Types of Numbers, Place Value and Face Value
Types of Numbers, Place Value and Face Value
Every Number System question in NTPC starts from one of three bases: which family a number belongs to, what a digit is worth at its position, or how digits behave when you reverse or rearrange them. Master these three and the first 1 to 2 Number System questions of every shift become 20-second marks.
1. Number Family: Complete Classification
| Type | Definition | Examples | Exam fact |
|---|---|---|---|
| Natural numbers (N) | Counting numbers 1, 2, 3, ... | 1, 7, 100 | Smallest natural number is 1; 0 is NOT natural |
| Whole numbers (W) | Natural numbers plus 0 | 0, 1, 2, ... | Smallest whole number is 0 |
| Integers (Z) | Whole numbers and their negatives | −3, 0, 5 | 0 is neither positive nor negative |
| Rational (Q) | Can be written as p/q, q ≠ 0, p and q integers | 3/4, −2, 0.75, 0.333... | Every terminating or recurring decimal is rational |
| Irrational | Cannot be written as p/q; decimal is non-terminating AND non-recurring | √2, √3, π, 0.1010010001... | π is irrational; 22/7 is only its rational approximation |
| Real (R) | Rational plus irrational | all of the above | Every point on the number line is real |
| Even | Divisible by 2 | 0, 2, 48 | 0 is even |
| Odd | Not divisible by 2 | 1, 15, 99 | Odd × odd = odd; even × anything = even |
| Prime | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11 | 2 is the only even prime; 1 is neither prime nor composite |
| Composite | More than two factors | 4, 6, 9, 91 | Smallest composite = 4; smallest odd composite = 9 |
| Co-prime | HCF of the two numbers is 1 | (8, 15), (9, 28) | Co-primes need not be prime themselves |
| Twin primes | Primes differing by 2 | (3, 5), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73) | 8 twin-prime pairs below 100 (add (5, 7)) |
| Perfect number | Sum of factors excluding itself = number | 6, 28, 496, 8128 | 6 = 1 + 2 + 3 |
Prime count facts: 25 primes from 1 to 100; 15 primes from 1 to 50; 10 primes from 51 to 100. Primes between 1 and 100: 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.
2. Rational vs Irrational: Operation Rules
| Operation | Result | Example |
|---|---|---|
| Rational ± rational | Always rational | 1/2 + 1/3 = 5/6 |
| Rational × rational (non-zero) | Always rational | 2/3 × 9 = 6 |
| Rational ± irrational | Always irrational | 3 + √2 |
| Non-zero rational × irrational | Always irrational | 5√3 |
| Irrational ± irrational | May be rational or irrational | (2 + √3) + (2 − √3) = 4 rational; √2 + √3 irrational |
| Irrational × irrational | May be rational or irrational | √8 × √2 = 4 rational; √2 × √3 = √6 irrational |
| √(positive rational that is not the square of a rational) | Irrational | √0.4, √12, √50 (but √0.04 = 0.2 and √2.25 = 1.5 are rational) |
✗ √0.4 = 0.2 | ✓ √0.04 = 0.2, but √0.4 = √(2/5) is irrational
✗ π = 22/7, so π is rational | ✓ 22/7 is a rational approximation; π itself is irrational
हिंदी नोट: परिमेय संख्या वह है जिसे p/q के रूप में लिखा जा सके और q शून्य न हो। सांत (terminating) या आवर्ती (recurring) दशमलव हमेशा परिमेय होते हैं, जबकि अनवसानी अनावर्ती दशमलव अपरिमेय होते हैं।
3. Place Value and Face Value
Face value of a digit is the digit itself. Place value = digit × value of its position. Place value of 0 is always 0, wherever it stands.
| Indian system | Value | International system |
|---|---|---|
| Unit | 1 | One |
| Ten | 10 | Ten |
| Hundred | 100 | Hundred |
| Thousand | 1,000 | Thousand |
| Ten thousand | 10,000 | Ten thousand |
| Lakh | 1,00,000 | Hundred thousand |
| Ten lakh | 10,00,000 | One million |
| Crore | 1,00,00,000 | Ten million |
| Ten crore | 10,00,00,000 | Hundred million |
| Arab | 1,00,00,00,000 | One billion |
Conversions to memorise: 1 million = 10 lakh; 1 billion = 100 crore; 1 crore = 10 million.
4. Digit and Series Formulas
Two-digit number with tens digit a and unit digit b = 10a + b; reversed = 10b + a
(10a + b) − (10b + a) = 9(a − b) and (10a + b) + (10b + a) = 11(a + b)
Three-digit abc − cba = 99(a − c), the middle digit never matters
Sum of first n natural numbers = n(n + 1)/2
Sum of first n odd numbers = n² and sum of first n even numbers = n(n + 1)
Sum of squares 1² + ... + n² = n(n + 1)(2n + 1)/6 and sum of cubes = [n(n + 1)/2]²
Number of terms from a to l with gap d = (l − a)/d + 1
Digits used to number pages: pages 1 to 9 use 9 digits, 10 to 99 use 180 digits, 100 to 999 use 2,700 digits. So a 250-page book uses 9 + 180 + (151 × 3) = 642 digits.
Shortcut for odd sums in a range: sum of odd numbers from 1 to (2n − 1) is n². So the sum of the odd numbers in a range = (number of odd numbers up to the last term)² − (number of odd numbers below the first term)². For 41 to 99: 50² − 20² = 2,100. This turns a 4-line AP sum into one subtraction.
Exam Pointer: In NTPC CBT-1 these appear as one-liners (place value minus face value, identify the irrational number, sum of odd numbers in a range, reversed-digit difference). The trap is always one notch away from the obvious: the Indian vs international comma, √0.4 vs √0.04, or forgetting that "between 40 and 100" excludes the end points.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Place value and face value combinations
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Find the difference between the place value and the face value of 8 in 7,48,36,915.
Read the number in the Indian system: 7 crore, 4 ten-lakh, 8 lakh. So 8 sits at the lakh place and its place value is 8,00,000. Face value is 8. Difference = 8,00,000 − 8 = 7,99,992.
Answer: 7,99,992
EXAMATLAS LEVEL
Q. In 5,38,26,437 the digit 3 appears twice. Find the difference between the sum of the place values of both 3s and the product of their face values.
Position the digits: 5 crore, 3 ten-lakh, 8 lakh, 2 ten-thousand, 6 thousand, 4 hundred, 3 tens, 7 units. The first 3 is worth 30,00,000 and the second 3 is worth 30, so the sum is 30,00,030. Product of face values = 3 × 3 = 9. Difference = 30,00,030 − 9 = 30,00,021. The trap is multiplying place values or adding face values; read the question word by word.
Answer: 30,00,021
Pattern 2: Identify rational or irrational
[PYQ: NTPC CBT-1 5-Jan-2021 Shift-1]
EXAM LEVEL
Q. Which of the following is irrational? (a) √1.44 (b) 0.333... (c) √12/√3 (d) √0.4
√1.44 = 1.2 is rational. 0.333... = 1/3 is rational. √12/√3 = √4 = 2 is rational. √0.4 = √(4/10) = √(2/5), and 2/5 is not a perfect square, so it is irrational. Students who confuse it with √0.04 = 0.2 lose this mark.
Answer: (d) √0.4
EXAMATLAS LEVEL
Q. Which of the following is rational? (a) (√5 + √3)² (b) (3 + √2)(3 − √2) (c) 7π/22 (d) √8 × √3
(a) expands to 5 + 3 + 2√15 = 8 + 2√15, irrational. (b) is a difference of squares: 9 − 2 = 7, rational. (c) would be 1 only if π were exactly 22/7; it is not, so 7π/22 is irrational. (d) √24 = 2√6, irrational. Only (b) survives, and the trap in (c) is the 22/7 habit.
Answer: (b)
Pattern 3: Reversed digits of a number
[PYQ: NTPC UG CBT-1 7-May-2026 Shift-1 | NTPC 2020-21 cycle]
EXAM LEVEL
Q. The sum of the digits of a two-digit number is 11. When the digits are reversed, the number increases by 27. Find the number.
Reversal changes a number by 9 times the digit difference, so 9(b − a) = 27 gives b − a = 3. With a + b = 11, b = 7 and a = 4. The number is 47 and 74 − 47 = 27 confirms it.
Answer: 47
EXAMATLAS LEVEL
Q. In a three-digit number the hundreds digit is 2 more than the units digit, the tens digit is twice the units digit and the sum of the digits is 14. Find the difference between the number and the number formed by reversing its digits.
Let the units digit be c. Then hundreds = c + 2 and tens = 2c, so (c + 2) + 2c + c = 14 gives 4c = 12 and c = 3. The number is 563 and its reverse is 365. The fast route: abc − cba = 99(a − c) = 99 × 2 = 198, and you could have written 198 the moment you read "hundreds digit is 2 more than units digit". The extra conditions are there only to slow you down.
Answer: 198
Pattern 4: Sum of consecutive odd, even or natural numbers
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Find the sum of all odd numbers from 1 to 59.
Odd numbers 1 to 59 are the first 30 odd numbers because (59 + 1)/2 = 30. Sum of first n odd numbers = n², so the sum is 30² = 900.
Answer: 900
EXAMATLAS LEVEL
Q. Find the sum of all odd numbers between 40 and 100.
Odd numbers up to 99 are the first 50 odd numbers, sum 50² = 2,500. Odd numbers up to 39 are the first 20, sum 20² = 400. Required sum = 2,500 − 400 = 2,100. Cross-check with the AP route: 41 to 99 has (99 − 41)/2 + 1 = 30 terms, average 70, sum 30 × 70 = 2,100.
Answer: 2,100
Pattern 5: Greatest and smallest numbers from given digits
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. Using each of the digits 0, 3, 5, 7 and 9 exactly once, find the difference between the greatest and the smallest five-digit numbers.
Greatest: arrange in descending order, 97,530. Smallest: ascending order but 0 cannot lead, so put the smallest non-zero digit first and 0 next: 30,579. Difference = 97,530 − 30,579 = 66,951.
Answer: 66,951
EXAMATLAS LEVEL
Q. Using the digits 1, 0, 4, 7 and 5 exactly once, find the sum of the smallest five-digit odd number and the greatest five-digit even number.
Smallest odd number: it must end in an odd digit and start with 1 (0 cannot lead). After 1 and 0, place 4, then the remaining 5 and 7 must be arranged so the number ends in an odd digit and stays smallest: 10,457 ends in 7 and beats 10,475. Greatest even number: it must end in 0 or 4. Ending in 0 gives 75,410; ending in 4 gives 75,104. So 75,410. Sum = 10,457 + 75,410 = 85,867. The common slip is writing 01,457 as a five-digit number.
Answer: 85,867
Pattern 6: Digits used in numbering pages
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. How many digits are needed to number the pages of a book with 340 pages?
Pages 1 to 9 need 9 digits, 10 to 99 need 90 × 2 = 180 digits, and 100 to 340 are 241 pages needing 241 × 3 = 723 digits. Total = 9 + 180 + 723 = 912.
Answer: 912
EXAMATLAS LEVEL
Q. A typist used 1,248 digits to number all the pages of a report. How many pages does the report have?
The first 99 pages use 9 + 180 = 189 digits. Remaining digits = 1,248 − 189 = 1,059, each three-digit page using 3 digits, so 1,059/3 = 353 three-digit pages. These start at page 100, so the last page is 100 + 353 − 1 = 452. The traps are 353 (three-digit pages only) and 100 + 353 = 453 (off by one). Three-digit pages run from 100 to 452, so the total is also 99 + 353 = 452.
Answer: 452 pages
60-Second Revision
- 1 is neither prime nor composite; 2 is the only even prime; 25 primes up to 100.
- Terminating or recurring decimal means rational; non-terminating non-recurring means irrational; π irrational, 22/7 rational.
- Place value = digit × position value; place value of 0 is 0; face value is the digit itself.
- Reversal: two-digit difference 9(a − b), sum 11(a + b); three-digit difference 99(a − c).
- Sum of first n odd numbers = n²; odd sum in a range = difference of two squares.
- Page numbering: 9 + 180 digits cover pages 1 to 99, then 3 digits per page.