CHAPTER-I: NUMBER SYSTEM

 1.NUMBERS

I. Numeral: A group of digits denoting a number. (i.e. 1,2,3,4,5,6,7,8,9).

example: 678467437

we represent this number as:

6 X 10⁸(ten crores)   = 600000000.

7 X 10⁷(crores)       =     70000000.

8 X 10⁶(ten lacs)     =       8000000.

4 X 10⁵(lacs)           =         400000.

6 X 10⁴(ten thousands)=      60000.

7 X 10³(thousands)  =            7000.

4 X 10²(hundreds)   =              400.

3 X 10¹(tens)           =                30.

7 X 10⁰(units)          =                  7.

                                       _________

                                       678467437

      We have to read it as: "Sixty-seven crores, eighty-four lacs, Sixty-seven thousand, four-hundred and thirty-seven.".

II Place Value or Local value:

  place value refers the value of a digit at that place.

ex: 6 X 10^8, 10^4, ...

6 X 10^8=600000000.

place value is 600000000.

III. Face value: it refers to the numeral value of the place.

ex: 6 X 10^8 = 600000000

the face value is 6.

IV. Type of Numbers:

1.Natural Numbers: all counting numbers (1,2,3,4,5, ....).

2.Whole numbers: all counting number together with zero. (0,1,2,3,4,5, ....).

(i) 0 is only whole number not a natural number.

(ii) every natural number is a whole number.

3.Intergers: All positive and negative natural numbers included with zero.

(i) Positive Integers: {0,1,2,3, 4, ....}

(ii)Negative Integers: {0, -1, -2, -3, -4, ....}

(iii) Zero is neither positive nor negative.

4.Even Numbers: a number divisible by 2. (2,4,6,8, ...).

5.Odd Numbers: a number not divisible by 2. (1,3,5,7,9, ...).

6.Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely and number itself.

Prime Numbers up o 100 are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Prime numbers greater than 100: Let p be a given number greater than 100. to find out whether it is prime or not, we use the following method:

Find a whole number nearly greater than square root of p.

Let k > sqrt(p). test whether p is divisible by any prime number less than k. if yes,then p is not a prime number.

example p=191.

then find sqrt(p).

sqrt(191)=13.82

which is less than 14

14 >sqrt(191).

then list out prime numbers than 14: 2,3,5,7,11,13.

then check 191 is divisible by any of these prime numbers (2,3,5,7,11,13).

if we check it 191 is not divisible by any of the prime numbers.

So, 191 is a prime number.

7.Composite Numbers: Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4,6,8,10,12.

Note: (i)1 is neither prime nor composite.

          (ii) 2 is the only even number with prime.

          (iii)There are 25 prime numbers between 1 and 100.

8.Co-primes: Two numbers a and b are said to be co-primes, if their H.C.F. is 1.e.g., (2,3), (4,5), (7,9), (8,11), etc. are co-primes.

V. Tests of Divisibility

1.Divisibiltiy by 2: A number is divisible by 2. if its last digit is 0,2,4,6,8.

   example: 4398 is divisible by 2.

                   4392 is divisible by 2.

                   4394 is divisible by 2.

                   4396 is divisible by 2.

                   4390 is divisible by 2.

2.Divisibility by 3: A number is divisible by 3. if its sum of digit is divisible by 3.

example: consider 396 as example.

             sum of its digits is (3+9+6) = 18.

             18 is divisible by 3.

             So, 396 is divisible by 3.

3.Divisibility of 4: A number is divisible by 4. if its last two digits are divisible by 4.

     example: consider 9716,

                     the last two digits is 16.

                     16 is divisible by 4.

                     So, 9716 is divisible by 4.

4.Divisibility of 5: A number is divisible by 5. if its last digit ends with 0 or 5.

      example:  45345 is divisible by 5.

                       34560 is divisible by 5.

5.Divisibility of 6: A number is divisible by 6. if it is divisible by both 2 and 3.

              example: consider a number 35256.

                              Since last digit is 6, it is divisible by 2.

                              (3+5+2+5+6) =21 is divisible by 3.

                              Then the number 35256 is divisible by 6.

6.Divisiblity of 8: A number is divisible by 8. If its last 3 digit is divisible by 8.

            example: consider a number 953360,

                           the last 3 digits are 360, it is divisible by 8.

                     So, the number 953360 is divisible by 8.

7.Divisibility of 9: A number is divisible by 9. if the sum of its digits is divisible by 9.

Consider 60732, (6+0+7+3+2) =18, which is divisible by 9.

So, 60732 is divisible by 9.

8.Divisibility of 10: A number is divisible by 10. if its last digit is 0.

example: 4596540.

9.Divisibility of 11: A number is divisible by 11.if its (sum of odd digit) -(sum of even digit)

is equal to zero(0) or a number divisible by 11.

formula: (Sum of odd digits)-(Sum of even digit) = 0 or number divisible by 11

example : consider a number 4832718.

(8+7+3+4)-(1+2+8)=11, which is divisible by 11.

So, the number 4832718 is divisible by 11.

10.Divisibility of 12: A number is divisible by 12. if the number is divisible by both 4 and 3.

         example: 34632, this number ends with 32. So, it is divisible by 4.

         (3+4+6+3+2)=18, it is divisible by 3.

          So, the number 34632 is divisible by 12.

11.Divisibility of 14: A number is divisible by 14, if it is divisible by both 2 and 7.

12.Divisibilityy of 15: A number is divisible by 15, if it is divisible by both 3 and 5.

13.Divisibility of 16: A number is divisible by 16, if its last 4 digit is divisible by 16.

example:7957536 is divisible by 16, 7536 is divisible by 16. So, this number is divisible by 16.

14.Divisibility of 24: A given number is divisible by 24, if it is divisible by both 3 and 8.

15.Divisibility of 40: A given number is divisible by 40, if its is divisible by both 5 and 8.

16.Divisibility of 80: A given number is divisible by 80, if it is divisible by both 5 and 16.

17. If a number is divisible by p and q. if p and q are co-primes then the number is divisible by pq.

example: 6 is divisible by 2 and 3. both are co-primes.

also 36 is divisible by 4 and 6, but 2X6=24, 36 is not divisible by 24, Since both 4 and 6 are not Co-primes.

VI. Multiplication by Shortcut Methods

I. Multiplication by Distributive law:

(i) a x (b + c) = (a x b) + (a x c).

(ii)a x (b-c) = (a x b) - (a x c).

example:  5696 x 999 = 5696 x (1000 - 1) = 5696000 - 5696 = 5690304.

VII. Basic Formulae

(i) (a + b) ² = a² + b² + 2(ab).

(ii) (a - b) ² = a² + b² - 2(ab).

(iii) (a + b) ² - (a - b) ² = 4(ab).

(iv) (a + b) ² + (a - b)² = 2(a² + b²).

(v) (a + b) ² = a² + b² + 2(ab).

(vi) (a + b) ² = a² + b² + 2(ab).

(vii) (a + b) ² = a² + b² + 2(ab).

(viii) (a + b) ² = ² + b² + 2(ab).

(ix) (a + b) ² = a² + b² + 2(ab).

(x) (a + b) ² = a² + b² + 2(ab).

VIII. Division Algorithm or Euclidean Algorithm

If we divide a given n umber by another number, then:

  Dividend= (Divisor x Quotient) + Remainder.

IX. (i)(xⁿ - aⁿ) is divisible by (x-a) for all odd values of n.

      (ii) (xⁿ - aⁿ) is divisible by (x+a) for all even values of n.

      (iii) (xⁿ + aⁿ) is divisible by (x + a) for all odd values of n.

X.PROGESSION

A succession of numbers formed and arranged in a definite sequence order according to a definition is called PROGRESSION.

1.      Arithmetic Progression (A.P.): if each term of a progression differs from its preceding term by a constant, then it is called arithmetic progession.

This constant difference is called common difference of AP.

A.P.:

A, A+D, A+2D….

The n^th term of this A.P. is given by T_n = a(n-1)d.

The Sum of n terms of this A.P.

S_n=n/2[2a+(n-1)d].

S_n=n/2[2a+(n-1)d]=n/2 [for (first term + last term)].

2.      Geometric Progression (G.P.) : A progression of numbers in which every term bears a constant ratio with its preceding term , is called Geometrical progession.

Common ratio of G.P. is r.

A, Ar, Ar²,…

In this G.P. T_n = ar^n-1

Sum of the n terms, S_n = a[1-rⁿ/(1-r)].