Number System and Binary Numbers

Why This Topic Matters for NDA

Every NDA Maths paper carries a handful of questions on the number system — classification of numbers, divisibility, and especially binary numbers. These are short, formula-light questions that you can solve in under a minute if your basics are clear.

Binary questions appear almost every year because the NDA syllabus explicitly lists the binary system of numbers. Examiners love it because it tests understanding, not memory. A candidate who has practised base conversions simply cannot get them wrong.

There is also a strategic reason to love this chapter. The NDA Maths paper has 120 questions to be solved in 150 minutes, so on average you get just over a minute per question. Spending that minute on a guaranteed binary conversion is far smarter than gambling it on a long integral. Strong students build their score on a base of such reliable topics and then attempt the harder ones with the time they have saved.

Classifying Numbers

Numbers are organised into nested sets. Each larger set contains the smaller ones inside it.

• Natural numbers (N): 1, 2, 3, … — the counting numbers.
• Whole numbers (W): 0, 1, 2, 3, … — naturals together with zero.
• Integers (Z): …, −2, −1, 0, 1, 2, … — positives, negatives and zero.
• Rational numbers (Q): any number of the form p÷q where p and q are integers and q ≠ 0. Their decimal form either terminates or repeats.
• Irrational numbers: non-terminating, non-repeating decimals such as √2, π and e.
• Real numbers (R): all rationals and irrationals together — everything on the number line.

A few facts about these sets are tested again and again. The number 0 is a whole number and an integer but is not a natural number. A terminating decimal such as 0.75 is rational because it equals 3÷4. A square root is irrational only when the number under the root is not a perfect square, so √9 = 3 is rational while √3 is irrational. Knowing these edge cases stops you from second-guessing in the exam hall.

Place Value and the Idea of a Base

Our everyday counting uses the decimal (base-10) system, with ten digits 0–9. The position of a digit decides its value. In 3_(rightmost) the digit is worth itself; one place left it is worth ten times more.

For example, take the number 235:

The same logic works for any base b. A number written in base b uses digits from 0 up to (b − 1), and each place is a power of b. Change the base and only the “b” in the powers changes.

• Binary: base 2, digits 0 and 1.
• Octal: base 8, digits 0–7.
• Hexadecimal: base 16, digits 0–9 then A, B, C, D, E, F for 10–15.

The Binary System

The binary system uses only two digits, 0 and 1, called bits. Computers use it because a switch is naturally either off (0) or on (1).

Each binary place is a power of 2. Reading from the right: 2⁰, 2¹, 2², 2³, and so on (1, 2, 4, 8, 16, 32, …).

Learn the first few powers of 2 by heart: 1, 2, 4, 8, 16, 32, 64, 128, 256. Almost every NDA binary question stays inside this range, so instant recall saves precious seconds.

Two more words you should know: the leftmost bit, which carries the largest place value, is the most significant bit (MSB), and the rightmost bit is the least significant bit (LSB). A group of 8 bits is called a byte, and the largest number a byte can hold is 11111111₂ = 255₁₀. Recognising this 8-bit, 0-to-255 range is handy because many computer-related general-awareness facts rest on it.

Converting Decimal to Binary

To convert a decimal number to binary, use the repeated division by 2 method:

1. Divide the number by 2 and write down the remainder (0 or 1).
2. Divide the quotient again by 2, noting the remainder.
3. Continue until the quotient becomes 0.
4. Read the remainders from bottom to top — that is your binary number.

Converting Binary to Decimal

This is the reverse and even easier. Write the place value (power of 2) above each bit, then add the place values wherever the bit is 1.

For binary fractions, places to the right of the binary point are 2⁻¹, 2⁻², … that is ½, ¼, ⅛. So 0.101₂ = ½ + ⅛ = 0.625₁₀.

This expansion method is the safest one for the exam because it never involves division. You simply line up the powers of 2 above the bits, tick off the ones, and add. With a little practice you will be able to read short binary numbers such as 1000 (= 8) or 1111 (= 15) at a glance, without writing anything down at all.

Binary Addition, Subtraction and Multiplication

Binary arithmetic follows the same column rules as decimal — you just carry whenever a column total reaches 2 (because 2 in binary is written 10).

Subtraction uses borrowing, where a borrow brings in a value of 2. Multiplication works exactly like decimal long multiplication but is simpler, because each partial product is either the number itself (multiply by 1) or zero (multiply by 0).

One reliable trick when you are short on time is to convert both binary numbers to decimal, do the arithmetic in decimal where you are fast, and then convert the result back to binary. This avoids carry mistakes under pressure. Use direct binary arithmetic only when the numbers are small and the column rules are fresh in your mind.

Octal and Hexadecimal Shortcuts

Octal (base 8) and hexadecimal (base 16) are popular because they convert to and from binary very quickly.

Binary to octal

Group the bits into sets of three from the right; each group becomes one octal digit (since 2³ = 8). Example: 11010 → group as 011 010 → 3 2 → 32₈.

Binary to hexadecimal

Group the bits into sets of four from the right; each group becomes one hex digit (since 2⁴ = 16). Example: 1011 1110 → B E → BE₁₆.

To go the other way, expand each octal or hex digit back into its bit group. The octal digit 7 becomes 111 and the hex digit C becomes 1100, because C is 12 and 12 = 8 + 4. Since these conversions are pure grouping, they take only a few seconds once the digit-to-bit table is memorised. If a question asks you to convert a large decimal number to octal or hex, the quickest route is often to convert it to binary first and then group the bits, rather than dividing repeatedly by 8 or 16.

Useful Number Properties

NDA also tests basic number properties drawn from the same chapter. Keep these handy.

• Even and odd: a number is even if it ends in an even digit; in binary, a number is even only if its last bit is 0.
• Divisibility by 2: last digit even. By 3: digit sum divisible by 3. By 5: ends in 0 or 5. By 9: digit sum divisible by 9.
• Prime numbers: have exactly two factors, 1 and themselves; 2 is the only even prime.
• HCF and LCM: for two numbers, HCF × LCM = product of the numbers.

Common Mistakes to Avoid

Finally, do not confuse a digit’s face value (the digit itself) with its place value (digit × power of the base). Many trick questions exploit exactly this difference.

Previous-Year Style Question

Quick Revision

Practise five conversions and two binary additions every day for a week, and this entire chapter becomes free marks in the NDA paper.