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Square roots & Cube roots

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visibility 250update 7 months agobookmarkshare

🎯 In this topic you will

  • Understand how square and cube numbers relate to square roots and cube roots
  • Find the squares of positive and negative integers and their corresponding square roots
  • Find the cubes of positive and negative integers and their corresponding cube roots
  • Recognise natural numbers, integers, and rational numbers
 

🧠 Key Words

  • cube number
  • cube root
  • consecutive
  • equivalent
  • index
  • natural numbers
  • rational numbers
  • square number
  • square root
Show Definitions
  • cube number: A number made by multiplying a whole number by itself twice (e.g., 2 × 2 × 2 = 8).
  • cube root: A number that when multiplied by itself twice gives the original number (e.g., ∛8 = 2).
  • consecutive: Numbers that follow one after another in order without gaps.
  • equivalent: Having the same value or amount, even if they look different (e.g., 1/2 = 2/4).
  • index: A number that shows how many times a base number is multiplied by itself (also called an exponent).
  • natural numbers: The counting numbers starting from 1, 2, 3, and so on.
  • rational numbers: Numbers that can be written as a fraction or ratio of two integers.
  • square number: A number made by multiplying a whole number by itself (e.g., 3 × 3 = 9).
  • square root: A number that when multiplied by itself gives the original number (e.g., √9 = 3).
 

🟣 Square numbers and square roots

1 × 1 = 1  2 × 2 = 4  3 × 3 = 9  4 × 4 = 16  5 × 5 = 25

The square numbers are 1, 4, 9, 16, 25, …

You use an index of 2 to show square numbers.

12 = 1  22 = 4  32 = 9  42 = 16  52 = 25

You read 12 as ‘1 squared’ and you read 22 as ‘2 squared’.

42 = 16 is equivalent to 4 = √16, which is read as ‘4 is the square root of 16’.

The symbol for square root is √.

 
📘 Worked example

Work out $\sqrt{100} - \sqrt{81}$.

Answer

$10^2 = 10 \times 10 = 100$ and $9^2 = 81$

So $\sqrt{100} = 10$ and $\sqrt{81} = 9$

$\sqrt{100} - \sqrt{81} = 10 - 9 = 1$

Recognize perfect squares: $100 = 10^2$ and $81 = 9^2$.

Find the square roots, then subtract: $\sqrt{100} - \sqrt{81} = 10 - 9 = 1$.

 

🔎 Reasoning Tip

Cube root symbol: The symbol for cube root is ∛.

 

🟠 Cube numbers and cube roots

1 × 1 × 1 = 1  2 × 2 × 2 = 8  3 × 3 × 3 = 27

The cube numbers are 1, 8, 27, …

You use an index of 3 and write 13 = 1, 23 = 8, 33 = 27, …

You read 13 as ‘1 cubed’ and you read 23 as ‘2 cubed’.

You say ‘2 is the cube root of 8’, which is written as 2 = ∛8.

 
📘 Worked example

Work out $\sqrt{64} \div \sqrt[3]{64}$.

Answer

$8^2 = 64$ and so $\sqrt{64} = 8$

$4^3 = 64$ and so $\sqrt[3]{64} = 4$

Hence, $\sqrt{64} \div \sqrt[3]{64} = 8 \div 4 = 2$

Identify the square root and cube root of 64: $\sqrt{64} = 8$, $\sqrt[3]{64} = 4$.

Then divide: $8 \div 4 = 2$.

 

🧠 Estimating square roots

You can estimate the square roots of integers that are not square numbers.

 
📘 Worked example

a) Show that 9 is the closest integer to $\sqrt{79}$.
b) Show that $\sqrt{215}$ is between 14 and 15.

Answer

a) $9^2 = 81$ and $8^2 = 64$
79 is between 64 and 81, so $\sqrt{79}$ is between 8 and 9.
79 is much closer to 81 than to 64, so 9 is the closest integer to $\sqrt{79}$.

b) $14^2 = 196$ and $15^2 = 225$
215 is between 196 and 225, so $\sqrt{215}$ is between 14 and 15.

To estimate a square root, find two perfect squares that the number lies between. Then determine which integer the square root is closest to based on proximity.

For example, $\sqrt{79}$ lies between 8 and 9, and since 79 is closer to $81 = 9^2$, the closest integer is 9.

 

❓ EXERCISES

1a. Copy and complete: $3^2 =$

👀 Show answer
$3^2 = 3 \times 3 = \boxed{9}$

1b. Copy and complete: $5^2 =$

👀 Show answer
$5^2 = 5 \times 5 = \boxed{25}$

1c. Copy and complete: $8^2 =$

👀 Show answer
$8^2 = 8 \times 8 = \boxed{64}$

1d. Copy and complete: $10^2 =$

👀 Show answer
$10^2 = 10 \times 10 = \boxed{100}$

1e. Copy and complete: $15^2 =$

👀 Show answer
$15^2 = 15 \times 15 = \boxed{225}$

2. An equivalent statement to $7^2 = 49$ is $\sqrt{49} = 7$. Write equivalent statements to your answers to Question 1.

👀 Show answer
$\sqrt{9} = 3$
$\sqrt{25} = 5$
$\sqrt{64} = 8$
$\sqrt{100} = 10$
$\sqrt{225} = 15$

3a. Find $\sqrt{36}$

👀 Show answer
$\sqrt{36} = \boxed{6}$

3b. Find $\sqrt{81}$

👀 Show answer
$\sqrt{81} = \boxed{9}$

3c. Find $\sqrt{121}$

👀 Show answer
$\sqrt{121} = \boxed{11}$

3d. Find $\sqrt{144}$

👀 Show answer
$\sqrt{144} = \boxed{12}$

4a. $1^3 =$

👀 Show answer
$1^3 = 1 \times 1 \times 1 = \boxed{1}$

4b. $2^3 =$

👀 Show answer
$2^3 = 2 \times 2 \times 2 = \boxed{8}$

4c. $3^3 =$

👀 Show answer
$3^3 = 3 \times 3 \times 3 = \boxed{27}$

4d. $4^3 =$

👀 Show answer
$4^3 = 4 \times 4 \times 4 = \boxed{64}$

4e. $5^3 =$

👀 Show answer
$5^3 = 5 \times 5 \times 5 = \boxed{125}$

5. Write equivalent statements to your answers to Question 4, using cube root notation.

👀 Show answer
$\sqrt[3]{1} = 1$
$\sqrt[3]{8} = 2$
$\sqrt[3]{27} = 3$
$\sqrt[3]{64} = 4$
$\sqrt[3]{125} = 5$

6a. Work out the integer closest to $\sqrt{15}$

👀 Show answer
$\sqrt{15} \approx 3.87 \Rightarrow$ closest integer = 4

6b. Work out the integer closest to $\sqrt{66}$

👀 Show answer
$\sqrt{66} \approx 8.12 \Rightarrow$ closest integer = 8

6c. Work out the integer closest to $\sqrt{150}$

👀 Show answer
$\sqrt{150} \approx 12.25 \Rightarrow$ closest integer = 12

7a. Show that $\sqrt{90}$ is between 9 and 10.

👀 Show answer
$9^2 = 81$, $10^2 = 100$ → $81 < 90 < 100$
Therefore, $\sqrt{90}$ is between 9 and 10.

7b. Find two consecutive integers to complete: $\sqrt{180}$ is between … and …

👀 Show answer
$13^2 = 169$, $14^2 = 196$ → $169 < 180 < 196$
So $\sqrt{180}$ is between 13 and 14.

7c. Find two consecutive integers to complete: $\sqrt[3]{90}$ is between … and …

👀 Show answer
$4^3 = 64$, $5^3 = 125$ → $64 < 90 < 125$
So $\sqrt[3]{90}$ is between 4 and 5.

8a. Use a calculator to find $17^2$.

👀 Show answer
$17^2 = 289$

8b. Complete this statement: $\sqrt{\,\boxed{\phantom{0}}\,} = 17$

👀 Show answer
$\sqrt{289} = 17$

9a. $\sqrt[3]{\boxed{\phantom{0}}} = 18$

👀 Show answer
$18^3 = 5832$

9b. $\sqrt[3]{\boxed{\phantom{0}}} = 20$

👀 Show answer
$20^3 = 8000$

9c. $\sqrt[3]{\boxed{\phantom{0}}} = 23$

👀 Show answer
$23^3 = 12167$

9d. $\sqrt[3]{\boxed{\phantom{0}}} = 26$

👀 Show answer
$26^3 = 17576$

10a. $\sqrt[3]{\boxed{\phantom{0}}} = 7$

👀 Show answer
$7^3 = 343$

10b. $\sqrt[3]{\boxed{\phantom{0}}} = 9$

👀 Show answer
$9^3 = 729$

10c. $\sqrt[3]{\boxed{\phantom{0}}} = 10$

👀 Show answer
$10^3 = 1000$

10d. $\sqrt[3]{\boxed{\phantom{0}}} = 12$

👀 Show answer
$12^3 = 1728$

11a. Show that 36 has nine factors.

👀 Show answer
The factors of 36 are:
1, 2, 3, 4, 6, 9, 12, 18, 36 → 9 factors

11b(i). Find the factors of 9.

👀 Show answer
Factors of 9: 1, 3, 9

11b(ii). Find the factors of 16.

👀 Show answer
Factors of 16: 1, 2, 4, 8, 16

11b(iii). Find the factors of 25.

👀 Show answer
Factors of 25: 1, 5, 25

11c. Explain why every square number has an odd number of factors.

👀 Show answer
Factors usually come in pairs (e.g., 2 × 6 = 12). But a square number has one repeated factor (like $6 \times 6 = 36$), which is only counted once — resulting in an odd total number of factors.

11d. Find a number that is not square and has an odd number of factors.

👀 Show answer
No such number exists.
Only perfect squares have an odd number of factors.

11e. Does every cube number have an odd number of factors? Give a reason.

👀 Show answer
No. Only perfect squares have an odd number of factors because only they have a repeated factor in the center (like 6 × 6 = 36). Cube numbers do not necessarily repeat a factor in the same way.

11f. Investigate how many factors different square numbers have.

👀 Show answer
  • $4$ has 3 factors: 1, 2, 4
  • $9$ has 3 factors: 1, 3, 9
  • $16$ has 5 factors: 1, 2, 4, 8, 16
  • $25$ has 3 factors: 1, 5, 25
  • $36$ has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
All have odd numbers of factors.
 

🧠 Think like a Mathematician

Question: What patterns can you find in the differences between consecutive square numbers and consecutive cube numbers?

Equipment: Pencil, paper, calculator (optional)

  1. Note: $1^2 = 1$ and $2^2 = 4$, so the difference is $4 - 1 = 3$.
  2. a) Copy and complete this diagram to find the differences between consecutive square numbers:

Square numbers:$1^2$, $2^2$, $3^2$, $4^2$, $5^2$, $6^2$

Differences:3, 5, 7, 9, 11

  1. b) Describe any pattern in your answers.
  2. c) Investigate the differences between consecutive cube numbers:

Cube numbers:$1^3$, $2^3$, $3^3$, $4^3$, $5^3$, $6^3$

Differences:7, 19, 37, 61, 91

Follow-up Questions:

1. What do you notice about the differences between square numbers?
2. What do you notice about the differences between cube numbers?
3. Can you describe the pattern or rule for each sequence?
👀 show answer
  • 1: The differences between consecutive square numbers increase by 2 each time: $3, 5, 7, 9, 11...$
  • 2: The differences between consecutive cube numbers increase in a quadratic pattern: $7, 19, 37, 61, 91...$
  • 3: The difference between consecutive squares is always an odd number. For cubes, the differences follow a pattern that increases more rapidly — the second differences are constant (i.e., the differences between the differences are equal).
 

❓ EXERCISES

12a(i). Work out: $\sqrt{1^3}$

👀 Show answer
$1^3 = 1$, so $\sqrt{1} = \boxed{1}$

12a(ii). Work out: $\sqrt{1^3 + 2^3}$

👀 Show answer
$1^3 + 2^3 = 1 + 8 = 9$, so $\sqrt{9} = \boxed{3}$

12a(iii). Work out: $\sqrt{1^3 + 2^3 + 3^3}$

👀 Show answer
$1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36$, so $\sqrt{36} = \boxed{6}$

12b. What do you notice about your answers to part a?

👀 Show answer
The square root of the sum of the cubes gives the sum of the base numbers:
$\sqrt{1^3 + 2^3} = 3$, and $1 + 2 = 3$.
$\sqrt{1^3 + 2^3 + 3^3} = 6$, and $1 + 2 + 3 = 6$.

12c. Does the pattern continue with more cubes? Explain.

👀 Show answer
Yes. The sum of the cubes of the first $n$ natural numbers is a square:
$1^3 + 2^3 + 3^3 + \dots + n^3 = \left( \dfrac{n(n+1)}{2} \right)^2$

13a. Add the first three odd numbers and find the square root of the answer.

👀 Show answer
$1 + 3 + 5 = 9$, and $\sqrt{9} = \boxed{3}$

13b. Add the first four odd numbers and find the square root of the answer.

👀 Show answer
$1 + 3 + 5 + 7 = 16$, and $\sqrt{16} = \boxed{4}$

13c. Can you generalise the results of parts a and b?

👀 Show answer
Yes. The sum of the first $n$ odd numbers is always $n^2$.

13d. Look at this diagram. How is it connected to the earlier parts?

Dot diagram showing growing square pattern using odd numbers

👀 Show answer
Each layer adds an odd number of dots, growing a square.
This shows that adding consecutive odd numbers builds perfect squares — just like in parts a–c.
 

⚠️ Be careful!

Don’t confuse $\sqrt{64}$ with $\sqrt[3]{64}$. The square root of 64 is $8$, but the cube root of 64 is $4$ — they are different operations with different results!

 

🍬 Learning Bridge

Now that you've explored square and cube numbers — and learned how to estimate their roots — you're ready to go deeper. You'll investigate what happens when roots involve negative numbers, discover why square roots can have two answers, and learn how these ideas help solve real equations.

 

🔁 Positive and negative roots

52 = 25

This means that the square root of 25 is 5. This can be written as $\sqrt{25} = 5$.

This is the only answer in the set of natural numbers.

However (–5)2 = –5 × –5 = 25

This means that the integer –5 is also a square root of 25.

Every positive integer has two square roots, one positive and one negative.

5 is the positive square root of 25 and –5 is the negative square root.

No negative number has a square root.

For example, the integer –25 has no square root because the equation $x^2 = -25$ has no solution.

53 = 125

This means that the cube root of 125 is 5. This can be written as $\sqrt[3]{125} = 5$.

You might think –5 is also a cube root of 125.

However (–5)3 = –5 × –5 × –5 = (–5 × –5) × –5 = 25 × –5 = –125

So $\sqrt[3]{-125} = -5$

Every number, positive or negative or zero, has only one cube root.

 

🔎 Reasoning Tip

Natural numbers: These are the counting numbers and zero.

 
📘 Worked example 1.3

Solve each equation.

a) $x^2 = 64$
b) $x^3 = 64$
c) $x^3 + 64 = 0$

Answer

a) 64 has two square roots. One is $\sqrt{64} = 8$ and the other is $-\sqrt{64} = -8$
So the equation has two solutions: $x = 8$ or $x = -8$

b) $\sqrt[3]{64} = 4$. This means $4^3 = 4 \times 4 \times 4 = 64$, and so $x = 4$

c) If $x^3 + 64 = 0$, then $x^3 = -64$. So $x = \sqrt[3]{-64} = -4$

Square roots have two answers (positive and negative), but cube roots only have one real answer.

To solve an equation, isolate the variable, then take the appropriate root.

 

❓ EXERCISES

14. Work out
a) $7^2$
b) $(-7)^2$
c) $7^3$
d) $(-7)^3$

👀 Show answer
a) 49
b) 49
c) 343
d) –343

15. Find
a) $\sqrt[3]{125}$
b) $\sqrt[3]{-27}$
c) $\sqrt[3]{1}$
d) $\sqrt[3]{8}$

👀 Show answer
a) 5
b) –3
c) 1
d) 2

16. Solve these equations:
a) $x^2 = 100$
b) $x^2 = 144$
c) $x^2 = 1$
d) $x^2 = 0$
e) $x^2 + 9 = 0$

👀 Show answer
a) $x = ±10$
b) $x = ±12$
c) $x = ±1$
d) $x = 0$
e) No real solution (complex roots)

17. Solve these equations:
a) $x^3 = 216$
b) $x^3 + 27 = 0$
c) $x^3 + 1 = 0$
d) $x^3 + 125 = 0$

👀 Show answer
a) $x = 6$
b) $x = -3$
c) $x = -1$
d) $x = -5$

18. Use the fact $27^2 = 9^3 = 729$ to find:
a) $\sqrt{729}$
b) $\sqrt{-729}$
c) $\sqrt[3]{729}$
d) $\sqrt[3]{-729}$

👀 Show answer
a) 27
b) Not a real number
c) 9
d) –9

19. a) A calculator shows that $\sqrt{82} - \sqrt{(-8)^2} = 0$.
Explain why this is correct.
b) Find the value of $\sqrt[3]{4^3} - \sqrt[3]{(-4)^3}$. Show your working.

👀 Show answer
a) $\sqrt{82} = \sqrt{64 + 18}$ and $(-8)^2 = 64$
So both values are approximately the same.
b) $4^3 = 64$ and $(-4)^3 = -64$
So $\sqrt[3]{64} - \sqrt[3]{-64} = 4 - (-4) = 8$

20. The square of an integer is 100.
What can you say about the cube of the integer?

👀 Show answer
The integer is either 10 or –10, so the cube is either 1000 or –1000.

21. The integer 1521 = $3^2 \times 13^2$
Use this fact to:
a) Find $\sqrt{1521}$
b) Solve the equation $x^2 = 1521$

👀 Show answer
a) $\sqrt{1521} = 3 \times 13 = 39$
b) $x = ±39$

22. a) How is $-5^2$ different from $(-5)^2$?
b) What is the difference between $-5^3$ and $(-5)^3$?

👀 Show answer
a) $-5^2 = -(5^2) = -25$, while $(-5)^2 = 25$
b) $-5^3 = -(5^3) = -125$, while $(-5)^3 = -125$ (no difference)

23. a) Show that $3^2 + 4^2 = 5^2$

b) Are these statements true or false? Give a reason for your answer each time:
 i) $(-3)^2 + (-4)^2 = (-5)^2$
 ii) $(-13)^2 = 12^2 + (-5)^2$
 iii) $8^2 = -10^2 - 6^2$

c) Show your work to a partner. Do they find your explanation clear?

👀 Show answer
a) $3^2 = 9$, $4^2 = 16$, so $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✅

b) i) $(-3)^2 + (-4)^2 = 9 + 16 = 25$, and $(-5)^2 = 25$ → True ✅
ii) $(-13)^2 = 169$, $12^2 = 144$, $(-5)^2 = 25$, $144 + 25 = 169$ → True ✅
iii) $8^2 = 64$, $-10^2 = -100$ (⚠️ actually equals $-100$ if negative outside the square), $6^2 = 36$, so RHS = $-100 - 36 = -136$ → False ❌

c) Answers may vary; students should explain each step clearly.
 

🧠 Think like a Mathematician

Question: How can we find two solutions to quadratic equations like $x^2 + x = n$? What patterns do you notice?

Equipment: Pencil, paper, calculator (optional)

Method:

  1. a) Here is an equation: $x^2 + x = 6$
    1. Show that $x = 2$ is a solution of the equation.
    2. Show that $x = -3$ is a solution of the equation.
  2. b) Here is another equation: $x^2 + x = 12$
    1. Show that $x = 3$ is a solution of the equation.
    2. Find a second solution to the equation.
  3. c) Find two solutions to this equation: $x^2 + x = 20$
  4. d) What patterns can you see in the answers to a, b and c?
    Find some more equations like this and write down the solutions.
  5. e) Compare your answers with a partner’s.

Follow-up Questions:

1. How can you check if a value is a solution to an equation?
2. Why do these equations seem to have two solutions?
3. What pattern do the solutions follow?
👀 show answer
  • a.i:$2^2 + 2 = 4 + 2 = 6$ ✅
  • a.ii:$(-3)^2 + (-3) = 9 - 3 = 6$ ✅
  • b.i:$3^2 + 3 = 9 + 3 = 12$ ✅
  • b.ii: Second solution is $x = -4$ because $(-4)^2 + (-4) = 16 - 4 = 12$
  • c: Solutions are $x = 4$ and $x = -5$ (since $4^2 + 4 = 20$ and $(-5)^2 + (-5) = 25 - 5 = 20$)
  • Pattern: The solutions are symmetrical: one positive, one negative, and they differ by 1 in absolute value.
 

❓ EXERCISES

24. a) Copy and complete this table.

$x$ $x - 1$ $x^3 - 1$ $x^2 + x + 1$
2 1 7  
3 2   13
4      
5      

b) What pattern can you see in your answers?
c) Add another row to see if the pattern is still the same.
d) Add three rows where $x$ is a negative integer.
  Is the pattern still the same if $x$ is a negative integer?

👀 Show answer
Completed table:
$x$ $x - 1$ $x^3 - 1$ $x^2 + x + 1$
2 1 7 7
3 2 26 13
4 3 63 21
5 4 124 31

For $x = 2$: $x^2 + x + 1 = 2^2 + 2 + 1 = 4 + 2 + 1 = 7$ ✅
For $x = 3$: $x^3 - 1 = 27 - 1 = 26$ ✅
For $x = 4$: $x - 1 = 3$, $x^3 - 1 = 64 - 1 = 63$, $x^2 + x + 1 = 16 + 4 + 1 = 21$
For $x = 5$: $x - 1 = 4$, $x^3 - 1 = 125 - 1 = 124$, $x^2 + x + 1 = 25 + 5 + 1 = 31$

Pattern: It appears that $x - 1$, $x^3 - 1$, and $x^2 + x + 1$ follow regular patterns that grow with $x$. Try $x = -1, -2, -3$ to test for negatives and observe whether symmetry or pattern holds.

25. Any number that can be written as a fraction is a rational number.
Examples are: $7\frac{3}{4}, -12, \frac{18}{25}, 6, \frac{1}{15}, -2\frac{9}{10}$

Here is a list of six numbers:
$5,\quad -\frac{1}{5},\quad -500,\quad 16,\quad -4.8,\quad 99\frac{1}{2}$
 

🔎 Reasoning Tip

Rational numbers: Integers and fractions are both part of the set of rational numbers.

Write:
a) all the integers in the list
b) all the natural numbers in the list
c) all the rational numbers in the list
👀 Show answer
a) Integers: 5, –500, 16
b) Natural numbers: 5, 16
c) Rational numbers: all of them — $5,\ -\frac{1}{5},\ -500,\ 16,\ -4.8,\ 99\frac{1}{2}$

26. This Venn diagram shows the relationship between natural numbers and integers.
N stands for natural numbers and I for integers.
 

🔎 Reasoning Tip

Integers and rational numbers: Remember, all integers are part of the rational number set.

a) Copy the Venn diagram.

 

Venn diagram showing circles labeled N and I

b) Write each of these numbers in the correct part of the diagram:
$1,\ -3,\ 7,\ -12,\ 41,\ -100,\ 2\frac{1}{2}$

c) Add another circle to your Venn diagram to show rational numbers.
d) Add these numbers to your Venn diagram:
$-8\frac{3}{7},\quad \frac{3}{5},\quad 0,\quad 6.3,\quad -\frac{10}{3}$

👀 Show answer
b) - Natural numbers (N): 1, 7, 41
- Integers (I but not N): –3, –12, –100
- Outside both (not integers): $2\frac{1}{2}$

d) - Rational numbers: all can be placed in the rational circle - Integers: 0
- Rational but not integer: $-8\frac{3}{7},\ \frac{3}{5},\ 6.3,\ -\frac{10}{3}$
 

📘 What we've learned

  • We learned how to calculate square numbers using $n^2$ and cube numbers using $n^3$, including for negative values.
  • We used square root $\sqrt{}$ and cube root $\sqrt[3]{}$ symbols to find roots of perfect squares and cubes.
  • We saw that every positive square number has two roots (positive and negative), while every cube number has exactly one real root.
  • We estimated square roots by comparing to the nearest perfect squares (e.g. $\sqrt{79}$ is closest to 9).
  • We solved equations involving squares and cubes by isolating the variable and applying the correct root.
  • We explored the differences between consecutive squares and cubes, discovering consistent patterns.
  • We evaluated and compared expressions involving squares and cubes, including those with negative signs.
  • We examined special cases like $(-5)^2$ vs -5² and confirmed that signs matter in powers.
  • We reviewed number sets: natural numbers, integers, and rational numbers, and sorted examples into Venn diagrams.
  • We reinforced that all integers and fractions are rational numbers, but only whole numbers starting from 0 or 1 are natural numbers.

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