Year 8 Maths: Binary Numbers

Binary numbers are a way of representing numbers using only two digits: 0 and 1. This system is also known as base-2, because it uses two digits, unlike our usual decimal (base-10) system, which uses ten digits (0 to 9). Binary numbers are fundamental to computing and digital systems, as computers operate using binary logic.

In the binary system, each place value represents a power of 2, starting from the rightmost digit. For example, the binary number $$1011_2$$ can be understood as:

$$ 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11_{10} $$

So, $$1011_2$$ in binary equals $$11_{10}$$ in decimal.

Key Concepts in Binary Numbers

1. Binary Place Value

In the binary system, each digit (bit) represents a power of 2. The rightmost digit represents $$2^0$$ (1), the next represents $$2^1$$ (2), the next represents $$2^2$$ (4), and so on. For example:

• Binary $$1001_2$$ is:
• $$ 1 \times 2^3 = 8 $$
• $$ 0 \times 2^2 = 0 $$
• $$ 0 \times 2^1 = 0 $$
• $$ 1 \times 2^0 = 1 $$ Therefore, $$1001_2 = 8 + 1 = 9_{10}$$ in decimal.

2. Converting Binary to Decimal

To convert binary numbers to decimal, you multiply each binary digit by its corresponding power of 2 and sum the results.

For example:
$$ 1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13_{10} $$

3. Converting Decimal to Binary

To convert a decimal number to binary, you repeatedly divide the number by 2, keeping track of the remainders. The binary number is formed by the remainders read from bottom to top.

For example:

• Convert 18 to binary:

1. $$ 18 \div 2 = 9 \text{ remainder } 0 $$
2. $$ 9 \div 2 = 4 \text{ remainder } 1 $$
3. $$ 4 \div 2 = 2 \text{ remainder } 0 $$
4. $$ 2 \div 2 = 1 \text{ remainder } 0 $$
5. $$ 1 \div 2 = 0 \text{ remainder } 1 $$ So, 18 in decimal is $$10010_2$$ in binary.

Practice Questions on Binary Numbers

Easy Level

1. Convert $$101_2$$ to decimal.
2. Convert $$11_2$$ to decimal.
3. Convert $$1000_2$$ to decimal.
4. Convert $$110_2$$ to decimal.
5. Convert $$100_2$$ to decimal.
6. Convert $$10_2$$ to decimal.
7. Convert $$111_2$$ to decimal.
8. Convert $$1010_2$$ to decimal.
9. Convert $$1001_2$$ to decimal.
10. Convert $$1100_2$$ to decimal.
11. Convert $$9_{10}$$ to binary.
12. Convert $$3_{10}$$ to binary.
13. Convert $$5_{10}$$ to binary.
14. Convert $$7_{10}$$ to binary.
15. Convert $$8_{10}$$ to binary.
16. Convert $$6_{10}$$ to binary.
17. Convert $$10_{10}$$ to binary.
18. Convert $$4_{10}$$ to binary.
19. Convert $$2_{10}$$ to binary.
20. Convert $$12_{10}$$ to binary.

Medium Level

1. Convert $$1011_2$$ to decimal.
2. Convert $$1111_2$$ to decimal.
3. Convert $$10011_2$$ to decimal.
4. Convert $$11010_2$$ to decimal.
5. Convert $$10110_2$$ to decimal.
6. Convert $$1001_2$$ to decimal.
7. Convert $$11001_2$$ to decimal.
8. Convert $$10101_2$$ to decimal.
9. Convert $$11011_2$$ to decimal.
10. Convert $$11100_2$$ to decimal.
11. Convert $$17_{10}$$ to binary.
12. Convert $$14_{10}$$ to binary.
13. Convert $$20_{10}$$ to binary.
14. Convert $$25_{10}$$ to binary.
15. Convert $$23_{10}$$ to binary.
16. Convert $$30_{10}$$ to binary.
17. Convert $$18_{10}$$ to binary.
18. Convert $$22_{10}$$ to binary.
19. Convert $$19_{10}$$ to binary.
20. Convert $$21_{10}$$ to binary.

Hard Level

1. Convert $$100101_2$$ to decimal.
2. Convert $$110101_2$$ to decimal.
3. Convert $$111001_2$$ to decimal.
4. Convert $$101110_2$$ to decimal.
5. Convert $$110110_2$$ to decimal.
6. Convert $$101101_2$$ to decimal.
7. Convert $$111111_2$$ to decimal.
8. Convert $$100110_2$$ to decimal.
9. Convert $$101001_2$$ to decimal.
10. Convert $$111010_2$$ to decimal.
11. Convert $$42_{10}$$ to binary.
12. Convert $$55_{10}$$ to binary.
13. Convert $$63_{10}$$ to binary.
14. Convert $$47_{10}$$ to binary.
15. Convert $$58_{10}$$ to binary.
16. Convert $$36_{10}$$ to binary.
17. Convert $$52_{10}$$ to binary.
18. Convert $$60_{10}$$ to binary.
19. Convert $$65_{10}$$ to binary.
20. Convert $$75_{10}$$ to binary.

Answers and Explanations

Easy Level

1. $$101_2 = 5_{10}$$

• $$ 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5 $$

1. $$11_2 = 3_{10}$$

• $$ 1 \times 2^1 + 1 \times 2^0 = 2 + 1 = 3 $$

1. $$1000_2 = 8_{10}$$

• $$ 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 8 $$

1. $$110_2 = 6_{10}$$

• $$ 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 = 6 $$

1. $$100_2 = 4_{10}$$

• $$ 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 4 $$

1. $$10_2 = 2_{10}$$

• $$ 1 \times 2^1 + 0 \times 2^0 = 2 $$

1. $$111_2 = 7_{10}$$

• $$ 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 4 + 2 + 1 = 7 $$

1. $$1010_2 = 10_{10}$$

• $$ 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 0 + 2 = 10 $$

1. $$1001_2 = 9_{10}$$

• $$ 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 1 = 9 $$

1. $$1100_2 = 12_{10}$$
   • $$ 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 8 + 4 = 12 $$
2. $$9_{10} = 1001_2$$
3. $$3_{10} = 11_2$$
4. $$5_{10} = 101_2$$
5. $$7_{10} = 111_2$$
6. $$8_{10} = 1000_2$$
7. $$6_{10} =

110_2$$

17. $$10_{10} = 1010_2$$
18. $$4_{10} = 100_2$$
19. $$2_{10} = 10_2$$
20. $$12_{10} = 1100_2$$

Medium Level

1. $$1011_2 = 11_{10}$$
2. $$1111_2 = 15_{10}$$
3. $$10011_2 = 19_{10}$$
4. $$11010_2 = 26_{10}$$
5. $$10110_2 = 22_{10}$$
6. $$1001_2 = 9_{10}$$
7. $$11001_2 = 25_{10}$$
8. $$10101_2 = 21_{10}$$
9. $$11011_2 = 27_{10}$$
10. $$11100_2 = 28_{10}$$
11. $$17_{10} = 10001_2$$
12. $$14_{10} = 1110_2$$
13. $$20_{10} = 10100_2$$
14. $$25_{10} = 11001_2$$
15. $$23_{10} = 10111_2$$
16. $$30_{10} = 11110_2$$
17. $$18_{10} = 10010_2$$
18. $$22_{10} = 10110_2$$
19. $$19_{10} = 10011_2$$
20. $$21_{10} = 10101_2$$

Hard Level

1. $$100101_2 = 37_{10}$$
2. $$110101_2 = 53_{10}$$
3. $$111001_2 = 57_{10}$$
4. $$101110_2 = 46_{10}$$
5. $$110110_2 = 54_{10}$$
6. $$101101_2 = 45_{10}$$
7. $$111111_2 = 63_{10}$$
8. $$100110_2 = 38_{10}$$
9. $$101001_2 = 41_{10}$$
10. $$111010_2 = 58_{10}$$
11. $$42_{10} = 101010_2$$
12. $$55_{10} = 110111_2$$
13. $$63_{10} = 111111_2$$
14. $$47_{10} = 101111_2$$
15. $$58_{10} = 111010_2$$
16. $$36_{10} = 100100_2$$
17. $$52_{10} = 110100_2$$
18. $$60_{10} = 111100_2$$
19. $$65_{10} = 1000001_2$$
20. $$75_{10} = 1001011_2$$

This set of questions and answers provides an understanding of how binary numbers work and how to convert between binary and decimal. It is tailored to the Key Stage 3 level, helping students build their understanding of the binary number system.