consider a scenario in which there are 4 apples and 0 persons. So, how can you divide 4 apples amongst (or by) 0 persons? You can think of any crazy answer and keep on arguing endlessly about it!!!!

(b) The cancellation law…

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IITJEE Foundation Maths: Tutorial Problems IV

  1. Resolve into factors: (a) 2x^{2}-3ab+(a-6b)x (b) 4x^{2}-4xy-15y^{2}
  2. In the expression, x^{3}-2x^{2}+3x-4, substitute a-2 for x, and arrange the result according to the descending powers of a.
  3. Simplify: (i) \frac{x}{1-\frac{1}{1+x}} (ii) \frac{x^{2}}{a+\frac{x^{2}}{a+\frac{x^{2}}{a}}}
  4. Find the HCF of 3x^{3}-11x^{2}+x+15 and 5x^{4}-7x^{3}-20x^{2}-11x-3
  5. Express in the simplest form: (i) \frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{y}-\frac{y+x}{x}} (ii) (\frac{x^{3}-1}{x-1} + \frac{x^{3}+1}{x+1})\div (\frac{1}{x-1} + \frac{1}{x+1})
  6. A person possesses Rs. 5000 stock, some at 3 per cent; four times as much at 3.5 %, and the rest at 4 %; find the amount of each kind of stock when his income is Rs. 178.
  7. Simplify the expression: -3[(a+b)-{(2a-3b) -( 5a+7b-16c)  - (-13a +2b -3c -5d)}], and find its value when a=1, b=2, c=3, d=4.
  8. Solve the following equations : (i) 11y-x=10 and 11x-101y=110 (ii) x+y-z=3, and x+z-y=5, and y+z-x=7.
  9. Express the following fractions in their simplest form: (i) \frac{32x^{3}-2x+12}{12x^{5}-x^{4}+4x^{2}} (ii) \frac{1}{x + \frac{1}{1+ \frac{x+3}{2-x}}}
  10. What value of a will make the product of 3-8a and 3a+4 equal to the product of 6a+11 and 3-4a?
  11. Find the LCM of x^{3}-x^{2}-3x-9 and x^{3}-2x^{2}-5x-12
  12. A certain number of two digits is equal to seven times the sum of its digits; if the digit in the units’ place be decreased by two and that in the tens place by one, and if the number thus formed be divided by the sum of its digits, the quotient is 10. Find the number.
  13. Find the value of \frac{6x^{2}-5xy-6y^{2}}{2x^{2}+xy-y^{2}} \times \frac{3x^{2}-xy-4y^{2}}{2x^{2}-5xy+3y^{2}} \div \frac{9x^{2}-6xy-8y^{2}}{2x^{2}-3xy+y^{2}}
  14. Resolve each of the following expressions into four factors: (i) 4a^{4}-17a^{2}b^{2}+4b^{4}; (ii) x^{8}-256y^{8}
  15. Find the expression of highest dimensions which will divide 24a^{4}b -2a^{2}b^{2}-9ab^{4} and 18a^{6}+a^{4}b^{2}-6a^{3}b^{3} without remainder.
  16. Find the square root of : (i) x(x+1)(x+2)(x+3)+1 (ii) (2a^{2}+13a+15)(a^{2}+4a-5)(2a^{2}+a-3)
  17. Simplify: x - \frac{2x-6}{x^{2}-6x+9} - 3 + \frac{x^{2}+3x-4}{x^{2}=x-12}
  18. A quantity of land, partly pasture and partly arable, is sold at the rate of Rs. 60 per acre for the pasture and Rs. 40 per acre for the arable, and the whole sum obtained at Rs. 10000. If the average price per acre were Rs. 50, the sum obtained would be 10 per cent higher; find how much of the land is pasture and how much is arable.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

Math from scratch

Most important out of all blogs…explains what is math and the nature of its rigour

Mathematics Hothouse

Math from scratch!

Math differs from all other pure sciences in the sense that it can be developed from *scratch*. In math jargon, it is called “to develop from first principles”. You might see such questions in Calculus and also Physics.

This is called Axiomatic-Deductive Logic and was first seen in the works of Euclid’s Elements (Plane Geometry) about 2500 years ago. The ability to think from “first principles” can be developed in high-school with proper understanding and practise of Euclid’s Geometry.

For example, as a child Albert Einstein was captivated to see a proof from *scratch* in Euclid’s Geometry that “the three medians of a triangle are concurrent”. (This, of course, does not need anyone to *verify* by drawing thousands of triangles and their mediansJ) That hooked the child Albert Einstein to Math, and later on to Physics.

An axiom is a statement which means “self-evident truth”. We accept…

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