scientific notation

scientific notation, method of writing large or small numbers in a shorter form. For example, 4,800,000,000,000 is written in scientific notation as 4.8 ⨉ 10¹². Similarly, 0.0000000000048 is written as 4.8 ⨉ 10^-12. These numbers are referred to as “4.8 times 10 to the power of 12” and “4.8 times 10 to the power of negative 12,” respectively. Scientific notation is used in fields such as engineering, chemistry, microbiology, physics, and astronomy where very large and very small numbers are common.

The basic formula used to write numbers in scientific notation ism ⨉ 10ⁿwhere m is a number greater than or equal to 1 but less than 10, 10 is the base number, and n is the exponent, or power of ten, which is a positive or negative whole number. The number 10ⁿ is equal to 10 multiplied by itself n times. For example, 10⁴ = 10 ⨉ 10 ⨉ 10 ⨉ 10 = 10,000.

To write a large number in scientific notation, one moves the decimal place to the left to form a number between 1 and 10. One then counts the number of places the decimal place was moved to the left. The base number 10 is raised to the power of the number of places the decimal was moved to the left. For example, if the number is 4,900,000,000, m = 4.9. The decimal place has been moved nine places to the left, so n = 9. The number 4,900,000,000 written in scientific notation is 4.9 ⨉ 10⁹.

For small numbers, the procedure is much the same. One moves the decimal place to the right to form a number between 1 and 10. One then counts the number of places the decimal place was moved to the right. The base number 10 is raised to the power of the negative number of places the decimal was moved to the right. For example, if the number is 0.00000036, m = 3.6. The decimal place has been moved seven places to the right, so n = -7. The number 0.00000036 written in scientific notation is 3.6 ⨉ 10^-7.

Scientific notation allows one to multiply, divide, add, and subtract very large or very small numbers more easily. For multiplication of two numbers m ⨉ 10ⁿ and a ⨉ 10^b,(m ⨉ 10ⁿ) ⨉ (a ⨉ 10^b)= (m ⨉ a) ⨉ 10^{n + b}Division works much the same:(m ⨉ 10ⁿ)/(a ⨉ 10^b)= (m/a) ⨉ 10^{n - b}

For addition and subtraction using scientific notation, the exponents must always be the same power of ten. That is, (m ⨉ 10ⁿ) ± (c ⨉ 10ⁿ)= (m ± c) ⨉ 10ⁿWhen the exponents are different powers, one of them can be changed to match the other. For example,8.2 ⨉ 10⁴ + 3.2 ⨉ 10³= 8.2 ⨉ 10⁴ + 0.32 ⨉ 10⁴since 3.2 = 0.32 ⨉ 10¹. Therefore, the solution is 8.52 ⨉ 10⁴.