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In a**binary** number (or base two) system, we use only two digits: 0 and 1.

The following diagrams show the place values for binary numbers and how to convert from binary to decimal.

### Convert From Binary To Decimal

101_{2} = 1 × 2^{2} + 0 × 2^{1} + 1 × 2^{0}

= 4 + 0 + 1

= 5_{10}

b)

1011_{2} = 1 × 2^{3} + 0 × 2^{2} + 1 × 2^{1} + 1 × 2^{0}

= 8 + 0 + 2 + 1

= 11_{10}

c)

0101_{2} = 1 × 2^{4} + 0 × 2^{3} + 1 × 2^{2} + 0 × 2^{1} + 1 × 2^{0}

= 16 + 0 + 4 + 0 + 1

= 21_{10}

**How To Convert Binary To Decimal?**

### Convert From Decimal To Binary

b)

78_{10} = 1001110_{2}

**Base 10 to Base 2 (Decimal to Binary) Conversion**

A demonstration of the repeated divide by 2 method for converting numbers from base 10 (or decimal) into base 2 (or binary) form.

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In a

The following diagrams show the place values for binary numbers and how to convert from binary to decimal.

Like base ten numbers, we can determine the value of a base two number by placing each digit in its respective place value, then writing its expanded notation.

Example:

Find the value of each of the following binary numbers, giving the values in decimal numbers:

a)101_{2}

b)1011_{2}

c)10101_{2}

Solution:

a)

2^{4} |
2^{3} |
2^{2} |
2^{1} |
2^{0} |

1 | 0 | 1 |

101

= 4 + 0 + 1

= 5

b)

2^{4} |
2^{3} |
2^{2} |
2^{1} |
2^{0} |

1 | 0 | 1 | 1 |

1011

= 8 + 0 + 2 + 1

= 11

c)

2^{4} |
2^{3} |
2^{2} |
2^{1} |
2^{0} |

1 | 0 | 1 | 0 | 1 |

0101

= 16 + 0 + 4 + 0 + 1

= 21

We can convert a decimal number into a binary number by repeatedly dividing the base ten number by two. Then, write the remainders from the bottom to the top as the answer:

**Example:**

Write each of the following base ten numbers as a binary number:

a)5_{10}

b)78_{10}

**Solution:**

5_{10} = 101_{2}

b)

78

A demonstration of the repeated divide by 2 method for converting numbers from base 10 (or decimal) into base 2 (or binary) form.

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