SCIENTIFIC NOTATION |

**Introduction**

When we work with very large or very small numbers, it is often best to put them into scientific notation first -- especially if we need to multiply or divide them. Since our number system is based on powers of ten, so that multiplying or dividing a number by some power of 10 simply moves the decimal place, we have developed a set of rules to standardize how numbers appear in science and math texts.

To put a very big or very small number into scientific form or notation, we have to rewrite it as a **single digit number with a decimal part multiplied by a power of 10.** The exponent we apply to 10 is equal to the number of places we moved the decimal to create a number with a SINGLE DIGIT LEFT OF THE DECIMAL multiplied by a power of 10. When we move the decimal point to the left of its original position, the power of 10 will be positive. When we move the decimal point to the right of its original position, the power of 10 will be negative.

**Examples:**

These numbers have been put in scientific notation:

a) 5 710 000.0 = 5.71 × 10^{6} (decimal moved 6 places left) |
b) 0.000359 = 3.59 × 10^{ - 4} (decimal moved 4 places right) |

This is one of the easiest topics in arithmetic because

all we have to do to succeed is follow the rules.

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**Rules for Scientific Notation**

1. There can only be a single digit left of the decimal point. |

2. If the decimal point moved to the left in step 1, the exponent on 10 is positive. |

3. If the decimal point moved to the right in step 1, the exponent on 10 is negative. |

4. The exponent applied to 10 equals the number of places the decimal moved. |

Don't forget that with whole numbers (integers), though the decimal is not visible, it is at the end of the number. So 5280 really means 5280.0 In scientific notation, 5280 = 5.28 × 10^{ 3} .

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**Practice**

**Exercise 1**

These large numbers are in scientific notation

Write the correct exponent in the blank.

a) 127.3 = 1.27 × 10^{ __ } |
b) 50 517 = 5.0517 × 10^{ __} |
c) 734 000 = 7.34 × 10^{ __} |

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**Exercise 2**

These small numbers are in scientific notation

Write the correct exponent in the blank.

a) 0.0175 = 1.75 × 10^{ __ } |
b) 0.0019 = 1.9 × 10^{ __} |
c) 0.00005 = 5 × 10^{ __} |

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**Exercise 3**

Put these numbers into scientific notation

a) 0.00556 = | b) 1 907 000 = | c) 275.23 = |

d) 473 000 000 = | e) 0.00000012 = | f) 0.075 = |

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**Solutions**

**Exercise 1**

These large numbers are in scientific notation

Write the correct exponent in the blank.

a) 127.3 = 1.27 × 10² | b) 50 517 = 5.0517 × 10^{ 4} |
c) 734 000 = 7.34 × 10^{ 5} |

**Exercise 2**

These small numbers are in scientific notation

Write the correct exponent in the blank.

a) 0.0175 = 1.75 × 10^{ - 2 } |
b) 0.0019 = 1.9 × 10^{ - 3} |
c) 0.00005 = 5 × 10^{ - 5} |

**Exercise 3**

a) 0.00556 = 5.56 × 10^{ - 3 } |
b) 1 907 000 = 1.907 × 10^{ 6} . |
c) 275.23 = 2.7523 × 10² |

d) 473 000 000 = 4.73 × 10^{ 8} |
e) 0.00000012 = 1.2 × 10^{ - 7} . |
f) 0.075 = 7.5 × 10^{ - 2} . |

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