2.1 Number System, BODMAS, Decimals & Fractions

Why This Matters

The number system is the bedrock of every arithmetic question on the RRB Group D CBT. Roughly one in four mathematics marks come from arithmetic and number-system topics, so a firm grasp here pays off across the entire 25-question Mathematics section. This section covers the classification of numbers, divisibility shortcuts, the BODMAS ordering rule, and the two-way conversion between decimals and fractions.

Classification of Numbers

Indian textbooks classify numbers in a strict hierarchy:

• Natural numbers (N): counting numbers 1, 2, 3, ... (no zero).
• Whole numbers (W): natural numbers plus 0, giving 0, 1, 2, 3, ...
• Integers (Z): whole numbers and their negatives, giving ..., -2, -1, 0, 1, 2, ...
• Rational numbers (Q): numbers of the form p/q where p and q are integers and q is not 0. Every integer is rational (write it over 1).
• Irrational numbers: cannot be written as p/q, for example the square root of 2, the square root of 3, and pi.
• Real numbers (R): all rational and irrational numbers together.

On the RRB CBT, expect a "which of the following is NOT a rational number" type question. Remember: a terminating decimal is always rational, and a recurring decimal is always rational; only non-recurring, non-terminating decimals are irrational.

Divisibility Rules

Divisibility rules let you test factors without long division. Memorise the table below, because at least one RRB question each cycle tests a divisibility rule directly.

Worked example (rule of 11): Is 5,183 divisible by 11? Alternating sum = 5 - 1 + 8 - 3 = 9. Since 9 is not 0 or a multiple of 11, 5,183 is not divisible by 11. Now try 2,937: 2 - 9 + 3 - 7 = -11, which is a multiple of 11, so 2,937 is divisible by 11.

BODMAS — The Order of Operations

When several operations appear together, evaluate strictly in this order:

Brackets, then Orders (powers and roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right).

A common RRB trap: division and multiplication have equal priority. Work them left to right, not multiplication first. The same applies to addition and subtraction.

Worked example. Simplify 8 + 4 / 2 * 3 - (6 - 2).

Step 1 (Brackets): (6 - 2) = 4. Expression becomes 8 + 4 / 2 * 3 - 4. Step 2 (Division and Multiplication, left to right): 4 / 2 = 2, then 2 * 3 = 6. Expression becomes 8 + 6 - 4. Step 3 (Addition and Subtraction, left to right): 8 + 6 = 14, then 14 - 4 = 10.

Answer: 10. A candidate who multiplies before dividing would compute 4 / (2 * 3) = 4 / 6 and then 8.667, which is wrong.

Decimals and Fractions

Decimal to fraction: write the decimal over its place value, then simplify. For 0.75: 75/100 = 3/4. For 0.045: 45/1000 = 9/200.

Fraction to decimal: divide numerator by denominator. 3/8 = 0.375; 7/20 = 0.35.

Recurring decimals, short method. Convert 0.363636... to a fraction. Let x = 0.363636..., multiply by 100: 100x = 36.363636..., subtract: 99x = 36, so x = 36/99 = 4/11. The rule for a pure recurring decimal with period length n is: denominator = 10^n - 1.

For a mixed recurring decimal like 0.1666... (one non-repeating digit, one repeating): let x = 0.1666..., 10x = 1.666..., 100x = 16.666..., subtract: 90x = 15, so x = 15/90 = 1/6. The general rule: the denominator is 9...90...0 with the count of 9s equal to the repeating block length and the count of 0s equal to the non-repeating block length.

Exam Scenarios and Common Traps

• Sign errors with integers: -5 - (-3) equals -5 + 3 = -2, not -8. Brackets flip the inner sign.
• Mis-applying BODMAS: questions like 12 / 3 * 2 appear every cycle. The answer is 8, not 2.
• Confusing 0 as natural: 0 is whole but not natural. The smallest whole number is 0; the smallest natural number is 1.
• Mixed recurring decimals: count only the repeating digits for the 9s and the non-repeating digits for the 0s.

Mastering these foundations means most arithmetic on the CBT becomes a 20-second mental calculation rather than a 90-second written one, crucial when you have only 90 minutes for 100 questions and negative marking punishes rushed guesses.