The number system forms the bedrock of the entire Mathematics section in MP TET. Questions from this topic test your conceptual clarity on types of numbers, their properties, and operations. Expect 3–5 direct questions in Varg-1/2/3 papers, plus this knowledge underpins nearly every other arithmetic topic.
Mastery here means understanding the hierarchy of numbers (natural → whole → integers → rational), fluency with place value for quick calculations, and confident handling of factors, multiples, HCF and LCM. The pedagogy angle often asks how to teach these concepts to children using concrete materials—connecting abstract numbers to real-world understanding.
Key Concepts
**Natural Numbers (N)**: Counting numbers starting from 1. {1, 2, 3, 4, ...}. Used for counting objects.
**Whole Numbers (W)**: Natural numbers plus zero. {0, 1, 2, 3, ...}. Zero represents "nothing" or empty set.
**Integers (Z)**: Whole numbers plus negative numbers. {..., -3, -2, -1, 0, 1, 2, 3, ...}. Introduced to represent loss, debt, temperature below zero.
**Rational Numbers (Q)**: Numbers expressible as p/q where p and q are integers and q ≠ 0. Includes all integers (since 5 = 5/1) and fractions. Every terminating or repeating decimal is rational.
**Place Value System**: The value of a digit depends on its position. In 4725: 4 is in thousands place (value = 4000), 7 in hundreds (700), 2 in tens (20), 5 in units (5).
**Face Value vs Place Value**: Face value is the digit itself; place value is face value × position value. In 3826, face value of 8 is 8, but place value is 800.
**Factors**: Numbers that divide a given number exactly (remainder = 0). Factors of 12: 1, 2, 3, 4, 6, 12.
**Multiples**: Products of a number with natural numbers. Multiples of 4: 4, 8, 12, 16, 20, ...
Formulas / Key Facts
| Concept | Key Point | |---------|-----------| | Number of factors of n | If n = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1) | | Sum of first n natural numbers | n(n+1)/2 | | Sum of first n whole numbers | Same as above (0 adds nothing) | | Product of two numbers | HCF × LCM = Product of the two numbers | | Divisibility by 2 | Last digit is 0, 2, 4, 6, or 8 | | Divisibility by 3 | Sum of digits divisible by 3 | | Divisibility by 4 | Last two digits form a number divisible by 4 | | Divisibility by 5 | Last digit is 0 or 5 | | Divisibility by 6 | Divisible by both 2 and 3 | | Divisibility by 9 | Sum of digits divisible by 9 | | Divisibility by 11 | Difference of sum of alternate digits is 0 or divisible by 11 | | Every integer is rational | Any integer n = n/1 | | Between any two rationals | Infinite rational numbers exist |
Worked Examples
**Example 1: Place Value** *Find the difference between the place value and face value of 6 in 46823.*
Place value of 6 = 6 × 1000 = 6000 Face value of 6 = 6 Difference = 6000 − 6 = **5994**
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**Example 2: Finding All Factors** *Find all factors of 36.*
**Example 3: Divisibility Test** *Is 2574 divisible by 6?*
Check divisibility by 2: Last digit is 4 (even) ✓ Check divisibility by 3: Sum of digits = 2+5+7+4 = 18, and 18÷3 = 6 ✓ Since divisible by both 2 and 3, **2574 is divisible by 6**.
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**Example 4: Rational Number Between Two Numbers** *Find a rational number between 1/3 and 1/2.*
Method 1: Average = (1/3 + 1/2) ÷ 2 = (5/6) ÷ 2 = 5/12 Method 2: Convert to like denominators: 1/3 = 2/6, 1/2 = 3/6. Between them: 5/12 (multiply both by 2 to get 4/12 and 6/12, middle = 5/12)
Answer: **5/12** (or any equivalent)
Common Mistakes
**Confusing place value with face value** → Remember: place value = digit × position weight. Face value is just the digit itself, regardless of position.
**Forgetting that 0 and 1 are special** → 0 is not a natural number but is a whole number. 1 is neither prime nor composite. 1 is a factor of every number.
**Assuming all decimals are irrational** → Only non-terminating, non-repeating decimals are irrational. Terminating decimals (0.25) and repeating decimals (0.333...) are rational.
**Missing factor pairs** → When listing factors, students often miss middle factors. Use prime factorisation or systematic pairing (1×n, 2×?, 3×?, ...) until pairs repeat.
**Applying wrong divisibility rule for 4** → Check last TWO digits, not just last digit. 312 → check 12 (divisible by 4 ✓). 322 → check 22 (not divisible by 4 ✗).
**Thinking negative numbers cannot be rational** → −3/4, −7, −2.5 are all rational. Rational numbers include positive, negative, and zero.
Quick Reference
**N ⊂ W ⊂ Z ⊂ Q** (Natural inside Whole inside Integers inside Rational)
**Factors are finite; multiples are infinite**
**1 is a factor of every number; 0 is a multiple of every number**
**Divisibility by 6 = check both 2 AND 3**
**Total factors formula: (a+1)(b+1)(c+1) for n = p^a × q^b × r^c**
**Between any two different rational numbers, infinitely many rationals exist**
**Place value questions: Always multiply digit by its position weight (units=1, tens=10, hundreds=100...)**
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What is the place value of the digit 7 in the number 45,732?
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What is the place value of the digit 7 in the number 45,732?
Q2 · Number System · MEDIUM
A number when divided by 12 leaves a remainder of 5. What will be the remainder when the same number is divided by 6?
Q3 · Number System · MEDIUM
The HCF of two numbers is 23 and their LCM is 1380. If one of the numbers is 92, what is the other number?
Q4 · Number System · HARD
A rational number p/q in its simplest form has q = 2^3 × 5^2. After how many decimal places will the decimal expansion of this rational number terminate?
Q5 · Number System · EASY
The sum of two consecutive odd numbers is 56. What is the smaller of the two numbers?