NUMB3RS NUMB3RS

Concept 1: Categorization of numbers

Real Imaginary

Rational Irrational

Non-recurring/non-terminating

Natural numbers Whole numbers Integers Fractions

Prime Composite

Proper Improper Mixed

Decimal Vulgar

Definitions:

Imaginary number: square root of any negative numberEg. √- 4 = √i24 = ± i√2 (where i2 = -1)

Real Number: Any number that can be marked on the number line.

Rational Number: Any number that can be written in the form of p/q where q ≠ 0.

Irrational number: defined as non-recurring and non-terminatingeg. Root of any prime number

Natural number N: N є 1, 2, 3……∞

Whole numbers W: W є 0, 1, 2, 3….. ∞

Integers I :I є -∞…-3, -2, -1, 0, 1, 2, 3….∞

Proper fraction:23

, 7

37 (denominator > numerator)

Improper fraction: 32

, 377

(numerator > denominator)

Mixed fraction: 123

, -3 7

37 (combination of integer and a proper fraction)

Concept 2: Recurring decimals as rational numbers

Let x = 0.77777… (1)

then 10x = 7.77777… (2)

Subtracting (1) from (2)… 9x = 7

So x = 79

= 0.7777….

Similarly 0.7272727272 = 7299

Concept 3: Prime & Composite numbersA prime number is a number that is divisible by 1 and itself only.Eg.2, 7, 17, 101 etc.

(Please note 1 is neither a prime number nor a composite number)

(The only even prime number is 2)

A composite number is one, which is a product of one or more primes.Eg.6, 9, 42 etc.

Concept 4: Factorials

Represented as n! = n.(n-1).(n-2).(n-3)…3.2.1Eg. 6! = 6.5.4.3.2.1 = 720 and 0! = 1

Factorials play an important role in permutation & combination.

Concept 5: The number lineA number line is a line with 0 at the center and numbered from 1 to ∞ at equal intervals towards the right of 0 and from -1 to -∞ towards the left of 0.

-∞ -3 -2 -1 0 1 2 3 ∞

NUMB3RS NUMB3RS

Concept 6: Absolute valueAbsolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies.

The absolute value of a number is never negative.

It is called absolute or mod value.

Mod x or absolute value of x is denoted as IxIEg. IxI = 5. Means x = ±5 IxI > 5 means -5 > x > 5 IxI < 5 means -5 < x < 5

Q: If I2x-7I = 5. Find x

DIVISIBILITY TESTSDivisibility Tests ExampleA number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.

168 is divisible by 2 since the last digit is 8.

A number is divisible by 3 if the sum of the digits is divisible by 3.

168 is divisible by 3 since the sum of the digits is 1+6+8=15, and 15 is divisible by 3.

A number is divisible by 4 if the last 2 digits of the number are divisible by 4.

316 is divisible by 4 since 16 is divisible by 4.

A number is divisible by 5 if the last digit is either 0 or 5.

195 is divisible by 5 since the last digit is 5.

A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3.

168 is divisible by 6 since it is divisible by 2 AND it is divisible by 3.

A number is divisible by 7 if the number obtained by adding5 times the number of units to the tens is divisible by 7

182 is divisible by 7 since 18 + 5x2 = 28 is divisible by 7.

A number is divisible by 8 if the last 3 digits of the number are divisible by 8.

7,120 is divisible by 8 since 120 is divisible by 8.

A number is divisible by 9 if the sum of the digits is divisible by 9.

549 is divisible by 9 since the sum of the digits is 5+4+9=18, and 18 is divisible by 9.

A number is divisible by 10 if the last digit is 0. 1,470 is divisible by 10 since last digit is 0.

A number is divisible by 11 if the difference between the sum of the digits in the odd place and the sum of the digits in the even place = 0 or a

1738 is divisible by 11 since (7+8) – (1+3) = 11 is divisible by 11.

multiple of 11

Concept 7: Brackets in interval notationx (5,∞) means x lies between 5 and ∞ not including 5 or ∞

x [5,∞) means x lies between 5 and ∞ including 5 but not including ∞

x (-∞, 4] U [8, 14] means lies between -∞ to 5 not including -∞ but including 5 and between 8 to 14 including both

Concept 8: Prime factors of a numberThe prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder.

Eg. 84 can be divided by the prime numbers 2, 3 and 7. So the prime factors of 84 are 2,3 and 7.Q: What are the prime factors of 3465.

Concept 9: Factors of a numberIf N = am x bn, where a & b are prime factors Then the number of factors of N = (m+1).(n+1)

Eg. 84 = 22 x 3 x 7No. of factors of 84 = (2+1)(1+1)(1+1) = 12

(Perfect squares have odd number of factors. The reverse is also true. If an integer has odd number of factors then the integer is a perfect square

Q: Find the number of odd factors of 6300

HCF & LCMConcept 10: HCF of numbers(HCF of a set of integers must be ≤ the smallest number of the set)

The highest common factor (HCF) of two or more numbers is the largest number that is a factor of all of the given numbers.

HCF is also known as GCD – Greatest Common Denominator

Eg. The HCF of 40, 48 and 5640 = 2 x 2 x 2 x 548 = 2 x 2 x 2 x 2 x 356 = 2 x 2 x 2 x 7

The common factors for all 3 numbers are 2 x 2 x 2 = 8

Alternately: Find the HCF of (48-40) and (56-48)

NUMB3RS NUMB3RS

Q:Find the HCF of 48, 84 and 132

Q:The maximum number of children amongst whom 1001 pens and 910 pencils can be distributed such that each student gets the same number of pens and the same number of pencils?

Concept 11: LCM of numbers(LCM of a set of integers must be ≥ the largest number of the set)

The least common multiple (LCM of two or more integers, is the smallest positive integer that is divisible by all the relevant integers

Eg. The LCM of 40, 48 and 5648 = 2 x 2 x 2 x 2 x 356 = 2 x 2 x 2 x 780 = 2 x 2 x 2 x 2 x 5

LCM = Product of (Prime factors common to three xPrime factors common to two x Remaining prime factorsLCM = (2 × 2 × 2) × (2) × (3 × 5 × 7)

Concept 12: Relation between HCF &LCM of two numbersa & b.Product of a and b = HCF(a,b) x LCM(a,b)

Concept 13: HCF of Fractions

The HCF of fractions = HCF of the Numerators

LCM of thedenominators

Eg. Find HCF of 12615

and10225

.

HCF of 126 and 102 = 6 and LCM of 15 and 25 = 75

So HCF of 12615

and 10225

= 6

75

Concept 14: LCM of Fractions

The LCM of fractions = LCM of the Numerators

HCF of the denominators

Eg. Find LCM of 12615

and 9025

.

LCM of 126 and 90 = 630 and HCF of 15 and 25 = 5

So LCM of 12615

and 9025

is 630

5 = 126

COMPARISONS BETWEEN NUMBERS

a> b then, a + c > b + c and a –c > b – c

a> b and c > 0 then,ac>bc and a/c > b/c

if a, b ≥ 0 and n > 0an>bn and 1/an< 1/bn

Ix – yI = Iy – xI

Ix.yI = IxI. IyI

Ix + yI ≤ IxI + IyI

Ix + yI ≥ IxI - IyI

INDICES & SURDS

Concept 15: Rules of Indices am × bm = (a.b)m

am × an = a(m+n)

NUMB3RS NUMB3RS am ÷ an = a(m-n)

(am)n = amn

am ≠(am)n

Eg.223

= 28whereas (22)3 = 26

Concept 16: SurdsA surd is an irrational number. All square roots of prime numbers are surds. √2 is a surd. √2 = 21/2

Cube root of 4 is written as 3√4= 41/3

√a .√b = √(ab) Eg. √2 ×√13 = √26

Complex surd – combination of a rational and irrational term Eg. 7 + 3√2

Conjugate of a surd Eg. If 7 + 3√2 is a surd, its conjugate is 7 - 3√2

Concept 17: Rationalization of Surds

If a surd is of the form (a+√ b)(c+√ d )

the concept of rationalization is to make the

denominator a rational number. So the numerator and the denominator are

multiplied by the conjugate of the denominator.

Eg.(7+3 √ 2)(4+2 √ 2)

= (7+3 √ 2)(4−2√ 2)(4+2 √ 2)(4−2 √ 2)

= (7+3 √ 2)(4−2 √ 2)

(42 – (2 √ 2)2) {(a+b)(a-b) = a2 – b2}

= (28−6 x2−14 √ 2+12 √ 2)

(16 – 4∗2)

= (28−12−14 √ 2+12 √ 2)

8

= (16−2√ 2)

8

= (8−√ 2)

4

Q: Rationalize (√ 3+√ 2)(√ 3−√ 2)

Concept 18: Square root of quadratic surdsA quadratic surd is of the form (√a + √b)2 = a + b + 2√ab

Eg. (√2 + √3)2 = 2 + 3 + 2√2√3 = 5 + 2√6

Conversely if a surd is of the form a + b + 2√ab, then its sq. root is √a + √b.

Eg. 7 + 4√3 = 7 + 2√(4.3) = √(7 + 2√12) = √4 + √3 = 2 + √3

Similarly √(7 - 4√3) = 2 -√3

Q: Find the square root of 8 - √60

Concept 19: Rule of Cyclicity for unit’s digitThe unit’s digit of successive power’s of n follow a cycle

Units digit of n Units digit of n, n2, n3, n4, n5, …. Cycle

1 1,1,1… 12 2, 4, 8, 6, 2, 4, …. 43 3, 9, 7, 1, 3, 9, …… 44 4, 6, 4, 6, …….. 25 5, 5, 5, …… 16 6, 6, 6, …….. 17 7, 9, 3, 1, 7, 9, …… 48 8, 4, 2, 6, 8, 4, ….. 49 9, 1, 9, 1, ….. 20 0, 0, 0…… 1

Eg. Find the units digit of 37254

The unit digit is 7 and the cycle for 7 is 4

So 254 = 4×63 + 2

i.e. 254 is 63 complete cycles and the 2nd part of the 64th cycle

The unit digit of the 2nd part of a cycle for n = 7 is 9

So unit digit of 37254 is 9

NUMB3RS NUMB3RSConcept 21: Rule of cyclicity for Remainders

Eg. Find the remainder when 4326 ÷ 7

41/7 R = 4, 42/7 R = 2, 43/7 R = 1

44/7 R = 4, 45/7 R = 2, 46/7 R = 1

Cycle for 4n/7 is 3

326 = 3×108 + 2

i.e 326 = 108 complete cycles and the 2nd part of the 109th cycle

So the remainder when 4326 ÷ 7 is 2

PROPERTIES OF NUMBERS

Concept 22: The product of n consecutive integers is divisible by n!

Concept 23: np – n is always divisible by p (where n is a whole number and p is a prime)Q: What’s the remainder when 1511 is divided by 11?

Concept 24: Odd + Odd = EvenOdd + Even = OddEven + Even = Even

Concept 25: Odd x Odd = OddOdd x Even = EvenEven x Even = Even

ALGEBRAIC RELATIONSHIPS

Concept 26: Some common formulae (a + b)(a - b) = a2 – b2

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

(a + b)3 = a3 +3ab(a + b) + b3

(a - b)3 = a3 – 3ab(a - b) – b3

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

Q: What is the value of 182 + 172 – 162 - 152

Concept 27: a3 + b3 + c3 -3abc = (a + b + c)(a2 + b2 + c2- ab – bc - ca)

So a3 + b3 + c3 = 3abc if a + b + c = 0

Concept 28: Divisibility an – bn is divisible by a - b for all n

an – bn is divisible by a + b for n even

an + bn is divisible by a + b for n odd

an + bn is not divisible a - b for any n

REMAINDER THEOREM & REMAINDERS

If a function of x f(x) is divided by x-a then the remainder R is a function of ‘a’ i.e. R = f(a)

Eg.x3 – 3x2 + 7x -8 is divided by x-2 Then R = f(a) = 23 – 3(2)2 + 7.2 – 8 = 2

Also if R = 0 then x-2 is a factor of f(x)

(If the sum of the coefficients of the function = 0 then x-1 is a factor)

(If the sum of the odd powers of x = sum of even powers of x, then x+1 is a factor)

Eg. If 222 x 224 x 226 is divided by 13 what is the remainder.

When 222 is divided by 13 the remainder is 1

When 224 is divided by 13 the remainder is 3

When 226 is divided by 13 the remainder is 5

NUMB3RS NUMB3RS So the problem can be treated as 1 x 3 x 5 ÷ 13 = 15 ÷ 13

Remainder = 2

Eg. If 225 + 227 + 229 is divided by 13 what is the remainder.

When 225 is divided by 13 the remainder is 4

When 227 is divided by 13 the remainder is 6

When 229 is divided by 13 the remainder is 8

So the problem can be treated as (4 +6 +8) ÷ 13 = 18 ÷ 13

Remainder = 5

BINOMIAL THEOREM(a + b)n = nC0.an + nC1 .an-1.b1 + nC2

.a n-2.b2 +…….nCn-1.a1.b n-1 + nCn.a0.bn

Alsoan – bn = (a – b)(an-1.b0 +an-2.b1 + an-3.b2+ …….a1.bn-2 + a0.bn-1)

Eg. What is the 40th term of (2x+y)42

40th term = 42C39.(2x)3.y39

40th term = 11,480.23.x3..y39

40th term = 91840 x3.y39

Eg. What is the remainder when 1717 is ÷ 61717 = (18-1)17

Expanding 1817 + 17C1.1816.(-1)1+ 17C21815.(-1)2 +….17C16.181.(-1)16 + (-1)17

Now, all the terms except the last term is divisible by 6, so the remainder is (-1)17 = -1hence positive remainder = 6 – 1 = 5

NUMBER BASED SYSTEMSDecimal system:

The decimal system has 10 as a base The decimal system has 10 digits i.e. digits 0 to 9

Concept 29: Mathematical notation of a numberA number xyz in a decimal system denotes x.102 + y.101 + z.100

xyz = 100x + 10y + z

Eg. 372 = 3.102 + 7.101 + 2.100 = 300 + 70 + 2

All number based systems follow the same notation. (256)8 denotes the number 256 to the base 8 and (256)8 = 2.82 +5.81 + 6.80 = 128 + 40 + 6 = (174)10

Concept 30: Other number bases The base number is always written as a subscript either as 5648 or (564)8. A base of n has digits 0 to n-1 i.e an octal system has digits 0 to 7. Base is 12, digits are 0, 1, 2…9, A, B (A denotes 10 and B denotes 11)

Concept 31: Conversion from decimal to other basesA number in the decimal system can also be obtained by arranging the remainders obtained by successive division of the number by 10 as below:

10 372 R

10 37 2 = 372

10 3 7

0 3

(372)10 is converted to another system (say, octal – 8) in the same manner

8 372 R

8 46 4 = (564)8 = (372)10

8 5 6 (Note: As the base decreases the number value increases)8 0 5

Concept 32: Addition and subtraction in other basesWhen 2 or more digits are added, what is in excess of base/multiples of base is written and the base/multiples of base carried to the next higher place.

NUMB3RS NUMB3RSEg. (4 7 6)8

+ (5 7 7)8

(1 2 7 5)8

Unit’s, 6 + 7 = 13, excess of base is 13 - 8 =5 write 5 carry 1

Ten’s, 1+7+7=15 excess of base 15–8=7 write 7 carry 1

Hundred’s, 1+4+5=10 excess of base 10-8=2 write 2 carry 1

Thousand’s, 1 write 1.

Concept 33: MultiplicationWhen two numbers are multiplied what is in excess of base/multiples of base is written and the base/multiples of base carried to the next higher place.Eg. (4 7) 8

× (5 7) 8

4 2 1 1 7 3

( 2 3 5 1) 8

Concept 34: Decimal to Binary conversion

Binary system (Base 2) – Digits 0 and 1

Conversion from decimal to binary – Methodology

Eg. Find 237 to base 2, 4 and 8

Find the nearest power of 2 larger than the number 28 = 256

28 – 1 in binary = 11111111 ( eight 1’s) = 255

To obtain 237, subtract 255-237 = 18 from the above i.e 16+2

1 1 1 1 1 1 1 1 = 111 0 1 1 0 1 = 23710

From base 2 to base 4, take digits 2 at a time (11)(10)(11)(01) = 32214

From base 2 to 8, take digits 3 at a time (11)(101)(101) = 3558