Subject: Compulsory Maths

A Binary Number is made up of only 0s and 1s. A method of representing numbers that has 2 as its base and uses only the digits 0 and 1. Each successive digit represents a power of 2. For example, 10011 represents (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (1 × 20), or 16 + 0 + 0 + 2 + 1, or 19.

It is necessary to review the decimal number system at first to understand more about binary number system. Decimal number system refers to base 10 positional notation. It uses ten different symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) arranged using positional notation. Positional notation is used when a number larger then 9 needs to be represented; each position of a digit signifies how many groups of 10, 100, 1000, etc. are contained in that number. For example,

4251

4 \(\times\) 1000 + 2 \(\times\) 100 + 5 \(\times\) 10 + 1

4\(\times\) 10^{3}+ 2\(\times\) 10^{2} + 5\(\times\) 10^{1} + 4\(\times\) 10^{2}

0 and 1 are used in binary number system which is arranged using positional notation (the digit 0 and 1 as a symbol). When a number larger than 1 needs to be represented, the positional notation is used to represent, the positional notation is used to represent how many groups of 2, 4, 8 are contained in the number. For example,

Let's consider the number 30

30 ÷ 2 = 15 Remainder 0

15 ÷ 2 = 7 Remainder 1

7 ÷ 2 = 3 Remainder 1

3 ÷ 2 = 1 Remainder 1

1 ÷ 2 = 0 Remainder 1

- A number is a mathematical object used to count, measure and label.
- The binary number system always uses only two different symbols ( the digit 0 and 1) that are arranged using positional notation.
- We can subtract a binary number from the another binary number.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Convert the following denary numbers into binary.

124_{10}

Solution:

2 | 124 |

2 | 62 |

2 | 31 |

2 | 15 |

2 | 7 |

2 | 3 |

2 | 1 |

0

Hence,124_{10}=1111100

Remainder

0

0

1

1

1

1

1↑

Convert the following binary numbers into the senary system.

11010

Solution:

11010_{2}

=1×2^{4}+1×2^{3}+0×2^{2}+1×2^{1}+0×2^{0}

=16+8+0+2+0

=26

Convert the following binary numbers into the senary system.

11010

Solution:

11010_{2}

=1×2^{4}+1×2^{3}+0×2^{2}+1×2^{1}+0×2^{0}

=16+8+0+2+0

=26

Convert the following binary numbers into the senary system.

101011

Solution:

101011_{2}

=1×2^{5}+0×2^{4}+1×2^{3}+0×2^{2}+1×2^{1}+1×2^{0}

=32+0+8+0+2+1

=43

Convert the following binary numbers into decimal numbers.

111

Solution:

1×2^{2}+1×2^{1}+1×2^{0}

=4+2+1

=7

Convert the following binary numbers into decimal numbers.

1010

Solution:

1010_{2}= 1×2^{3}+0×2^{2}+1×2^{1}+0×2^{0}

=8+0+2+0

=10

Convert the following binary numbers into decimal numbers.

1110

Solution:

1110_{2} = 1×2^{3}+1×2^{2}+1×2^{1}+0×2^{o}

=8+4+2+0

= 14

Convert the following binary numbers into decimal numbers.

10101

Solution:

1×2^{4}+0×2^{3}+1×2^{2}+0×2^{1}+1×2^{o}

=16+0+4+0+1

=21

Convert the following binary numbers into decimal numbers.

101010

Solution:

=1×2^{5}+0×2^{4}+1×2^{3}+0×2^{2}+1×2^{1}+0×2^{0}

^{=}32+0+8+0+2+0

=42

Convert the following binary numbers into decimal numbers.

110011

Solution:

1×2^{5}+1×2^{4}+0×2^{3}+0×2^{2}+1×2^{1}+1×2^{0}

= 32+16+0+0+2+1

=51

Convert the following into decimal system

212_{5}

Solution:

212_{5}

= 2×5^{2}+1×5^{1}+2×5^{0}

= 2×25+1×5+2×1

= 50+5+2

= 57_{10}

Convert the following into decimal system:

314_{5}

Solution:

314_{5}

=3×5^{2}+1×5^{1}+4×5^{0}

= 3×25+1×5+5+4

= 75+5+4

= 84_{10}

Convert the following numbers into base-ten numbers.

24_{5}

Solution:

24_{5}

= 2×5^{1}+4×5°

= 2×5+4×1

=10+4

=14_{10}

Convert the following numbers into base-ten numbers.

324_{5}

Solution:

354_{5}

= 3×5^{2}+2×5^{1}+4×5°

= 75+10+4

= 89_{10}

Convert the decimal number into binary number.

105

Solution:

2 | 105 | 1 |

2 | 52 | 0 |

2 | 26 | 0 |

2 | 13 | 1 |

2 | 6 | 0 |

2 | 3 | 1 |

2 | 1 | 1 |

0 |

∴ 105_{10} = 1101001_{2}

Convert this Decimal number into binary number.

1234

Solution:

2 | 1234 | 0 |

2 | 617 | 1 |

2 | 308 | 0 |

2 | 154 | 0 |

2 | 77 | 1 |

2 | 38 | 0 |

2 | 19 | 1 |

2 | 9 | 1 |

2 | 4 | 0 |

2 | 2 | 0 |

2 1 1

0

∴ 1234_{10} = 10011010010_{2}

Convert this decimal number into binary number.

975_{10}

Solution:

2 | 975 | 1 |

2 | 487 | 1 |

2 | 243 | 1 |

2 | 121 | 1 |

2 | 60 | 0 |

2 | 30 | 0 |

2 | 15 | 1 |

2 | 7 | 1 |

2 | 3 | 1 |

1 |

∴ 975_{10} = 1111001111_{2}

Convert the following binary number into decimal numbers.

11111111

Solution:

11111111_{2}

= 1×2^{7}+ 1×2^{6} + 1×2^{5}+ 1×2^{4} + 1×2^{3} + 1×2^{2} +1×2^{1} + 1×2^{0}

= 128 + 64 + 32 + 16 + 8 + 4 +2 + 1

= 225

Convert the Binary numbers into decimal numbers.

11011011001_{2}

Solution:

11011011001_{2}

= 1×2^{10}+1×2^{9}+0×2^{8}+1×2^{7}+1×2^{6}+0×2^{5}+1×2^{4}+1×2^{3} +0×2^{2}+0×2^{1}+1×2^{0}

=1050 +512 +0 +128 +64 +0 +16 +8 +0 +0 +1

= 1751

∴ 11011011001_{2} = 1751_{10}

Convert into decimal number.

100000001_{2}

Solution:

= 1×2^{8} + 0×2^{7} + 0×2^{6} + 0×2^{5} + 0×2^{4} + 0×2^{3} + 0×2^{2} + 0×2^{1} + 1×2^{0}

= 256 + 0 + 0 +0 +0 +0 +0 +0 +1

= 257

∴ 100000001_{2} = 257_{10}

Convert the binary numbers into decimal numbers.

1010101110_{2}

Solution:

= 1× 2^{9} + o×2^{8} + 1×2^{7} + 0×2^{6} +1×2^{5} +0×2^{4} + 1×2^{3} + 1×2^{2} + 1×2^{1} + 0×2^{0}

= 512 + 0 + 128 +0 + 32 + 0 + 8 + 4 + 2 +0

= 512 + 0 + 128 + 0 + 32 + 0 + 8 + 4 + 2 + 0

= 686

∴ 1010101110_{2} = 686_{10}

Convert the binary number into decimal numbers.

1000010000_{2}

Solution:

= 1×2^{9}+0×2^{8}+0×2^{7}+0×2^{6}+0×2^{5}+1×2^{4}+0×2^{3}+0×2^{2}+0×2^{1} +0×2^{0}

= 512 +0 +0+0+0+16+0+0+0 +0

=528

∴ 1000010000_{2} = 528_{10}

Convert the binary numbers into decimal numbers.

101110111_{2}

Solution:

101110111_{2}

= 1×2^{8}+ 0×2^{7} + 1×2^{6} + 1×2^{5} + 1×2^{4} +0×2^{3} + 1×2^{2} + 1×2^{1} + 1×2^{0}

= 256 + 0 + 64 + 32 + 16 + 0 + 4 + 2 +1

=359

∴ 101110111_{2} = 359_{10}

Convert the binary numbers into decimal numbers.

100000001_{2}

Solution :

100000001_{2}

= 1×2^{8} + 0×2^{7}+ 0×2^{6} + 0×2^{5} + 0×2^{4} + 0×2^{3} + 0×2^{2} + 0×2^{1} + 1×2^{0}

= 256 + 0 +0 +0 +0 +0 +0+0 +1

= 257

∴ 100000001_{2} = 257_{10}

Convert the binary numbers into decimal numbers.

1100110011_{2}

Solution:

1100110011_{2}

= 1×2^{9} + 1×2^{8} + 0×2^{7} + 0×2^{6} + 1×2^{5} + 1×2^{4} + 0×2^{3} + 0×2^{2} + 1×2^{1} + 1×2^{0}

= 512 + 256 + 0 +0 + 32 + 16 + 0 +0 +2 +1

= 819

∴ 1100110011_{2} = 819_{10}

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