# Expanded form

Expanded form is a method for writing numbers that breaks the number down into the value of each of its digits. There are a few ways to write a number in expanded form. Take the number 127 as an example. "127" is the standard way of writing the number one hundred twenty-seven. One way to write 127 in expanded form is:

(1) 127 = 100 + 20 + 7

This is based on the fact that our numeral system, the decimal numeral system is a positional numeral system. This means that the value of a numeral is based on both the digits in the numeral, and the position of each digit in the numeral.

The system we use is a base 10 system, meaning that each digit represents a power of 10. To the left of the decimal point, the first position is the ones place, followed by the hundreds place, thousands place, ten-thousands place, and so on based on powers of 10. To the right of the decimal point is the tenths, hundredths, thousandths, ten-thousandths, and so on, again based on powers of 10, though in this case they are negative powers.

What this means is that our example above can be written as the product of the digit and the power of 10 that its position indicates. In 127, the 1 is in the hundreds place, the 2 is in the 10s place, and the 7 is in the ones place. Thus, another version of the number 127 in expanded form is:

(2) 127 = (1 × 100) + (2 × 10) + (7 × 1)

which is just based on the powers of 10 being written as their values, rather than as powers of 10. Thus, another way that 127 can be written in expanded form is:

(3) 127 = (1 × 10^{2}) + (2 × 10^{1}) + (7 × 10^{0})

All of the above are equal, though the third version most clearly shows how our base-ten positional numeral system works.

Below is a table showing place values from millions to millionths. The term in the "Name" column is what is used to refer to the specific position in the numeral in the "Place" column; the 1 in 1,000,000 is in the "millions place" of the numeral. The 0 directly after the 1 in 1,000,000 is in the hundred thousands place, and so on.

Name | Place (position) |
---|---|

Millions | 1,000,000 |

Hundred thousands | 100,000 |

Ten thousands | 10,000 |

Thousands | 1,000 |

Hundreds | 100 |

Tens | 10 |

Ones | 1 |

Tenths | 0.1 |

Hundredths | 0.01 |

Thousandths | 0.001 |

Ten thousandths | 0.0001 |

Hundred thousandths | 0.00001 |

Millionths | 0.000001 |

Example

1. Write the number 1,567,234 in expanded form using form (3) described above:

(1 × 10^{6}) + (5 × 10^{5}) + (6 × 10^{4}) + (7 × 10^{3}) + (2 × 10^{2}) + (3 × 10^{1}) + (4 × 10^{0})

2. Write the number 2.07894 in expanded form using form (3) described above:

(2 × 10^{0}) + (0 × 10^{-1}) + (7 × 10^{-2}) + (8 × 10^{-3}) + (9 × 10^{-4}) + (4 × 10^{-5})

In this example, we just need to keep track of the decimal place, which comes after the 2 in the ones place. Any product of a digit and a negative power of 10 comes after the decimal point in its respective position.