Types of Numbers, Place Value and Decimals

The Number Family

SSC loves asking which "family" a number belongs to. It came up often in the 2025 papers, so get these sets perfectly clear.

Natural numbers ($\mathbb{N}$): the counting numbers 1, 2, 3, 4, … There is no 0 and nothing negative.

Whole numbers ($\mathbb{W}$): the natural numbers plus 0, so 0, 1, 2, 3, …

Integers ($\mathbb{Z}$): the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …

Rational numbers ($\mathbb{Q}$): any number that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Every integer is rational (5 = 5/1), and so are fractions like 3/4, terminating decimals like 0.75 and recurring decimals like 0.333…

Irrational numbers: numbers that cannot be written as $\frac{p}{q}$. Their decimals go on forever without repeating. Examples: $\sqrt{2}$, $\sqrt{7}$, $\pi$, 0.101001000…

Real numbers ($\mathbb{R}$): all the rational and irrational numbers together. Every point on the number line is a real number.

The sets sit inside one another like boxes:

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$

Irrational numbers are inside $\mathbb{R}$ too, but completely outside $\mathbb{Q}$. A number is either rational or irrational, never both.

Where does each number belong?

Key Point: 0 is a whole number and an integer, but not a natural number. Every integer is rational, and every number in this syllabus is real.

The Traps in Number-Type Questions

Most of these questions are free marks if you slow down for five seconds and test each option against the definition. These are the traps that catch people:

• A root that is secretly a whole number. $\sqrt{16} = 4$, $\sqrt{0.25} = 0.5$ and $\sqrt{\frac{9}{4}} = \frac{3}{2}$ are all rational. Only the root of a number that is not a perfect square, like $\sqrt{7}$ or $\sqrt{0.4}$, is irrational. Always simplify the root first.
• 22/7 is not $\pi$. 22/7 is a fraction, so it is rational. It is only a handy approximation of $\pi$, which is irrational.
• Recurring means rational. 0.333… and 3.142857142857… both repeat, so both are rational. "Goes on forever" is not enough to make a number irrational. It has to go on forever without repeating.
• "Positive" and "non-negative" are different. Natural numbers are the positive integers. Whole numbers are the non-negative integers, because they include 0.
• "Integer but not natural" means 0 or a negative integer. "Integer but not whole" means a negative integer only.

Statement questions ("Which of the following is true?") follow the box picture. A smaller set is always inside the bigger one, never the other way round.

[SSC Tip] When the options are names of sets ("Integers", "Real numbers") rather than numbers, look for one member that breaks the condition. For "the set that contains only positive numbers", integers fail because of −1, and rational and real numbers fail because of −1/2. Only natural numbers work.

Adding and Multiplying Rational and Irrational Numbers

Why "can be either"? Look at these:

• $\sqrt{2} + \sqrt{3}$ is irrational, but $\sqrt{2} + (-\sqrt{2}) = 0$ is rational.
• $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ is irrational, but $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ is rational.

One more catch: 0 is rational, and $0 \times \sqrt{2} = 0$. That is why the rule says non-zero rational times irrational.

[SSC Tip] Simplify before you decide. $7\sqrt{5} \times \sqrt{20} = 7\sqrt{100} = 70$ is rational, even though both parts had roots in them. And $(3 + \sqrt{5}) + (4 - \sqrt{5}) = 7$ is rational because the roots cancel out.

Terminating and Recurring Decimals

Every rational number has a decimal form that either stops (terminating) or repeats a block forever (recurring). Irrational numbers never do either.

When does a fraction terminate? Write it in its lowest terms first. If the denominator has no prime factors other than 2 and 5, the decimal terminates. Otherwise it recurs.

• $\frac{7}{40}$: 40 = $2^3 \times 5$, only 2s and 5s, so it terminates: 0.175.
• $\frac{7}{30}$: 30 = 2 × 3 × 5. The 3 makes it recur: 0.2333…

Always reduce first. $\frac{9}{30} = \frac{3}{10} = 0.3$ terminates, even though 30 has a 3 in it.

How many decimal places? For a terminating fraction in lowest terms, write the denominator as $2^m \times 5^n$. The number of decimal places is the larger of $m$ and $n$. For $\frac{7}{40}$, the denominator is $2^3 \times 5^1$, so there are 3 decimal places.

The bar notation. A bar goes over the part that repeats: $0.\overline{3}$ = 0.333…, $0.\overline{45}$ = 0.4545…, $0.1\overline{6}$ = 0.1666…

Turning a Recurring Decimal into a Fraction

Pure recurring (the repeat starts straight after the decimal point): put the repeating digits on top, and as many 9s as there are repeating digits underneath.

$$0.\overline{3} = \frac{3}{9} = \frac{1}{3} \qquad 0.\overline{45} = \frac{45}{99} = \frac{5}{11} \qquad 0.\overline{123} = \frac{123}{999} = \frac{41}{333}$$

Mixed recurring (some digits come before the repeat starts): on top, write all the digits after the point minus the digits that don't repeat. Underneath, write one 9 for each repeating digit, followed by one 0 for each non-repeating digit.

$$0.1\overline{6} = \frac{16 - 1}{90} = \frac{15}{90} = \frac{1}{6}$$

$$0.2\overline{35} = \frac{235 - 2}{990} = \frac{233}{990}$$

If there is a whole-number part, keep it aside and add it at the end: $3.\overline{6} = 3 + \frac{6}{9} = 3\frac{2}{3}$.

[SSC Tip] $0.\overline{9} = \frac{9}{9} = 1$. It looks strange, but 0.999… and 1 are the same number, and options sometimes use this.

To add or subtract recurring decimals, change them to fractions first. Adding 0.777… and 0.555… digit by digit goes wrong at the carry; $\frac{7}{9} + \frac{5}{9} = \frac{12}{9}$ does not.

Even, Odd, Prime, Composite and Co-prime

Even numbers are divisible by 2 and odd numbers are not. 0 is even.

• Even ± even = even, odd ± odd = even, even ± odd = odd.
• Odd × odd = odd. Any product that has an even number in it is even.

Prime numbers have exactly two factors, 1 and the number itself: 2, 3, 5, 7, 11, 13, … Composite numbers have more than two factors: 4, 6, 8, 9, 10, …

• 1 is neither prime nor composite, because it has only one factor.
• 2 is the only even prime.
• There are 15 primes up to 50 and 25 primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
• Twin primes are primes that differ by 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).

Is it prime? Find roughly the square root of the number, then check divisibility by every prime up to that root. If none of them divides it, the number is prime. For 179: $\sqrt{179}$ is a little over 13, so check 2, 3, 5, 7, 11 and 13. None divides 179, so 179 is prime.

Co-prime numbers have HCF 1, so their only common factor is 1. They don't have to be prime themselves: 8 and 15 are co-prime.

• Any two consecutive numbers are co-prime.
• Any two different primes are co-prime.

Perfect numbers equal the sum of their factors other than themselves: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. They rarely come up, but the term sometimes appears in options.

Place Value, Face Value and Digit Questions

The face value of a digit is the digit itself. The place value is the digit multiplied by the value of its position. In 4,72,315 the digit 7 has face value 7 and place value 70,000, because it sits in the ten-thousands place.

Largest and smallest number from given digits. Arrange the digits in descending order for the largest and ascending order for the smallest, but remember a number cannot start with 0. Using 0, 3, 5 and 8 once each: largest 8530, smallest 3058.

Rearranging digits. Recent CGL papers have given a seven-digit number, asked for its digits in ascending (or descending) order, and asked for the difference with the original. This is really a careful subtraction test: write both numbers one under the other and subtract column by column.

Reversing a two-digit number. If the number is $10a + b$, the reversed number is $10b + a$, so

$$(10a + b) - (10b + a) = 9(a - b) \qquad (10a + b) + (10b + a) = 11(a + b)$$

A number minus its reverse is always a multiple of 9, and a number plus its reverse is always a multiple of 11.

Counting digits. To number the pages of a book from 1 to 150 you need 9 one-digit numbers, 90 two-digit numbers and 51 three-digit numbers, so 9 + 180 + 153 = 342 digits.

Binary Numbers [Tier 2]

Binary uses only the digits 0 and 1, and each place is a power of 2 instead of a power of 10. Reading from the right, the places are 1, 2, 4, 8, 16, 32, 64, …

Binary to decimal: add up the place values wherever there is a 1.

$$101101_2 = 32 + 8 + 4 + 1 = 45$$

Decimal to binary: keep dividing by 2 and write the remainders from the last one to the first. For 13: 13 ÷ 2 = 6 remainder 1, 6 ÷ 2 = 3 remainder 0, 3 ÷ 2 = 1 remainder 1, 1 ÷ 2 = 0 remainder 1. Reading the remainders upwards, 13 = $1101_2$.

Binary questions show up in Tier 2 and in other SSC papers more often than in CGL Tier 1, so a quick check like this is all you need.

Solved Examples

Question 1

Which of these is an integer but not a whole number: 0, 7, −4, 3/2?

Answer. Whole numbers start at 0 and go up, so 0 and 7 are whole numbers. 3/2 is not an integer at all. −4 is an integer, but it is negative, so it can't be a whole number. The answer is −4.

Question 2

Which number belongs to all four sets (whole numbers, integers, rational numbers and real numbers) and is not zero? (a) −2 (b) 0 (c) 9 (d) 2.5

Answer. −2 is not a whole number, 0 is ruled out by the question, and 2.5 is not an integer. 9 is whole, an integer, rational (9/1) and real. The answer is (c) 9.

Question 3

Which of these sets contains only positive numbers? (a) Integers (b) Rational numbers (c) Natural numbers (d) Real numbers

Answer. I look for one negative number in each set. Integers have −1, rational numbers have −1/2, and real numbers have −1 too. Natural numbers start at 1, so every one of them is positive. The answer is (c) Natural numbers.

Question 4

Sort these into rational and irrational: $\sqrt{49}$, $\sqrt{50}$, 22/7, $0.\overline{36}$, $\pi$.

Answer. $\sqrt{49} = 7$, so it is rational. 50 is not a perfect square, so $\sqrt{50} = 5\sqrt{2}$ is irrational. 22/7 is a fraction, so it is rational, even though it is close to $\pi$. $0.\overline{36}$ repeats, so it is rational. $\pi$ is irrational.

So the rational ones are $\sqrt{49}$, 22/7 and $0.\overline{36}$, and the irrational ones are $\sqrt{50}$ and $\pi$.

Question 5

Which statement is true? (a) Every rational number is an integer. (b) Every integer is a rational number. (c) Every real number is rational. (d) Some integers are irrational.

Answer. Integers sit inside the rational numbers, so (b) is true. (a) fails for 1/2, (c) fails for $\sqrt{2}$, and (d) is impossible because no number is both rational and irrational. The answer is (b).

Question 6

Is $7\sqrt{5} \times \sqrt{20}$ rational or irrational?

Answer. I multiply under the root: $7\sqrt{5 \times 20} = 7\sqrt{100} = 7 \times 10 = 70$. That is a whole number, so it is rational. The roots looked scary, but they disappear.

Question 7

A student says, "The sum of two irrational numbers is always irrational." Which pair proves him wrong? (a) $\sqrt{2}$ and $\sqrt{3}$ (b) $3 + \sqrt{2}$ and $3 - \sqrt{2}$ (c) $\pi$ and $\sqrt{2}$ (d) $2\sqrt{3}$ and $\sqrt{3}$

Answer. I need two irrational numbers whose sum comes out rational. In (b), $(3 + \sqrt{2}) + (3 - \sqrt{2}) = 6$, because the $\sqrt{2}$ parts cancel. Both numbers are irrational, but their sum is 6. The answer is (b).

Question 8

Which fraction gives a terminating decimal? (a) 7/12 (b) 9/40 (c) 11/30 (d) 5/14

Answer. All four are already in lowest terms, so I just look at the denominators. 12 has a 3, 30 has a 3 and 14 has a 7, so those three will recur. 40 = $2^3 \times 5$ has only 2s and 5s. So (b) 9/40 terminates: 9/40 = 0.225.

Question 9

Write $0.\overline{45}$ as a fraction in lowest terms.

Answer. Two digits repeat, so I put 45 over 99: $\frac{45}{99}$. Both divide by 9, which gives $\frac{5}{11}$.

Question 10

Write $2.1\overline{6}$ as a fraction.

Answer. I keep the 2 aside and work on $0.1\overline{6}$. The digits after the point are 16 and the part that doesn't repeat is 1, so the top is 16 − 1 = 15. One repeating digit and one non-repeating digit give 90 at the bottom. So $0.1\overline{6} = \frac{15}{90} = \frac{1}{6}$. Adding the 2 back gives $2\frac{1}{6} = \frac{13}{6}$.

Question 11

Find $0.\overline{7} + 0.\overline{5}$.

Answer. $0.\overline{7} = \frac{7}{9}$ and $0.\overline{5} = \frac{5}{9}$. Adding them gives $\frac{12}{9} = \frac{4}{3}$, which is $1.\overline{3}$.

Question 12

How many prime numbers are there between 50 and 80?

Answer. I list them: 53, 59, 61, 67, 71, 73, 79. That makes 7 primes. Numbers like 51 (3 × 17), 57 (3 × 19) and 77 (7 × 11) look prime but are not.

Question 13

Is 221 a prime number?

Answer. $\sqrt{221}$ is just under 15, so I check the primes up to 13. 221 is odd, its digit sum is 5 so 3 doesn't divide it, and it doesn't end in 0 or 5. It is not divisible by 7 or 11, but 221 ÷ 13 = 17. So 221 = 13 × 17, and it is not prime.

Question 14

Which pair is co-prime? (a) 12 and 18 (b) 15 and 28 (c) 21 and 35 (d) 16 and 24

Answer. Co-prime means the only common factor is 1. 12 and 18 share 6, 21 and 35 share 7, and 16 and 24 share 8. 15 = 3 × 5 and 28 = 2 × 2 × 7 share nothing. So (b) 15 and 28 are co-prime, even though neither of them is prime.

Question 15

Find the difference between the place value and the face value of 7 in 4,72,315.

Answer. The 7 is in the ten-thousands place, so its place value is 70,000. Its face value is just 7. The difference is 70,000 − 7 = 69,993.

Question 16

The digits of 8462913 are arranged in ascending order. What is the difference between the original number and the new number?

Answer. The digits 8, 4, 6, 2, 9, 1, 3 in ascending order are 1, 2, 3, 4, 6, 8, 9, which makes 1234689. Now I subtract: 8462913 − 1234689 = 7228224.

Question 17

A two-digit number is 27 more than the number formed by reversing its digits. What is the difference between its digits?

Answer. A number minus its reverse is always 9 × (difference of the digits). So 9 × (difference) = 27, which means the difference is 3.

Question 18 [Tier 2]

Convert the binary number 110011 to decimal.

Answer. From the right, the place values are 1, 2, 4, 8, 16 and 32. The 1s sit at 32, 16, 2 and 1. Adding them: 32 + 16 + 2 + 1 = 51.