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Scientific Notation 

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Scientific notation has been one of the oldest mathematical approaches. If numbers are too big or too small, it can be simply calculated, using scientific notations. Scientific notation is defined as a standard way of writing very large or very small numbers so that they are easier to compare and use in computations.
The standard form of scientific notation is given as: N $\times$ 10$^a$ where 'N' is a number between 1 and 10, but not 10 itself, and 'a' is an integer number.Scientific notation helps in simplifying, reading, writing and computing with large and small numbers. A positive or negative number whose absolute value is 10 or greater has a positive exponent when expressed in scientific notation. It is easier to write a number in scientific notation. We move the decimal Point of a number until the new form, and then record the exponent (a) as the number of places the decimal point was moved. Whether we move the decimal to right or to left then the power of 10 is positive or negative depends on it.
With the help of scientific notations we can represent any large value which contains decimal in very simpler form. A very good knowledge of powers which are over a given number is required.
Suppose, if we write a number like 5 * 10$^2$, it means that 5 * 100 = 500, so as we see that if power is 2 over ten that means two zeros in front of two. If we are having a expression 4 * 10$^{-2}$ it means that .004, so whenever we have any power in minus then we have to put two zeros in left with a decimal.

Conversion of Scientific Notation

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Changing from standard form to scientific notation.
  1. Increase the power of ten by one, decimal point should be placed such that there is one non zero digit to the left of the decimal point.
  2. Count the number of decimal places the decimal has moved from the original number. This will be the exponent of 10.
  3. If the original number is less than one, exponent will be negative and if the original number is greater than one, exponent will be positive.
Changing from scientific notation to standard form.
  1. Exponents tells us how many places to move.
  2. For positive exponents of 10 move the decimal point to the right.
  3. For negative exponents of 10 move the decimal point to the left.

Rules For Scientific Notation

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Given below are the rules for scientific notation for addition, subtraction, multiplication and division.

Steps for adding numbers in scientific notation
Step1: The given numbers should be written in scientific notation.

Step 2: Expand it to similar exponents only if the values of the exponents are not same, except the exponent portion if it is unchanged.

Step 3: Add the new coefficients.

Step 4: The coefficient should be between 1 and 10, if not convert again into scientific notation.

Steps for subtracting numbers in scientific notation
Step 1: Determine the number by which to increase the smaller exponent by so it is equal to the larger exponent.

Step 2: Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left of the same number of places.

Step 3: Difference the new coefficients.

Step 4: The coefficient should be between 1 and 10, if not convert again into scientific notation.

Steps for multiplying numbers in scientific notation
Step 1: The coefficient of the given numbers should be multiplied.

Example : a * 10$^{b}$ * c * 10$^{d}$

Step 2: Add the numbers 10 to the power.
Example : ac * 10$^{b+d}$

Step 3: If the coefficient of the result is greater than 10, the result will be in the form ac * 10 $^{(b+d+1)}$

Steps for dividing numbers in scientific notation
Step 1: For the given problem difference the exponents of the base.

Step 2: With the subtracted new exponent write the number.

Step 3: Shift the decimal point to the right if the resulting number is less than 1, according to the decimal places shifted decrease the power. The result should be written in scientific notation.

Problems on Scientific Notation

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Example 1: Write 622,000,000,000 in scientific notation
Step 1: Move the decimal place to the left.

For example: we write 622,000,000,000 in decimal. The decimal point is placed at the end of the number 622,000,000,000 and written as:
N = 6.22

Step 2: Now, determine how many times we moved the decimal. In this example, we moved the decimal 11 times and as the exponent is positive, in this we also move the decimal to left. Therefore, a = 11, and so we get 10 $^{11}$.

Step 3: Lastly, put the number in the correct form of scientific notation that is, N x 10$^{a}$ ,$\therefore$ 622,000,000,000 = 6.22 x 10 $^{11}$.
In this way, we write the scientific notation of any number.

Example 2: Convert 2.84 * 10$^6$ from scientific notation to standard notation
Solution:
Step 1: Given 2.84 * 10$^6$ in scientific notation
Exponent = 6 (Positive)

Step 2: Move the decimal place 6 places to the right because exponent is positive
2.84 * 10$^6$ = 2.84 * 1,000,000 = 28,400,000

Example 3: Divide (0.27 x 10$^8$) by (3.2 x 10$^3$)
Solution:
Step 1: Divide the decimal numbers
0.27 รท 3.2 = 0.084375

Step 2: Subtract the powers of the base 10
8 - 3 = 5

Step 3: Write the number with the subtracted new exponent
0.084375 x 10$^5$

Step 4: Now the decimal number is less than 1, so right shift the decimal point by 2 places and decrease the power by 2
0.084375 = 8.84375
0.08846 x 10$^{5}$ = 8.84375 x 10$^{3}$

Step 5: Write the result in scientific notation
Hence, dividing scientific notation (0.27 x 10$^8$) by (3.2 x 10$^3$) we get 8.84375 x 10$^{3}$