NUMBER SYSTEM

Natural Numbers: These are in numerical form of -1, 2, 3, 4, 5, 6, 7, 8, 9, 10..........denoted by N.

Whole Numbers: These are in numerical form of – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10..........denoted by W.

Integers: -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 denoted by Z.

Rational Numbers: All the numbers that can be mathematically written in the form p/q, q ≠0 are known as rational numbers where p and q refers to the integers.

Irrational Numbers: A number‘s’ is known as irrational, if it cannot be mathematically written in the form p/q where p and q are integers and q ≠ 0.

• • = 3.1666666, 219921​ = 2.33333
• Irrational Numbers
  •  It is a number that cannot be written as a ratio xyyx​ form (or fraction). An Irrational numbers are non-terminating and non-periodic fractions. For example: 22​ = 1.414
• Complex Numbers
  • The complex numbers are the set {a+bi}, where, a and b are real numbers and ‘i’  is the imaginary unit.
• Imaginary Numbers
  • A number does not exist on the number line is called imaginary number. For example square root of negative numbers are imaginary numbers. It is denoted by ‘i’ or ‘j.
• Even Numbers
  • A number divisible by 2 is called an even number.
  • For example: 2, 6, 8, 14, 18, 246, etc.
• Odd Numbers
  • A number not divisible by 2 is called an odd number.
  • For example: 3, 7, 9, 15, 17, 373, etc.
• Prime numbers
  • A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
  • For example: 2, 3, 5, 7, 11, 13, 17, etc.
• Composite numbers
  • Numbers greater than 1 which are not prime, are known as composite numbers. For example: 4, 6, 8, 10, etc.

Important Formulas of Number System

1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

(1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1)/6

(1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/2)²

Entirety of first n odd numbers = n²

Entirety of first n even numbers = (n + 1)

Mathematical Formulas

1. (a + b)(a – b) = (a² – b²)

2. (a + b)² = (a² + b² + 2ab)

3. (a – b)² = (a² + b² – 2ab)

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

5. (a³ + b³) = (a + b)(a² – ab + b²)

6.(a³ – b³) = (a – b)(a² + ab + b²)

7. (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)

8. when a + b + c = 0, then a³ + b³ + c³ = 3abc

For ‘a’ and ‘b’ positive real numbers the following entities will be true:

• √ab = √a √b
• √(a/b) =  √a / √b
• (√a + √b) (√a - √b) = a - b 
• (√a + √b)2 = a + 2√ab + b
• (a + √b) (a - √b) = a2 - b
• (a + b) (a - b) = a2 - b2

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Ratio and Proportion Formula

The Ratio and Proportion Formula is used to express the relationship between two quantities or compare their sizes.

The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. The sign used to denote a ratio is ‘:’

Important Properties of Proportion

The following are the important properties of proportion:

• Addendo – If a : b = c : d, then a + c : b + d
• Subtrahendo – If a : b = c : d, then a – c : b – d
• Dividendo – If a : b = c : d, then a – b : b = c – d : d
• Componendo – If a : b = c : d, then a + b : b = c+d : d
• Alternendo – If a : b = c : d, then a : c = b: d
• Invertendo – If a : b = c : d, then b : a = d : c
• Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

Fourth, Third and Mean Proportional

If a : b = c : d, then:

• d is called the fourth proportional to a, b, c.
• c is called the third proportion to a and b.
• Mean proportional between a and b is √(ab).

Comparison of Ratios

If (a:b)>(c:d) = (a/b>c/d)

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios

If a:b is a ratio, then:

• a2:b2 is a duplicate ratio
• √a:√b is the sub-duplicate ratio
• a3:b3 is a triplicate ratio

The proportion can be classified into the following categories, such as:

• Direct Proportion
• Inverse Proportion
• Continued Proportion

Now, let us discuss all these methods in brief:

Direct Proportion

The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b.

Inverse Proportion

The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).

Continued Proportion

Consider two ratios to be a: b and c: d.

Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.

For the given ratio, the LCM of b & c will be bc.

Thus, multiplying the first ratio by c and the second ratio by b, we have

First ratio- ca:bc

Second ratio- bc: bd

Thus, the continued proportion can be written in the form of ca: bc: bd

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Time and Work

Important Time and Work Formula

Knowing the formulas can completely link you to a solution as soon as you read the question. Thus, knowing the formula for any numerical ability topic make the solution and the related calculations simpler.

Given below are a few such important time and work formulas for your reference:

• Work Done = Time Taken × Rate of Work
• Rate of Work = 1 / Time Taken
• Time Taken = 1 / Rate of Work
• If a piece of work is done in x number of days, then the work done in one day = 1/x
• Total Wok Done = Number of Days × Efficiency
• Efficiency and Time are inversely proportional to each other
• X:y is the ratio of the number of men which are required to complete a piece of work, then the ratio of the time taken by them to complete the work will be y:x
• If x number of people can do W1 work, in D1 days, working T1 hours each day and the number of people can do W2 work, in D2 days, working T2 hours each day, then the relation between them will be

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Speed, Time & Distance

Speed, Time & Distance Conversions

• To convert from km / hour to m / sec, we multiply by 5 / 18. So, 1 km / hour = 5 / 18 m / sec
• To convert from m / sec to km / hour, we multiply by 18 / 5. So, 1 m / sec = 18 / 5 km / hour = 3.6 km / hour
• Similarly, 1 km/hr = 5/8 miles/hour
• 1 yard = 3 feet
• 1 kilometer= 1000 meters = 0.6214 mile
• 1 mile= 1.609 kilometer
• 1 hour= 60 minutes= 60*60 seconds= 3600 seconds
• 1 mile = 1760 yards
• 1 yard = 3 feet
• 1 mile = 5280 feet
• 1 mph = (1 x 1760) / (1 x 3600) = 22/45 yards/sec
• 1 mph = (1 x 5280) / (1 x 3600) = 22/15 ft/sec
• For a certain distance, if the ratio of speeds is a : b, then the ratio of times taken to cover the distance would be b : a and vice versa.

The concept of Speed, Time and Distance is used extensively for questions relating to different topics such as motion in a straight line, circular motion, boats and streams, races, clocks, etc.

Relationship Between Speed, Time & Distance

1. Speed = Distance/Time – This tells us how slow or fast an object moves. It describes the distance travelled divided by the time taken to cover the distance. 
2. Speed is directly Proportional to Distance and Inversely proportional to Time. Hence,
3. Distance = Speed X Time, and 
4. Time = Distance / Speed, as the speed increases the time taken will decrease and vice versa. 

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