Natural Numbers, Whole Numbers and Integers
Natural Numbers, Whole Numbers and Integers
Part-III carries 80 questions, split roughly 40 for mathematics and 40 for science, and every one of those marks counts towards your merit. Number system alone can yield 4 to 5 questions. This opening topic fixes the vocabulary the rest of the paper assumes you already know.
The Number Families
| Set | Symbol | Members | Smallest member |
|---|---|---|---|
| Natural | N | 1, 2, 3, ... | 1 |
| Whole | W | 0, 1, 2, 3, ... | 0 |
| Integers | Z | ..., -2, -1, 0, 1, 2, ... | None |
| Rational | Q | p/q where q is not 0 | None |
Each set contains the one before it. Every natural number is a whole number, every whole number is an integer, and every integer is rational because it can be written over 1. The reverse is never true, and that one-way relationship is what statement-based questions test.
✗ Every whole number is a natural number | ✓ Every natural number is whole, but 0 is whole and not natural
Properties of Operations
Four properties are asked by name, and the pattern of exceptions is the actual question.
| Property | Addition | Subtraction | Multiplication | Division |
|---|---|---|---|---|
| Closure on N | Yes | No | Yes | No |
| Commutative | Yes | No | Yes | No |
| Associative | Yes | No | Yes | No |
| Identity element | 0 | — | 1 | — |
Subtraction and division are the troublemakers throughout. 5 − 3 is a natural number but 3 − 5 is not, so natural numbers are not closed under subtraction. Zero is the additive identity and one is the multiplicative identity.
a × (b + c) = a × b + a × c — the distributive property
Zero, and Why Division by It Fails
Zero behaves differently in each operation and the paper exploits this.
- a + 0 = a, and a − 0 = a.
- a × 0 = 0 for every a.
- 0 ÷ a = 0 when a is not zero.
- a ÷ 0 is not defined — not infinity, not zero.
- 0 ÷ 0 is also not defined.
The reason matters if you are going to teach it: division asks what number multiplied by the divisor gives the dividend. With a divisor of zero, no such number exists, because anything times zero is zero.
Integers on the Number Line
Rules for signs are best remembered through the number line rather than as a list.
- Adding a positive moves right; adding a negative moves left.
- Subtracting a negative is the same as adding a positive: 7 − (−3) = 10.
- Product or quotient of two like signs is positive; of two unlike signs is negative.
- For a product of several negatives, an even count gives a positive and an odd count gives a negative.
✗ −3 is greater than −2 because 3 is greater than 2 | ✓ −3 < −2 — on the number line −3 lies further left
शून्य से भाग परिभाषित नहीं है — अनंत नहीं, शून्य भी नहीं।
TRE pointer: Part-III mathematics gives you about 40 questions, five times the maths weight of Part-II, so this subject is where your merit is actually built. Expect property-based statements here rather than calculation — which set is closed under which operation, and what division by zero equals. The single most repeated trap is treating negative numbers as if their size decides their order. Remember also that with five options and −1/3 for a wrong answer, a property you cannot justify is safer left to option E.
60-Second Recap
- N is inside W is inside Z is inside Q; the reverse containment is false.
- Natural numbers are not closed under subtraction or division.
- Addition and multiplication are commutative and associative; subtraction and division are not.
- 0 is the additive identity; 1 is the multiplicative identity.
- a ÷ 0 is not defined, and neither is 0 ÷ 0.
- An even number of negative factors gives a positive product.
Frequently Asked Questions
Is zero a natural number?
No. Natural numbers begin at one and are the counting numbers. Zero belongs to the whole numbers, which is exactly the set of natural numbers with zero added. This distinction is asked directly and is also the point at which many class 6 students first go wrong.
Why is division by zero not defined?
Because division asks which number, multiplied by the divisor, gives the dividend. If the divisor is zero, every product is zero, so either no answer exists or every answer works. Neither case gives a single value, so the operation is left undefined rather than called infinity.
Are natural numbers closed under subtraction?
No. Closure means the result always stays inside the set, and 3 minus 5 gives negative two, which is not a natural number. Natural numbers are closed under addition and multiplication only. Integers, by contrast, are closed under subtraction as well.
How many maths questions are there in BPSC TRE 4.0 Part-III?
Part-III has 80 questions for the concerned subject. For Mathematics and Science candidates that is about 40 mathematics and 40 science. Since Part-I is only qualifying, Part-II and Part-III together decide your merit, which makes these 40 marks the most valuable in the paper.