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Demonstrate a general understanding of significant digits and the use of reasonable numerical estimates in performing measurements and calculations

**Significant digits** (or significant figures) are indicative of the degree of approximation in a measurement. In determining the significant digits of a value, all non-zero digits are counted along with trailing zeros to the right of the decimal point and all zeros between significant digits. As calculations are performed, a consistent number of significant digits will properly convey the levels of approximation that were used in the initial values.

Examples:

- 200
- the two trailing zeros are not to the right of the (implicit) decimal point, therefore the 2 is the only significant digit
- 200.0
- the 2 and the trailing zero to the right of the decimal point are significant, therefore making the zeros in between also significant for a total of 4 significant digits
- 0.200
- 3 significant digits
- 0.2
- 1 significant digit

When performing calculations, the result should be rounded to the least significant figures of the values involved. For example, 15.2 x 5 = 76 ≈ 80.

Reasonable estimates should remain in the same order of magnitude and close to the number of significant digits. Using 15.2 x 5 again, estimating 15.2 as 15 only reduces its significant digits by one and gives a result of 75, but estimating 15.2 as 20 now reduces its significant digits by two and gives a result of 100, which is much farther off from the actual result of 76. More accurate estimations of calculations can also be gotten by rounding values in opposite directions for multiplication (one up, one down) and the same direction for division (both up or both down).

Interpreting numbers using **scientific notation** can make relationships between values easier to comprehend and simpler to calculate, especially for very small or very large values. **A value is transformed into scientific notation by representing its significant digits as a number between 1 and 10 (in absolute value), multiplied by a power of base 10 to make it equal to the initial value.** Thus 4,300 becomes 4.3 x 10^{3}, and -0.000901 becomes -9.01 x 10^{-4}. The value represented by scientific notation is less than 1 (in absolute value) when the exponent is negative, and greater than 10 (in absolute value) when the exponent is positive.

Because the coefficient for numbers in scientific notation is always in the same power of magnitude by being between 1 and 10 (in absolute value), quick comparisons can be made based on the exponent on the power of 10. For example, the number with the greater exponent is the greater number regardless of the value of the coefficient. For example, 1.0 x 10^{4} is greater than 8.0 x 10^{3}. Indeed, 10,000 is greater than 8,000, which is quickly discerned by comparing 10^{4} to 10^{3}, regardless of the coefficients 1.0 and 8.0. Conversely, if the exponents are the same, then it falls to the coefficient to determine the greater number. For example, **8.0** x 10^{-2} is greater than **1.0** x 10^{-2}, and indeed 0.08 is greater than 0.01.

Calculations on numbers in scientific notation can be done quickly using addition of the exponents in the case of multiplication and subtraction of exponents in the case of subtraction.

Usage | Example |
---|---|

values with positive exponents are large numbers | 2.0 x 10^{3} = 2,000-2.0 x 10^{3} = -2,000 |

values with negative exponents are small numbers | 2.0 x 10^{-3} = 0.002-2.0 x 10^{-3} = -0.002 |

value with greater exponent is the greater value (in absolute value) | 1.23 x 10^{-2} > 7.89 x 10^{-3}-7.89 x 10^{-3} > -1.23 x 10^{-2} |

multiply coefficients and add exponents when multiplying | 2.0 x 10^{15} * 4.0 x 10^{-2}= (2.0 * 4.0) x 10 ^{15 + -2}= 8.0 x 10 ^{13} |

divide coefficients and subtract exponents when dividing | 2.0 x 10^{15} / 4.0 x 10^{-2}= (2.0 / 4.0) x 10 ^{15 - -2}= 0.50 x 10 ^{17} |

remove a magnitude from exponent when adding a magnitude to coefficient | 0.50 x 10^{17}= 5.0 x 10 ^{17 - 1}= 5.0 x 10 ^{16} |