Basic Mathematics

Number System

Systems to deal with different types of numbers is termed as 'number system'. Number (Numerals) is the symbolical representation in mathematical language, which are classified in further ways: 

Types of number

Natural numbers                            all counting numbers except zero are called natural numbers like 1, 2, 3.... etc

Whole numbers                               all natural numbers along with zero are called whole numbers like 0, 1, 2, 3 ... etc

Even numbers                                  all counting numbers which are divisible by two are cold even numbers like 2, 4, 6, 8 ....etc

Odd numbers                                    all counting numbers which are not divisible by two are called odd numbers like 1, 3, 5, 7, .....etc

Prime numbers                                a counting number is called prime number, if it has only two factors i.e. 1 and the number it's self, like 2, 3, 5, 7, 11, 13 ...... etc.

Composite numbers                      a natural number which is not a prime number is called composite number like   4, 6, 8, 9 – – – etc.

Integers                                                whole numbers along with negative numbers is called integers e.g. -3, -2, -1, 0, 1, 2, 3, ...... etc.

Rational numbers                            a number which can be expressed in the form of \frac{p}{q} , Where p and q are integer and q ≠ 0 is Called rational number.

Irrational number                           a number which cannot be expressed in the form of \frac{p}{q}, where p and q are integer and q ≠ 0 like √2, √3, √7 ......etc

Real numbers                                     real numbers comprises of both rational and irrational numbers, like \frac{7}{9}, √2, √5, π, \frac{8}{9} ........ etc.

Division on numbers

Let D and d be two numbers, then \frac{D}{d} is called the operation of division, where D is the dividend and d is the divisor. A number which tells how many times a divisor 'd' exists in the dividend 'D' is called the quotient 'Q'. Thus

Dividend = (Divisor) X (Quotient)

If dividend D is not a multiple of divisor d, then D is not exactly divisible by d and in this case remainder R is obtained. Thus,

Dividend = (Divisor) X (Quotient) + Remainder

Some Points to remember

  • If p divides q and r, then p will divide q ± r as well.
  • When  x^{n} + k is divided by (x - 1), then the remainder is
    • 1 + k when k < (x - 1)
    • 1 + p when k > (x -1), where p is the remainder obtained by dividing k by (x - 1).
  • If N is a composite number of the form N =  a^{p}. b^{q}. c^{r}, where a, b, c are prime numbers, then the number of divisiors of N represented by m is given by m = (p + 1)(q + 1)(r + 1)

Divisibility

Testing divisibility of a large number by a given divisor is dying consuming process, if we do it by performing actual division. Also, there is no well defined general rule to test The divisibility of numbers. Test of divisibility maybe derived from the properties of multiples of specific divisors.

  • A number formed by repeating a digit six six times will be divisible by 7, 11, 13. e.g. 888-8888.
  • A number formed by repeating or two digit number three times will be divisible by 7.  e.g. 323232.
  • A number formed by repeating a three digit number two times will be divisible by 7, 11 and 13 e.g. 632632.
  • If a number is divisible by two co-prime numbers a and b then N is also divisible by ab.

Rules for the visibility of a number.

A number is divisible

  • By 2, if the digit at units please is even or 0 e.g. 16, 84, 476 etc.
  • By 3, if the sum of all digits of the number is divisible by 3 e.g. 1473.
  • By 4, if the number formed by the last two digits is divisible by 4 e.g. 63848.
  • By 5, if the digit and the unit place is 0 or 5 e.g. 450, 45, 3400 etc.
  • By 6, if the number is divisible by 2 and 3 both. e.g. 720, 1440 etc.
  • By 7, if the difference between twice the digit at the unit place and the number formed by the other digits is either 0 or a multiple of 7 e.g. 784.
  • By 8, if the number formed by the last three digits is divisible by eight or the last three or more digits are zero e.g. 57832, 6730000.
  • By 9, if the sum of all digits of the number is divisible by 9. e.g. 986391.
  • By 10, if the digit at the unit place is 0. e.g 108350.
  • By 11, if the difference between the sum of its digits at odd places and add even places is either 0 or a multiple of 11. e.g. 30426.
  • By 12, if the number is divisible by 3 and 4. e.g. 2268.
  • By 15, if the number is divisible by 3 and 5 both. e.g. 1305.
  • By 25, if it's last two digits are either 0 or are divisible by 25. 

Points to remember

  • ( x^{m}- a^{m}) is divisible by (x - a) for all values of m.
  •  ( x^{m}- a^{m}) is divisible by (x + a) for all values of m.
  • ( x^{m}+ a^{m}) is divisible by (x + a) for all values of m.

Progression

There are three types of progression as follows:

  1. Arithmetic Progression
  2. Geometric Progression
  3. Harmonic Progression

Arithmetic Progression

If there is a common difference between any two consecutive numbers, then the progression is called Arithmetic Progression.

Example: 3, 7, 11, 15 ..... ( The difference between any two consecutive number is 4)

  •  n^{th} term of an Arithmetic Progression (T_{{n}}) = a + (n - 1)d = l, where l = last term, a = primary term, n = number of terms, d = difference between two consecutive terms.
  • The sum of AP up to terms =  S_{{n}} = \frac{n}{2}\times [2a+(n-1)\times d]=\frac{n}{2}(a+l)

Geometric Progression

geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where r common ratio
a1 first term
a2 second term
a3 third term
an-1 the term before the n th term
an the n th term

The geometric sequence is sometimes called the geometric progression or GP, for short.

For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.

The geometric sequence has its sequence formation: 

To find the nth term of a geometric sequence we use the formula:

where r common ratio
a1 first term
an-1 the term before the n th term
n number of terms

 

Sum of Terms in a Geometric Progression

Finding the sum of terms in a geometric progression is easily obtained by applying the formulas:

nth partial sum of a geometric sequence

sum to infinity

where Sn sum of GP with n terms
S sum of GP with infinitely many terms
a1 the first term
r common ratio
n number of terms

Harmonic Progression

If inverse of a sequence follows rule of an A.P. then it is said to be in harmonic
progression.
e.g. 1,1/2, 1/3, 1/4, 1/5 ...............
1/10, 1/7, 1/4, 1, – 1/2, ...........
In general
1/a, 1/a+d, 1/a+2d, ..................
Note:
Three convenient numbers in H.P. are
1/a–d, 1/a, 1/a+d
Four convenient numbers in H.P. are
1/a–3d, 1/a–d, 1/a+d, 1/a+3d
Five convenient numbers in H.P. are
1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d
Harmonic mean between two numbers a and b
Let H be the harmonic mean between two and number a and b.
So, a, H, b are in H.P.
or, 1/a, 1/H, 1/b are in A.P.
or, 1/H – 1/a = 1/b – 1/H.
or, 2/H = 1/a + 1/b = a+b/ab
∴ H =2ab/a+b
Similarly, we can find two harmonic mean between two number.

Points to remember

  • If a, b and c are in harmonic progression, then b=\frac{2ac}{a+c}
  • If a, b and c are in arithmetic progression, then b=\frac{a+c}{2}
  • If a, b and c are in geometric progression, then b=\sqrt ac

 

 

 

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