Number System Signed Arithmetic and BODMAS

What you will learn

Read number types and operation signs accurately, explain a calculation one step at a time, and recognise when division has no defined answer. By the end you should be able to:

• Distinguish natural numbers, whole numbers, integers and rational numbers without treating these as disjoint groups
• Compare signed integers and calculate with positive, negative and zero values
• Apply BODMAS to clearly written expressions, including equal-precedence operations
• Explain why −6² and (−6)² differ, and why a zero divisor is not allowed
• Translate a short situation into a grouped expression and check the result

This is the first numbers-and-calculation lesson in the RRB Group D Mathematics course. Number system and BODMAS are named in CEN 09/2025 §14.1. This lesson does not predict question counts or cover the whole Mathematics syllabus. Fraction arithmetic, HCF/LCM and square-root methods are developed in later lessons.

1 Before you begin

You need to read whole numbers and the symbols +, −, ×, ÷, =, < and >, and perform one-operation whole-number arithmetic. Negative-number rules, powers and order of operations are taught below; they are not entry requirements. Use paper to keep the steps visible. First try these four checks without looking at the feedback.

1. What is the place value of 4 in 407?
2. Calculate 68 + 27.
3. Calculate 9 × 6.
4. Calculate 84 ÷ 7.

Entry feedback and a short repair

1. 400. The digit 4 is in the hundreds place: 407 = 400 + 0 + 7. A digit and its place value are different. Retry: the place value of 6 in 603 is 600; the place value of 5 in 251 is 50.
2. 95. Break 27 into 20 + 7: 68 + 20 + 7 = 88 + 7 = 95. Retry: 47 + 36 = 47 + 30 + 6 = 83; 82 − 25 = 82 − 20 − 5 = 57.
3. 54. Nine taken six times is 54. Use 10 × 6 − 6 = 60 − 6 = 54 to check. Retry: 8 × 7 = 10 × 7 − 2 × 7 = 56; 6 × 4 = 24.
4. 12. Division reverses multiplication: 7 × 12 = 84. Retry: 72 ÷ 8 = 9 because 8 × 9 = 72; 96 ÷ 6 = 16 because 6 × 16 = 96.

If a check was difficult, cover its retry answers and work them again before moving to a long expression. There is no pass cutoff or time limit. The aim is to identify which building block needs attention.

2 Number types are connected

Convention: in this lesson natural numbers begin at 1. Some mathematical contexts include 0 in that name; use the convention stated here.

• Natural numbers: 1, 2, 3, …, used for counting
• Whole numbers: 0, 1, 2, 3, …
• Integers: …, −3, −2, −1, 0, 1, 2, 3, …
• Rational numbers: numbers expressible as p/q, where p and q are integers and q ≠ 0

Each natural number is a whole number; each whole number is an integer; each integer is rational because it can be written with denominator 1. The reverse implications do not always hold. A fraction can represent an integer, so the written appearance alone does not decide the number type.

You will later meet irrational numbers, which cannot be expressed as such a p/q. Rational and irrational numbers together form the real numbers. For now we only recognise the rational notation; fraction operations and irrational square-root examples come later.

Worked example 1 Classification and nested number families

Classify 0, −6, 9 and 5/2 as natural, whole, integer and rational numbers. A number may belong to more than one set.

1. 0 = 0/1: whole, integer, rational
2. −6 = −6/1: integer, rational
3. 9 = 9/1: natural, whole, integer, rational
4. 5/2: rational; not an integer

Use natural numbers 1, 2, 3, … in this lesson. Whole numbers also include 0. A rational number is p/q for integers p and q with q ≠ 0, so every integer is rational. The families overlap.

Error check: Do not mark 0 as “no type”, or assume a rational number must have a non-integer value.

3 Use the number line for signed addition and subtraction

A number line has equally spaced unit marks. Numbers increase as you move right: −9 < −3 < 0 < 4. Zero is neither positive nor negative. Of two negative numbers, the one nearer zero is greater: −3 > −9, even though 3 < 9.

Adding a positive integer moves right; adding a negative integer moves left. To add two numbers with the same sign, add their sizes and keep the sign: −4 + (−5) = −9. For opposite signs, compare their sizes: −9 + 14 is five units on the positive side of zero.

The opposite of a nonzero number has the same distance from zero and the other sign. The opposite of 4 is −4; the opposite of −4 is 4. Zero is its own opposite: −0 = 0. Subtracting a number means adding its opposite. This gives a meaning to the rule a − b = a + (−b), rather than asking you to erase minus signs by sight.

Worked example 2 Signed addition and subtraction

Calculate −9 + 14 and −7 − (−4). Explain the direction or sign change.

1. −9 + 14 = 5
2. −7 − (−4) = −7 + 4 = −3

For the first, move 14 units right from −9. For the second, subtracting −4 adds its opposite, +4. This rule is about subtracting a number; it is not a rule to erase any two minus signs seen nearby.

Error check: −7 − (−4) is not −11. Keep the operation sign separate from the sign of −4.

4 Explain the multiplication and division signs

For multiplication, equal signs give a positive product and different signs give a negative product. Multiplication by zero gives zero. A short pattern explains the less familiar case:

• (−3) × 2 = −6
• (−3) × 1 = −3
• (−3) × 0 = 0
• (−3) × (−1) = 3
• (−3) × (−2) = 6

As the second factor decreases by 1, each product increases by 3. Continuing this pattern preserves the distributive rule and explains why a negative times a negative is positive. It is not the same rule as adding two negative numbers.

Division uses the multiplication check: if a ÷ b = c, then b × c = a, provided b ≠ 0. Thus −24 ÷ (−4) = 6 because (−4) × 6 = −24. For nonzero values, division has the same equal-sign/different-sign rule as multiplication.

When multiplication and division occur together with no further grouping, work from left to right; neither symbol has higher priority. This is one part of the complete order of operations below.

Worked example 3 Signs in multiplication and division

Evaluate (−8) × 3 ÷ (−4).

1. (−8) × 3 ÷ (−4) = −24 ÷ (−4) = 6
2. Check: 6 × (−4) = −24

Different signs give a negative product; dividing two negative numbers gives a positive quotient. Multiplication and division share a precedence level, so use the leftmost operation first here.

Error check: Do not carry a minus sign into the final answer just because the first number was negative.

5 Read the base before evaluating a power

A positive integer exponent tells how many equal factors to multiply. For example, 4³ = 4 × 4 × 4 = 64. The base is 4 and the exponent is 3. It does not mean 4 × 3. Squaring uses two equal factors: 6² = 6 × 6.

Parentheses specify the entire base when a negative number is raised to a power. Without those parentheses, a minus placed before a squared positive number asks for the opposite of that square.

Worked example 4 Power base and unary minus

Compare −6², (−6)² and −(6²).

1. −6² = −(6 × 6) = −36
2. (−6)² = (−6) × (−6) = 36
3. −(6²) = −36

In −6², only 6 is squared, then its opposite is taken. In (−6)² the brackets make −6 the base, so both factors are −6. The leading minus is not a negative exponent.

Error check: “Every expression with a square is positive” is false. First identify exactly what is squared.

6 BODMAS has four stages, not six separate priority levels

BODMAS is a memory aid for brackets, orders, division, multiplication, addition and subtraction. Here “orders” means powers; roots will be introduced later. Use the following procedure:

1. Grouping: start with the innermost brackets and evaluate the expression inside using these same rules. All bracket shapes group expressions; their position inside one another matters, not their shape.
2. Powers: evaluate powers with their bases correctly identified.
3. Multiplication and division: they have equal precedence. Work from left to right among the remaining operations at this level.
4. Addition and subtraction: they also have equal precedence. Work from left to right at this level.

For example, 5 + 2 × 6 = 5 + 12 = 17. Doing the addition first would instead calculate (5 + 2) × 6 = 42, a different expression. Do not simply work left to right through every symbol before dealing with the higher-priority operations.

“Left to right” is a safe evaluation procedure within a precedence level. Later you can use valid rewrites, such as adding signed terms in a convenient order, but you must preserve every term and sign. It never permits swapping operands in a subtraction or division without changing the expression.

Worked example 5 Equal precedence error analysis

A learner says 42 ÷ 7 × 3 = 2 and 23 − 8 + 4 = 11. Correct both calculations.

1. 42 ÷ 7 × 3 = 6 × 3 = 18
2. 23 − 8 + 4 = 15 + 4 = 19

The first wrong answer treats 7 × 3 as a grouped denominator even though no brackets group it. The second wrongly adds 8 and 4 before subtracting. D and M have equal precedence; A and S have equal precedence. Within either level, evaluate left to right.

Error check: BODMAS does not mean “all divisions before every multiplication” or “all additions before every subtraction”.

Worked example 6 Nested grouping and powers

Evaluate 54 − [18 ÷ 3 + (5 − 2)²].

1. 54 − [18 ÷ 3 + (5 − 2)²]
2. = 54 − [18 ÷ 3 + 3²]
3. = 54 − [6 + 9]
4. = 54 − 15
5. = 39

Start inside the innermost group. Within the square brackets, evaluate the power and division before addition. The bracket shape does not create a permanent ranking: nesting tells us which group is inner.

Error check: 3² means 3 × 3, not 3 × 2. Do not drop the subtraction before the square bracket.

Keep the notation clear

Write multiplication as × in this lesson. Use (18 + 6) ÷ (5 − 2), not an ungrouped chain of slashes. It evaluates to 24 ÷ 3 = 8. A horizontal fraction bar also groups the whole numerator above it and the whole denominator below it; each is evaluated before division. When typing a fraction on one line, retain the parentheses around a multi-term numerator or denominator.

We do not use puzzles such as 6 ÷ 2(1 + 2), whose mixed division and implicit-multiplication notation invites conflicting readings. The point is to evaluate a clear expression, not guess what its writer intended. In a later fraction lesson, a phrase such as “half of the sum” will be translated into a product with the sum grouped. Do not invent an extra universal priority for the word “of”.

7 Check that division is defined

A zero numerator is allowed when the divisor is nonzero: 0 ÷ 9 = 0 because 9 × 0 = 0. A zero divisor is not allowed. Always simplify a grouped divisor before dividing; its value might be zero even when none of the written numbers is zero.

Worked example 7 Zero and undefined division

Which expressions are defined: 0 ÷ 9, 9 ÷ 0, 0 ÷ 0 and 10 ÷ (7 − 7)?

1. 0 ÷ 9 = 0
2. 9 ÷ 0: undefined
3. 0 ÷ 0: undefined
4. 10 ÷ (7 − 7) = 10 ÷ 0: undefined

Division asks for a unique number that gives the dividend when multiplied by the divisor. No number multiplied by 0 gives 9; every number multiplied by 0 gives 0, so 0 ÷ 0 cannot select a unique answer. Evaluate a grouped denominator before dividing.

Error check: Division by zero is neither zero nor a real-number answer called infinity.

8 Let the situation choose the grouping

Write what each multiplication or addition counts before calculating. Keep money values in the same unit, then check that the result makes sense: change should not exceed the payment when a positive purchase cost is being subtracted.

Worked example 8 Translate a short situation and check the result

Four notebooks cost ₹27 each. One pen costs ₹13. If ₹150 is paid, how much change is due?

1. Cost = 4 × 27 + 13 = 108 + 13 = 121
2. Change = 150 − (4 × 27 + 13) = 150 − 121 = ₹29
3. Check: 121 + 29 = 150

First group the entire cost, then subtract it from the payment. Only the notebooks are bought four times; the single pen is not multiplied by four.

Error check: 150 − 4 × 27 + 13 adds the pen price back instead of subtracting it.

9 Independent practice

Attempt all twelve questions before viewing the key. Choose exactly one option per question and write at least one reason or calculation. These are original learning questions, without a time limit, negative marking or pass cutoff. They are not a full RRB CBT simulation.

1. Which is rational but not an integer?

A. −8 B. 0 C. 13/4 D. 11

2. Which number is greatest?

A. −11 B. −4 C. −9 D. −7

3. Evaluate −13 + 8.

A. −5 B. 5 C. −21 D. 21

4. Evaluate 6 − (−9).

A. −15 B. −3 C. 3 D. 15

5. Evaluate (−7) × (−4).

A. −28 B. 28 C. −11 D. 11

6. Evaluate (−42) ÷ 7.

A. −6 B. 6 C. −35 D. 35

7. Evaluate 63 ÷ 9 × 2.

A. 3 B. 7 C. 14 D. 18

8. Evaluate 31 − 12 + 5.

A. 14 B. 18 C. 22 D. 24

9. What are the values of −5² and (−5)², respectively?

A. −25, 25 B. 25, 25 C. −25, −25 D. 25, −25

10. Evaluate 47 − [24 ÷ 6 + (7 − 4)²].

A. 30 B. 34 C. 38 D. 42

11. Evaluate 8 ÷ (3 − 3).

A. 0 B. 1 C. 8 D. Undefined

12. Three notebooks cost ₹32 each and one pen costs ₹17. How much change is due from ₹150?

A. ₹27 B. ₹33 C. ₹37 D. ₹47

10 Explained answer key

1. C. 13/4 has integer numerator and nonzero integer denominator, but is not an integer. The other three are integers and therefore also rational; the question requires both conditions.

2. B. −4 lies furthest right on the number line. Among negative numbers, the one closer to zero is greater; comparing 11 and 4 while ignoring the signs reverses the conclusion.

3. A. Move 8 units right from −13 to reach −5.

4. D. 6 − (−9) = 6 + 9 = 15.

5. B. Two negative factors give a positive product: 7 × 4 = 28.

6. A. The signs differ, so the quotient is negative; 42 ÷ 7 = 6.

7. C. 63 ÷ 9 × 2 = 7 × 2 = 14, using equal precedence from left to right. Multiplying the divisor 9 by 2 would create a different, grouped denominator.

8. D. 31 − 12 + 5 = 19 + 5 = 24. Adding 12 and 5 first would change the expression to 31 − (12 + 5), whose value is 14.

9. A. −5² = −(5 × 5) = −25; (−5)² = (−5) × (−5) = 25.

10. B. 47 − [24 ÷ 6 + 3²] = 47 − [4 + 9] = 34. The exponent applies to the value of the entire inner group, 7 − 4.

11. D. The denominator is 3 − 3 = 0. Division by zero is undefined.

12. C. 150 − (3 × 32 + 17) = 150 − (96 + 17) = ₹37. Check: 113 + 37 = 150. The pen is bought once, so do not multiply ₹17 by three.

11 Use your errors to choose what to revisit

• Number-family or comparison error: revisit sections 2–3
• Sign error in an operation: revisit sections 3–4
• Wrong base or exponent: revisit section 5
• Incorrect next operation or missing grouping: revisit section 6
• Zero-divisor error: revisit section 7
• Wrong expression for a situation: revisit section 8

Cover the key, retry the missed question and explain why the incorrect step changes the calculation. Before moving on, explain in your own words why 42 ÷ 7 × 3 is 18, why −6² differs from (−6)², and why 0 ÷ 0 is not assigned a value. There is no numerical pass gate.

Sources and next step

The teaching prose and practice are original. Sources establish the concepts and notation:

• RRB CEN 09/2025, §14.1, printed p.28: Mathematics topic scope; no topic-frequency claim is made here
• NIOS Mathematics 211 Chapter 1, §§1.2–1.3, pp.4–9, and §1.7, pp.15–21: number families and arithmetic; Hindi Chapter 1 supports terminology
• NIOS Mathematics 211 Chapter 2, §2.1, pp.40–43: base and positive-integer exponent
• NCERT Ganita Prakash Grade 7 Chapter 2, reprint 2026–27, §2.2, pp.26–35: grouping, signed terms and expression meaning
• OpenStax Prealgebra 2e §2.1, Order of Operations: equal precedence; §10.2, Example 10.13: leading minus versus a parenthesised negative base. These are supplementary notation references, not RRB syllabus documents

Next, Factors Divisibility and Prime Factorisation explains how a positive whole number can be divided exactly and built from primes. No recruitment dates or eligibility advice are part of this lesson.

Analogy

Brackets are a label around a whole group

Imagine two trays. Each tray holds three cups, and one extra cup stands outside the trays. The total is 2 × 3 + 1 = 7 cups. If the instruction changes to “each of the two trays holds three cups and one extra cup”, the total is 2 × (3 + 1) = 8 cups.

The bracket labels everything repeated for each tray. It does not decorate the expression: it changes which quantity is multiplied. Read the words, mark the group and only then calculate. In the notebook example, brackets label the whole bill that must be subtracted from the payment.

This comparison explains grouping, not every arithmetic rule. For operation order and negative signs, use the rules and worked calculations in the lesson.

Quick reference

Number types

Natural numbers start at 1 in this lesson. Whole numbers also include 0. Integers consist of the whole numbers and the opposites of positive whole numbers. A rational number is p/q for integers p,q with q ≠ 0. Every integer is rational; not every rational number is an integer. Zero is neither positive nor negative.

Signed calculation

• Addition: same signs → add sizes and keep the sign; opposite signs → subtract sizes and take the sign of the larger size; equal opposite sizes sum to 0
• Subtraction: add the opposite, a − b = a + (−b); the opposite of 0 is 0
• Multiplication/division of nonzero values: equal signs → positive; different signs → negative
• Multiplication by 0 gives 0; division by 0 is undefined, including 0 ÷ 0
• Check division with divisor × quotient = dividend

Four operation-order stages

1. Innermost grouping, applying the same rules inside
2. Powers, with the base identified
3. × and ÷ together, left to right
4. + and − together, left to right

−6² = −36, but (−6)² = 36. A² means A × A, not A × 2. Bracket shape does not determine priority; nesting does. Use explicit × and group multi-term denominators. A fraction bar groups its entire numerator and denominator.

Before accepting an answer

Check the signs, each equal-precedence step, the full scope of every bracket and whether any divisor becomes 0. For a word problem, check the unit and substitute the result back into the situation. A learning score does not impose a pass gate.