Question

A grain of sand has a mass approximately of \[0.00000003\] grams. How would this number be expressed in scientific notation?

Answer 1

Hint:The mass of a grain of sand is given in the form of a decimal number whose non-zero digits reside at the end of the number. Its unit is grams.To express this given number in scientific notation, the last non-zero digit is separated from the second last digit (it can be zero or non-zero) by a decimal point. In other words, the decimal point is shifted backwards to the non-zero digits. The zero digits are then indicated in powers of 10.

Complete answer:
Scientific notation is sometimes referred to as the standard index form. The general representation of scientific notation is: \[a{\text{ }} \times {\text{ }}{10^{b\;\;}}\;\] where \[1{\text{ }} \leqslant {\text{ }}a{\text{ }} < {\text{ }}10\] and \[b\] can be any integer. The number \[b\] is known as the order of magnitude while the number \[a\] is referred to as the mantissa or significant. The number \[a\] is the coefficient of the scientific notation and is normally greater than or equal to 1 and less than 10.
The given number is \[.00000003\] grams. The non-zero digit is 3. Then according to the condition of significant numbers, this digit can be separated from the zero digit by a decimal point. Thus we can write it as 0.3 or \[a = 0.3\]and the remaining zeros can be written as power of 10. Thus, \[0.00000003 = \dfrac{{0.3}}{{10000000}} = 0.3 \times {10^{ - 7}}\]. The power factor appears to be \[{10^{ - 7}}\]. Therefore, \[b = - 7\]. This is a negative exponent.
Therefore, \[.00000003\] would be written in scientific notation as \[0.3 \times {10^{ - 7}}\] grams. The unit grams can be converted into micrograms by writing \[0.00000003{\text{ grams}} = 0.03 \times {10^{ - 6}}{\text{grams}} = 0.03{\text{ micro - grams}}\]. (Since \[{10^{ - 6}}\] denotes micro)

Note: Note that scientific notation is preferred for very large or very small numbers.Since \[0.00000003\] is a very small number, therefore the preferred scientific notation for it is \[0.3 \times {10^{ - 7}}\]. Note that \[0.00000003{\text{ grams}}\] is written in terms of smaller units as \[0.03{\text{ micro - grams}}\]. The prefix unit micro denotes \[{10^{ - 6}}\]. The prefix micro is denoted by the symbol \[\mu \]. Therefore \[0.03{\text{ micro - grams}}\] can also be written as \[0.03{\text{ }}\mu {\text{ - grams}}\].