Number System & Simplification
The ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits, which can represent any number.
- These are the numbers (1, 2, 3, etc.) that are used for counting. It is denoted by N.
- There are infinite natural numbers and the smallest natural number is one (1).
- Natural numbers which are divisible by 2 are even numbers. It is denoted by E.
- E = 2, 4, 6, 8, ….
- Smallest even number is 2. There is no largest even number.
- Natural numbers which are not divisible by 2 are odd numbers.
- It is denoted by O.
- O = 1, 3, 5, 7, ….
- Smallest odd number is 1.
- There is no largest odd number.
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Based on divisibility, there could be two types of natural numbers:
Prime and Composite
- Prime Numbers: Natural numbers which have exactly two factors, i.e., 1 and the number itself are called prime numbers. The lowest prime number is 2. 2 is also the only even prime number.
- Composite Numbers: It is a natural number that has atleast one divisor different from unity and itself.
Every composite number can be factorized into its prime factors.
For example: 24 = 2 × 2 × 2 × 3. Hence, 24 is a composite number.
The smallest composite number is 4.
- The natural numbers along with zero (0), from the system of whole numbers.
- It is denoted by W.
- There is no largest whole number and
- The smallest whole number is 0.
- The number system consisting of natural numbers, their negative and zero is called integers.
- It is denoted by Z or I.
- The smallest and the largest integers cannot be determined.
- All numbers that can be represented on the number line are called real numbers.
- It is denoted by R.
- R+: Positive real numbers and R– : Negative real numbers.
Real numbers = Rational numbers + Irrational numbers.
- Rational numbers:
- Any number that can be put in the form of , where p and q are integers and q 0, is called a rational number.
- It is denoted by Q.
- Every integer is a rational number.
- Zero (0) is also a rational number. The smallest and largest rational numbers cannot be determined. Every fraction (and decimal fraction) is a rational number
2. Irrational numbers:
- The numbers which are not rational or which cannot be put in the form of , where p and q are integers and q 0, is called irrational number.
- It is denoted by Q’ or Qc.
Fraction: A fraction is a quantity which expresses a part of the whole.
TYPES OF FRACTIONS:
- Proper fraction : If numerator is less than its denominator, then it is a proper fraction.
- Improper fraction: If numerator is greater than or equal to its denominator, then it is a improper fraction.
- Mixed fraction: it consists of an integer and a proper fraction.
- Equivalent fraction/Equal fractions: Fractions with same value.
- Like fractions: Fractions with same denominators.
- Unlike fractions: Fractions with different denominators.
- Simple fractions: Numerator and denominator are integers.
- Complex fraction: Numerator or denominator or both are fractional numbers.
- Decimal fraction: Denominator with the powers of 10.
- Vulgar fraction: Denominators are not the power of 10.
Operations: The following operations of addition, subtraction, multiplication and division are valid for real numbers.
- Commutative property of addition: a + b = b + a
- Associative property of addition: (a + b) + c = a + (b + c)
- Commutative property of multiplication: a * b = b * a
- Associative property of multiplication: (a * b) * c = a * (b * c)
- Distributive property of multiplication with respect to addition (a + b) * c = a * c + b * c
- A number of the form a + bi, where a and b are real number and i = (imaginary number) is called a complex number.
- It is denoted by C.
- For Example: 5i (a = 0 and b = 5), + 3i (a = and b = 3)
- Divisibility by 2: A number is divisible by 2 if it’s unit digit is even or 0.
- Divisibility by 3: A number is divisible by 3 if the sum of it’s digit are divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the last 2 digits are divisible by 4, or if the last two digits are 0’s.
- Divisibility by 5: A number is divisible by 5 if it’s unit digit is 5 or 0.
- Divisibility by 6: A number is divisible by 6 if it is simultaneously divisible by 2 and 3.
- Divisibility by 7: We use osculator (-2) for divisibility test.
- Divisible by 11: In a number, if difference of sum of digit at even places and sum of digit at odd places is either 0 or multiple of 11, then no. is divisible by 11.
- Divisible by 13: we use (+4) as osculator.
- Divisible by 17: We use (-5) as osculator.
- Divisible by 19: We use (+2) as osculator.
- Divisibility by a Composite number: A number is divisible by a given composite number if it is divisible by all factors of composite number.
Dividend = (Divisor × Quotient) + Remainder where, Dividend = The number which is being divided Divisor = The number which performs the division process Quotient = Greatest possible integer as a result of division Remainder = Rest part of dividend which cannot be further divided by the divisor.
- A complete remainder is the remainder obtained by a number by the method of successive division.
- Complete reminder = [I divisor × II remainder] + I remainder
- Two different numbers x and y when divided by a certain divisor D leave remainder r1 and r2 When the sum of them is divided by the same divisor, the remainder is r3. Then, divisor D = r1 + r2 – r3
- Method to find the number of different divisors (or factors) (including 1 and itself) of any composite number N:STEP I: Express N as a product of prime numbers as N = xa × yb × zcSTEP II: Number of different divisors (including l and itself) = (a + 1) (b + 1) (c +1) …..
Counting Number of Zeros
- Sometimes we come across problems in which we have to count number of zeros at the end of factorial of any numbers. for example- Number of zeros at the end of 10!
- 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 ×1
- Here basically we have to count number of fives, because multiplication of five by any even number will result in 0 at the end of final product. In 10! we have 2 fives thus total number of zeros are 2.
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In number system,
LAW OF SURDS – all 3 formula
LAW OF INDICES – 4th formula
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