Question

Express each of the following numbers in scientific notation: a. \(19,306,183\) (population of New York state) b. \(6,602,224,175\) (world population) c. \(0.00000000007461 \mathrm{~m}\) (length of a hydrogenhydrogen chemical bond) d. \(0.000015 \mathrm{~m}\) (diameter of a human hair)

Short answer

a. 1.9306183 x 10^7, b. 6.602224175 x 10^9, c. 7.461 x 10^-11 m, d. 1.5 x 10^-5 m

Step-by-step solution

Convert 19,306,183 into Scientific Notation

To convert the number into scientific notation, move the decimal point so that it is after the first non-zero digit. Here, the decimal point is moved left by 7 places, giving us 1.9306183. Therefore, in scientific notation, 19,306,183 is written as 1.9306183 x 10^7.

Convert 6,602,224,175 into Scientific Notation

Move the decimal point left so that it comes after the first non-zero digit, which is done by moving the decimal point 9 places to the left, giving us 6.602224175. The number in scientific notation is 6.602224175 x 10^9.

Convert 0.00000000007461 m into Scientific Notation

Move the decimal point right so that it comes after the first non-zero digit. The decimal point moves 11 places giving us 7.461. The number in scientific notation is 7.461 x 10^-11 m.

Convert 0.000015 m into Scientific Notation

Move the decimal point right to come after the first non-zero digit, moving it 5 places. This gives us 1.5. Hence, in scientific notation, the number is 1.5 x 10^-5 m.

Key concepts

Expressing Large Numbers

Dealing with huge numbers can be daunting, but scientific notation simplifies it by using powers of ten. This method is essential in fields such as astronomy, physics, and engineering where massive figures are common. Take the New York state population of 19,306,183 as an example. Rather than writing all those digits, scientific notation allows us to condense the number to 1.9306183 x 10^7.

This approach is not just about brevity; it also provides clarity in calculation and consistency in the representation of large data sets. To express a large number in scientific notation, identify the first non-zero digit, place a decimal point after it, and count how many places you move the decimal to reach its original position. This count becomes the exponent over the base ten. Such a format makes comparisons and computations of large values much more manageable.

Expressing Small Numbers

Just like their colossal counterparts, very tiny numbers are elegantly handled using scientific notation. It's particularly favored in chemistry and physics, where exceedingly small measurements are the norm. Reflect on the length of a hydrogen-hydrogen chemical bond, a miniscule 0.00000000007461 meters. The scientific notation for this is 7.461 x 10^-11 m, showing how the notation efficiently deals with such diminutive dimensions.

When expressing a small number in scientific notation, we again locate the first non-zero digit. We then move the decimal point to position it right after this digit and tally the number of moves. This time, however, the count of moves becomes a negative exponent of ten. This form not only simplifies the number but also makes it easier to understand and use in mathematical operations.

Significant Figures

The concept of significant figures plays a crucial role when using scientific notation, as it reflects the precision of the numbers. Significant figures are the digits in a number that are known with certainty plus the first uncertain or estimated digit. For instance, 1.9306183 x 10^7 has eight significant figures, indicating a very precise population count for New York state.

In scientific notation, the number of significant figures is noted before the x 10^ part, which does not affect the count of significant digits. Correct use of significant figures is important as it directly correlates with the reliability of the data. Overestimating or underestimating can lead to loss of accuracy and could significantly impact analytical results, particularly in scientific and engineering calculations.