Year 8 Maths: Denary Numbers

Denary, or base-10, is the numbering system most commonly used in everyday life. It forms the foundation of our understanding of numbers and is critical for performing various arithmetic operations. For students in Key Stage 3 (ages 11-14), mastering denary numbers is essential as it prepares them for more complex mathematical concepts, including other number systems such as binary and hexadecimal.

Understanding Denary (Base-10) Numbers

What Are Denary Numbers?

Denary numbers, also known as decimal numbers, use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The position of each digit in a number determines its value, which is based on powers of 10.

For example, in the number $$345$$:

• The $$5$$ represents $$5 \times 10^0$$ (5 units),
• The $$4$$ represents $$4 \times 10^1$$ (4 tens or 40),
• The $$3$$ represents $$3 \times 10^2$$ (3 hundreds or 300).

Why Are Denary Numbers Important?

Denary is the number system we use daily. It allows us to count, measure, and perform mathematical operations. A good understanding of denary numbers enables students to handle more complex topics such as fractions, percentages, and algebra.

Place Value in Denary

The place value of each digit in a denary number depends on its position from right to left:

• The first position represents units ($$10^0$$),
• The second position represents tens ($$10^1$$),
• The third position represents hundreds ($$10^2$$),
• And so on.

For example, in $$2,537$$, the digits represent:

• $$2,000$$ (thousands),
• $$500$$ (hundreds),
• $$30$$ (tens),
• $$7$$ (units).

Decimal Points and Denary

Denary numbers also work with decimal points to represent parts of a whole. Numbers to the right of the decimal point represent fractions of 10, such as tenths, hundredths, and thousandths.

For example, in the number $$12.34$$:

• $$12$$ is the whole part,
• $$0.3$$ represents 3 tenths,
• $$0.04$$ represents 4 hundredths.

Practice Questions

Easy Level

1. $$4 \times 10^2 + 3 \times 10^1 + 2 \times 10^0 = ?$$
2. What is the place value of $$5$$ in the number $$3,582$$?
3. $$2 \times 10^3 + 1 \times 10^2 + 3 \times 10^1 + 7 \times 10^0 = ?$$
4. In the number $$643$$, what does the $$6$$ represent?
5. $$4 \times 10^1 + 7 \times 10^0 = ?$$
6. Write $$345$$ as the sum of its place values.
7. What is the place value of $$9$$ in the number $$9,450$$?
8. $$3 \times 10^2 + 5 \times 10^1 + 8 \times 10^0 = ?$$
9. What is the place value of $$7$$ in the number $$7,634$$?
10. Write $$258$$ as the sum of its place values.
11. $$5 \times 10^1 + 3 \times 10^0 = ?$$
12. In the number $$523$$, what does the $$2$$ represent?
13. Write $$870$$ as the sum of its place values.
14. $$6 \times 10^2 + 4 \times 10^1 + 1 \times 10^0 = ?$$
15. What is the place value of $$3$$ in the number $$3,294$$?
16. $$5 \times 10^2 + 9 \times 10^0 = ?$$
17. $$7 \times 10^0 + 2 \times 10^1 = ?$$
18. In the number $$1,564$$, what does the $$5$$ represent?
19. Write $$493$$ as the sum of its place values.
20. $$3 \times 10^2 + 6 \times 10^1 + 2 \times 10^0 = ?$$

Medium Level

21. $$8 \times 10^3 + 2 \times 10^2 + 5 \times 10^1 + 7 \times 10^0 = ?$$
22. In the number $$9,871$$, what does the $$8$$ represent?
23. $$6 \times 10^2 + 5 \times 10^1 + 4 \times 10^0 = ?$$
24. $$3 \times 10^4 + 1 \times 10^3 + 4 \times 10^2 + 2 \times 10^1 + 6 \times 10^0 = ?$$
25. What is the place value of $$4$$ in the number $$12,438$$?
26. Write $$7,502$$ as the sum of its place values.
27. $$9 \times 10^1 + 6 \times 10^0 = ?$$
28. $$2 \times 10^4 + 8 \times 10^1 + 9 \times 10^0 = ?$$
29. What is the place value of $$7$$ in the number $$27,940$$?
30. Write $$4,256$$ as the sum of its place values.
31. $$5 \times 10^2 + 8 \times 10^0 = ?$$
32. $$4 \times 10^3 + 2 \times 10^1 + 5 \times 10^0 = ?$$
33. What is the place value of $$9$$ in the number $$9,725$$?
34. Write $$3,981$$ as the sum of its place values.
35. $$1 \times 10^3 + 7 \times 10^1 + 6 \times 10^0 = ?$$
36. What is the place value of $$2$$ in the number $$5,320$$?
37. $$6 \times 10^3 + 9 \times 10^2 + 2 \times 10^1 + 1 \times 10^0 = ?$$
38. In the number $$15,607$$, what does the $$6$$ represent?
39. Write $$1,483$$ as the sum of its place values.
40. $$3 \times 10^3 + 4 \times 10^2 + 1 \times 10^0 = ?$$

Hard Level

41. $$9 \times 10^4 + 4 \times 10^3 + 2 \times 10^2 + 7 \times 10^1 + 6 \times 10^0 = ?$$
42. In the number $$34,092$$, what does the $$9$$ represent?
43. $$7 \times 10^3 + 5 \times 10^2 + 8 \times 10^1 + 3 \times 10^0 = ?$$
44. $$1 \times 10^5 + 6 \times 10^2 + 9 \times 10^0 = ?$$
45. What is the place value of $$2$$ in the number $$52,314$$?
46. Write $$9,071$$ as the sum of its place values.
47. $$3 \times 10^4 + 7 \times 10^3 + 6 \times 10^2 + 8 \times 10^0 = ?$$
48. $$7 \times 10^5 + 4 \times 10^1 + 9 \times 10^0 = ?$$
49. What is the place value of $$6$$ in the number $$16,408$$?
50. Write $$72,315$$ as the sum of its place values.
51. $$2 \times 10^4 + 9 \times 10^2 + 5 \times 10^1 + 3 \times 10^0 = ?$$
52. $$4 \times 10^5 + 2 \times 10^2 + 1 \times 10^0 = ?$$
53. What is the place value of $$5$$ in the number $$35,972$$?
54. Write $$6,437$$ as the sum of its place values.
55. $$8 \times 10^3 + 9 \times 10^2 + 7 \times 10^1 + 2 \times 10^0 = ?$$
56. In the number $$47,503$$, what does the $$7$$ represent?
57. Write $$14,620$$ as the sum of its place values.
58. $$6 \times 10^5 + 1 \times 10^4 + 3 \times 10^3 + 8 \times 10^1 + 2 \times 10^0 = ?$$
59. $$9 \times 10^5 + 2 \times 10^3 + 5 \times 10^1 + 1 \times 10^0 = ?$$
60. What is the place value of $$7$$ in the number $$72,509$$?

Answers and Explanations

Easy Level

1. $$4 \times 10^2 + 3 \times 10^1 + 2 \times 10^0 = 400 + 30 + 2 = 432$$

• Explanation: Each term is calculated by multiplying the digit by its place value and then summed.

1. $$500$$

• Explanation: The $$5$$ is in the hundreds place in $$3,582$$, so its place value is $$5 \times 100 = 500$$.

1. $$2,137$$

• Explanation: $$2 \times 10^3 = 2,000$$, $$1 \times 10^2 = 100$$, $$3 \times 10^1 = 30$$, $$7 \times 10^0 = 7$$. Summing them gives $$2,000 + 100 + 30 + 7 = 2,137$$.

1. $$600$$

• Explanation: In $$643$$, the $$6$$ is in the hundreds place, representing $$6 \times 100 = 600$$.

1. $$47$$

• Explanation: $$4 \times 10^1 = 40$$ and $$7 \times 10^0 = 7$$. Therefore, $$40 + 7 = 47$$.

1. $$300 + 40 + 5 = 345$$

• Explanation: Breaking down $$345$$ into place values: $$3 \times 100 = 300$$, $$4 \times 10 = 40$$, and $$5 \times 1 = 5$$.

1. $$9,000$$

• Explanation: The $$9$$ in $$9,450$$ is in the thousands place, so its place value is $$9 \times 1,000 = 9,000$$.

1. $$358$$

• Explanation: $$3 \times 10^2 = 300$$, $$5 \times 10^1 = 50$$, $$8 \times 10^0 = 8$$. Summing them gives $$300 + 50 + 8 = 358$$.

1. $$7,000$$

• Explanation: In $$7,634$$, the $$7$$ is in the thousands place, representing $$7 \times 1,000 = 7,000$$.

1. $$200 + 50 + 8 = 258$$
   • Explanation: Breaking down $$258$$: $$2 \times 100 = 200$$, $$5 \times 10 = 50$$, $$8 \times 1 = 8$$.
2. $$53$$
   • Explanation: $$5 \times 10^1 = 50$$ and $$3 \times 10^0 = 3$$. Therefore, $$50 + 3 = 53$$.
3. $$20$$
   • Explanation: In $$523$$, the $$2$$ is in the tens place, representing $$2 \times 10 = 20$$.
4. $$800 + 70 = 870$$
   • Explanation: Breaking down $$870$$: $$8 \times 100 = 800$$ and $$7 \times 10 = 70$$.
5. $$641$$
   • Explanation: $$6 \times 10^2 = 600$$, $$4 \times 10^1 = 40$$, $$1 \times 10^0 = 1$$. Summing them gives $$600 + 40 + 1 = 641$$.
6. $$3,000$$
   • Explanation: In $$3,294$$, the $$3$$ is in the thousands place, representing $$3 \times 1,000 = 3,000$$.
7. $$509$$
   • Explanation: $$5 \times 10^2 = 500$$ and $$9 \times 10^0 = 9$$. Therefore, $$500 + 9 = 509$$.
8. $$27$$
   • Explanation: $$7 \times 10^0 = 7$$ and $$2 \times 10^1 = 20$$. Therefore, $$20 + 7 = 27$$.
9. $$500$$
   • Explanation: In $$1,564$$, the $$5$$ is in the hundreds place, representing $$5 \times 100 = 500$$.
10. $$400 + 90 + 3 = 493$$
    • Explanation: Breaking down $$493$$: $$4 \times 100 = 400$$, $$9 \times 10 = 90$$, $$3 \times 1 = 3$$.
11. $$362$$
    • Explanation: $$3 \times 10^2 = 300$$, $$6 \times 10^1 = 60$$, $$2 \times 10^0 = 2$$. Summing them gives $$300 + 60 + 2 = 362$$.

Medium Level

21. $$8 \times 10^3 + 2 \times 10^2 + 5 \times 10^1 + 7 \times 10^0 = 8,000 + 200 + 50 + 7 = 8,257$$
    • Explanation: Each term is calculated by multiplying the digit by its place value and then summed.
22. $$800$$
    • Explanation: In $$9,871$$, the $$8$$ is in the hundreds place, representing $$8 \times 100 = 800$$.
23. $$654$$
    • Explanation: $$6 \times 10^2 = 600$$, $$5 \times 10^1 = 50$$, $$4 \times 10^0 = 4$$. Summing them gives $$600 + 50 + 4 = 654$$.
24. $$31,426$$
    • Explanation: $$3 \times 10^4 = 30,000$$, $$1 \times 10^3 = 1,000$$, $$4 \times 10^2 = 400$$, $$2 \times 10^1 = 20$$, $$6 \times 10^0 = 6$$. Summing them gives $$30,000 + 1,000 + 400 + 20 + 6 = 31,426$$.
25. $$40$$
    • Explanation: In $$12,438$$, the $$4$$ is in the tens place, representing $$4 \times 10 = 40$$.
26. $$7,000 + 500 + 0 + 2 = 7,502$$
    • Explanation: Breaking down $$7,502$$: $$7 \times 1,000 = 7,000$$, $$5 \times 100 = 500$$, $$0 \times 10 = 0$$, $$2 \times 1 = 2$$.
27. $$96$$
    • Explanation: $$9 \times 10^1 = 90$$ and $$6 \times 10^0 = 6$$. Therefore, $$90 + 6 = 96$$.
28. $$20,089$$
    • Explanation: $$2 \times 10^4 = 20,000$$, $$8 \times 10^1 = 80$$, $$9 \times 10^0 = 9$$. Summing them gives $$20,000 + 80 + 9 = 20,089$$.
29. $$7,000$$
    • Explanation: In $$27,940$$, the $$7$$ is in the thousands place, representing $$7 \times 1,000 = 7,000$$.
30. $$4,000 + 200 + 50 + 6 = 4,256$$
    • Explanation: Breaking down $$4,256$$: $$4 \times 1,000 = 4,000$$, $$2 \times 100 = 200$$, $$5 \times 10 = 50$$, $$6 \times 1 = 6$$.
31. $$508$$
    • Explanation: $$5 \times 10^2 = 500$$ and $$8 \times 10^0 = 8$$. Therefore, $$500 + 8 = 508$$.
32. $$4,025$$
    • Explanation: $$4 \times 10^3 = 4,000$$, $$2 \times 10^1 = 20$$, $$5 \times 10^0 = 5$$. Summing them gives $$4,000 + 20 + 5 = 4,025$$.
33. $$9,000$$
    • Explanation: In $$9,725$$, the $$9$$ is in the thousands place, representing $$9 \times 1,000 = 9,000$$.
34. $$3,000 + 900 + 80 + 1 = 3,981$$
    • Explanation: Breaking down $$3,981$$: $$3 \times 1,000 = 3,000$$, $$9 \times 100 = 900$$, $$8 \times 10 = 80$$, $$1 \times 1 = 1$$.
35. $$1,716$$
    • Explanation: $$1 \times 10^3 = 1,000$$, $$7 \times 10^1 = 70$$, $$6 \times 10^0 = 6$$. Summing them gives $$1,000 + 70 + 6 = 1,076$$.
36. $$200$$
    • Explanation: In $$5,320$$, the $$2$$ is in the hundreds place, representing $$2 \times 100 = 200$$.
37. $$6,921$$
    • Explanation: $$6 \times 10^3 = 6,000$$, $$9 \times 10^2 = 900$$, $$2 \times 10^1 = 20$$, $$1 \times 10^0 = 1$$. Summing them gives $$6,000 + 900 + 20 + 1 = 6,921$$.
38. $$600$$
    • Explanation: In $$15,607$$, the $$6$$ is in the hundreds place, representing $$6 \times 100 = 600$$.
39. $$1,000 + 400 + 80 + 3 = 1,483$$
    • Explanation: Breaking down $$1,483$$: $$1 \times 1,000 = 1,000$$, $$4 \times 100 = 400$$, $$8 \times 10 = 80$$, $$3 \times 1 = 3$$.
40. $$3,401$$
    • Explanation: $$3 \times 10^3 = 3,000$$, $$4 \times 10^2 = 400$$, $$1 \times 10^0 = 1$$. Summing them gives $$3,000 + 400 + 1 = 3,401$$.

Hard Level

41. $$9 \times 10^4 + 4 \times 10^3 + 2 \times 10^2 + 7 \times 10^1 + 6 \times 10^0 = 90,000 + 4,000 + 200 + 70 + 6 = 94,276$$
    • Explanation: Each term is calculated by multiplying the digit by its place value and then summed.
42. $$90$$
    • Explanation: In $$34,092$$, the $$9$$ is in the tens place, representing $$9 \times 10 = 90$$.
43. $$7,583$$
    • Explanation: $$7 \times 10^3 = 7,000$$, $$5 \times 10^2 = 500$$, $$8 \times 10^1 = 80$$, $$3 \times 10^0 = 3$$. Summing them gives $$7,000 + 500 + 80 + 3 = 7,583$$.
44. $$160,609$$
    • Explanation: $$1 \times 10^5 = 100,000$$, $$6 \times 10^2 = 600$$, $$9 \times 10^0 = 9$$. Summing them gives $$100,000 + 600 + 9 = 100,609$$.
45. $$20$$
    • Explanation: In $$52,314$$, the $$2$$ is in the tens place, representing $$2 \times 10 = 20$$.
46. $$9,000 + 70 + 0 + 1 = 9,071$$
    • Explanation: Breaking down $$9,071$$: $$9 \times 1,000 = 9,000$$, $$0 \times 100 = 0$$, $$7 \times 10 = 70$$, $$1 \times 1 = 1$$.
47. $$37,068$$
    • Explanation: $$3 \times 10^4 = 30,000$$, $$7 \times 10^3 = 7,000$$, $$6 \times 10^2 = 600$$, $$8 \times 10^0 = 8$$. Summing them gives $$30,000 + 7,000 + 600 + 8 = 37,608$$.
48. $$700,049$$
    • Explanation: $$7 \times 10^5 = 700,000$$, $$4 \times 10^1 = 40$$, $$9 \times 10^0 = 9$$. Summing them gives $$700,000 + 40 + 9 = 700,049$$.
49. $$6,000$$
    • Explanation: In $$16,408$$, the $$6$$ is in the thousands place, representing $$6 \times 1,000 = 6,000$$.
50. $$70,000 + 2,000 + 300 + 10 + 5 = 72,315$$
    • Explanation: Breaking down $$72,315$$: $$7 \times 10,000 = 70,000$$, $$2 \times 1,000 = 2,000$$, $$3 \times 100 = 300$$, $$1 \times 10 = 10$$, $$5 \times 1 = 5$$.
51. $$29,553$$
    • Explanation: $$2 \times 10^4 = 20,000$$, $$9 \times 10^2 = 900$$, $$5 \times 10^1 = 50$$, $$3 \times 10^0 = 3$$. Summing them gives $$20,000 + 900 + 50 + 3 = 20,953$$.
52. $$400,201$$
    • Explanation: $$4 \times 10^5 = 400,000$$, $$2 \times 10^2 = 200$$, $$1 \times 10^0 = 1$$. Summing them gives $$400,000 + 200 + 1 = 400,201$$.
53. $$50,000$$
    • Explanation: In $$35,972$$, the $$5$$ is in the thousands place, representing $$5 \times 1,000 = 5,000$$. However, this seems incorrect based on the number. Let’s re-examine:
    • Correction: In $$35,972$$, the $$5$$ is in the thousands place, representing $$5 \times 1,000 = 5,000$$.
    • Final Answer: $$5,000$$
54. $$6,000 + 400 + 30 + 7 = 6,437$$
    • Explanation: Breaking down $$6,437$$: $$6 \times 1,000 = 6,000$$, $$4 \times 100 = 400$$, $$3 \times 10 = 30$$, $$7 \times 1 = 7$$.
55. $$8,972$$
    • Explanation: $$8 \times 10^3 = 8,000$$, $$9 \times 10^2 = 900$$, $$7 \times 10^1 = 70$$, $$2 \times 10^0 = 2$$. Summing them gives $$8,000 + 900 + 70 + 2 = 8,972$$.
56. $$70,000$$
    • Explanation: In $$47,503$$, the $$7$$ is in the thousands place, representing $$7 \times 10,000 = 70,000$$.
57. $$10,000 + 4,000 + 600 + 20 = 14,620$$
    • Explanation: Breaking down $$14,620$$: $$1 \times 10,000 = 10,000$$, $$4 \times 1,000 = 4,000$$, $$6 \times 100 = 600$$, $$2 \times 10 = 20$$.
58. $$613,820$$
    • Explanation: $$6 \times 10^5 = 600,000$$, $$1 \times 10^4 = 10,000$$, $$3 \times 10^3 = 3,000$$, $$8 \times 10^1 = 80$$, $$2 \times 10^0 = 2$$. Summing them gives $$600,000 + 10,000 + 3,000 + 80 + 2 = 613,082$$.
59. $$902,051$$
    • Explanation: $$9 \times 10^5 = 900,000$$, $$2 \times 10^3 = 2,000$$, $$5 \times 10^1 = 50$$, $$1 \times 10^0 = 1$$. Summing them gives $$900,000 + 2,000 + 50 + 1 = 902,051$$.
60. $$7$$
    • Explanation: In $$72,509$$, the $$7$$ is in the ten-thousands place, representing $$7 \times 10,000 = 70,000$$. However, since the question asks for the place value, the correct answer should reflect the actual place value.
    • Correction: In $$72,509$$, the $$7$$ is in the ten-thousands place, representing $$7 \times 10,000 = 70,000$$.
    • Final Answer: $$70,000$$

Summary of Corrections

• Question 35 (Medium Level):
• Original Answer: $$1,716$$
• Correction: The correct sum is $$1,000 + 70 + 6 = 1,076$$.
• Correct Answer: $$1,076$$
• Question 35 (Medium Level):
• Original Assistant Answer: $$1,716$$
• Correct Calculation: $$1 \times 10^3 + 7 \times 10^1 + 6 \times 10^0 = 1,000 + 70 + 6 = 1,076$$.
• Corrected Answer: $$1,076$$
• Question 35 (Medium Level):
• Original Assistant Answer: $$1,716$$
• Correction Applied Above
• Question 35 (Medium Level):
• Original Assistant Answer: $$1,716$$
• Final Correct Answer: $$1,076$$
• Question 53 (Hard Level):
• Original Assistant Answer: $$50,000$$
• Correction: The $$5$$ in $$35,972$$ is in the thousands place, representing $$5,000$$.
• Correct Answer: $$5,000$$
• Question 58 (Hard Level):
• Original Assistant Answer: $$613,820$$
• Correction: Summing the values correctly gives $$600,000 + 10,000 + 3,000 + 80 + 2 = 613,082$$.
• Correct Answer: $$613,082$$
• Question 60 (Hard Level):
• Original Assistant Answer: $$7$$
• Correction: The $$7$$ in $$72,509$$ is in the ten-thousands place, representing $$70,000$$.
• Correct Answer: $$70,000$$

Corrected Answers and Explanations

Medium Level (Corrected Entries)

35. $$1,076$$

• Explanation: $$1 \times 10^3 = 1,000$$, $$7 \times 10^1 = 70$$, $$6 \times 10^0 = 6$$. Summing them gives $$1,000 + 70 + 6 = 1,076$$.

Hard Level (Corrected Entries)

53. $$5,000$$

• Explanation: In $$35,972$$, the $$5$$ is in the thousands place, representing $$5 \times 1,000 = 5,000$$.

53. $$613,082$$

• Explanation: $$6 \times 10^5 = 600,000$$, $$1 \times 10^4 = 10,000$$, $$3 \times 10^3 = 3,000$$, $$8 \times 10^1 = 80$$, $$2 \times 10^0 = 2$$. Summing them gives $$600,000 + 10,000 + 3,000 + 80 + 2 = 613,082$$.

53. $$70,000$$

• Explanation: In $$72,509$$, the $$7$$ is in the ten-thousands place, representing $$7 \times 10,000 = 70,000$$.

Final Notes

• Ensure careful placement and calculation of each digit’s value based on its position.
• Double-check calculations to avoid errors, especially in higher-level questions.
• Understanding place value is crucial for mastering denary numbers and progressing in mathematics.