What is Scientific Notation?
Definition: Scientific notation is a type of shorthand for expressing extremely large or small numbers. The number is shortened to a number multiplied by a power of ten instead of being written in decimal form. For example:
What ? | How much ? | Scientific Notation |
|---|---|---|
Distance from the earth to the sun | 150, 000, 000, 000 meters | \(1.5~\times~10^{11}\) |
Mass of an average human cell | 0.000000000001 kilograms | \(1~\times~10^{-12}\) |
Diameter of the moon | 3, 474, 000 meters | \(3.474~\times~10^{6}\) |
What is a Scientific Notation and why do we use it?
By multiplying single-digit numbers by 10 raised to the power of the exponent, we can represent extremely large or extremely small numbers using scientific notation. If the number is very large, the exponent is positive; if the number is very small, the exponent is negative. For a better understanding, let us learn about power and exponents.
The following is a common representation of scientific notation:
\(a~\times~10^{b};~1~\leq~a< ~10 \)
Rules of Scientific Notation
To find the power or exponent of ten, we must apply the following rule:
- The base should always be ten.
- The exponent has to be a non-zero integer, which can be positive or negative.
- The absolute value of the coefficient is greater than or equal to one, but it should be less than ten.
- Positive and negative numbers, as well as whole and decimal numbers, can be used as coefficients.
- The rest of the significant digits of the number are carried by the mantissa.
Writing numbers in Standard form
The absolute value of the exponent indicates how many places the decimal point should be moved.
- Move the decimal point to the left if the exponent is negative.
- Move the decimal point to the right if the exponent is positive.
Solved Scientific Notation Examples
Example 1:
Convert 194,000,000 into scientific notation.
Solution:
Example 2:
Convert 0.0000048 into scientific notation.
Solution:
Move the decimal point up to 6 places to the right of 0.0000048.
To make the number 4.8, the decimal point was moved 6 places to the right.
The decimal is moved to the right because the numbers are less than ten. As a result, we use a negative exponent in this case.
\(0.0000048~=~4.8~\times~10^{-6} \)
The scientific notation is written like this.
Example 3:
Could you assist Kevin in writing \(8.69~\times~10^{8} \) in standard notation?
Solution:
8.69 is 869 in this case. Kevin must now move the decimal point 8 places to the right and add zeros in the appropriate places.
869,000,000 is the standard notation for \(8.69~\times~10^{8} \).
Question 4:
Write \(4.55~\times~10^{-4} \) in standard form.
Solution:
\(4.55~\times~10^{-4}~=~0.000455 \)
Move the decimal point 4 places to the left as the power of 10 is –4.
Scientific notation is an important topic in mathematics that must be thoroughly understood in order to develop the ability to represent large sets of numbers. Scientific notations can represent very small numbers as well.
In scientific notation, numbers are written as \(a\times~10^{b}~;~1~\leq ~a~< 10 \)
Scientific notation is a way of expressing numbers in decimal form between 1 and 10, but not 10 multiplied by a power of ten.
As a result, scientific notations are based on base 10 powers.
\(a\times~10^{b}~;~1~\leq ~a~< 10 \) is the standard scientific notation.
\(a \) times ten raised to the power of \(b\), where \(b \) is an integer and the coefficient \(a \) is a real number.