In mathematics, standard form and scientific notation are used to reduce the difficulty of reading and writing a massive number or a small number. Scientific notation is also referred to as standard form. The standard form and scientific notation have the same work to reduce the difficulty of studying or writing a larger number or smaller number.
In this post, we will discuss both terms with examples.
What is Standard Form?
In mathematics, the standard form is defined as an example or notation of that specific element. It is based upon the problem whether or not or now no longer it’s far numbers, an equation, or a line. Any number that we are able to write as a decimal number amongst 1.0 and 10.0 multiplied by the power of 10 is known as Standard form e.g., 8.898×1036.
In mathematics, the amount of a number remains the same when we convert it into standard form. For example, we write 3 thousand in digits as 3000, and in standard form, we write as 3×103 each the values denote three thousand. It is tough to observe or write a number in standard form e.g., 9876543210000000, 0r 0.0000000012345. In order to make it simple to read or write we use the standard form above stated numbers may be written in standard form as 9.877×1015 and 1.2345×10-9.
Let us take some examples to understand more accurately.
Example 1
Convert 9876543210000000 to the standard form.
Solution
Step 1: write down the number.
9876543210000000
Step 2: Identify the decimal place.
- 9876543210000000. The decimal point is after the last digit
Step 3: Now move the decimal point to the first non-zero digit.
9.876543210000000
Step 4: count the movement of the decimal point and write it in the power of 10 and multiply by the number.
9.876543210000000x1015
Step 5: Round the number up to two decimal places.
9.88 x 1015
This is the required standard form of 9876543210000000.
Example 2
Convert 0.0000000012345 to the standard form.
Solution
Step 1: write down the number.
0.0000000012345
0.0000000012345
Step 3: Now move the decimal point to the first non-zero digit.
0000000001.2345
Step 4: count the movement of the decimal point and write it in the power of 10 and multiply by the number.
0000000001.2345x10-9
Step 5: Ignore the leading zeros.
1.2345×10-9
This is the required standard form of 0.0000000012345.
What is Scientific Notation?
Scientific notation is a scientific method of writing a number. This is a notation used to denote a larger number or a smaller number in the form of b x 10a, here “b” is a number or any decimal point number, and “a” is the power of ten that can be positive or negative. The addition, subtraction, multiplication, or division can be done easily with two or more scientific notations.
How to calculate Scientific Notation?
We can add, subtract, multiply, and divide two or more scientific notations. Let us discuss addition, subtraction, multiplication, and division of scientific notations.
Addition of Scientific Notations
For the addition of two or more scientific notations, the exponents must be the same then add the coefficient and the power of ten which is the same of all the number remains same. If the exponents are not the same then first of all make the exponents the same then add the scientific notations. Let us take some examples in order to understand this concept.
Example 1
Add the scientific notations 6.3 x 104 and 15 x 104.
Solution
Step 1: Put an addition sign between the numbers.
6.3 x 104 + 15 x 104
Step 2: As the exponents are the same add the coefficients.
6.3 + 15 = 21.3
Step 3: Write the result with the exponent.
21.3 x 104
It can also be written as.
2.13 x 105
Example 2
Add the scientific notations 9.5 x 106 and 18 x 105.
Solution
Step 1: Put an addition sign between the numbers.
9.5 x 106 +18 x 105
Step 2: Exponents are not the same, make them the same.
We have to make any exponent the same as the other.
9.5 x 106 +1.8 x 106
Or
95 x 105 +18 x 105
Take any one form above. Let us take,
95 x 105 +18 x 105
Step 3: As the exponents are the same add the coefficients.
95 + 18 = 113
Step 4: Write the result with the exponent.
113 x 105
It can also be written as.
1.13 x 107
Subtraction of Scientific Notations
For the subtraction of two or more scientific notations, the exponents must be the same then subtract the coefficient, and the power of ten which is the same of all the numbers remains the same. If the exponents are not the same then first of all make the exponents the same then subtract the scientific notations. Let us take some examples in order to understand this concept.
Example 1
Subtract the scientific notations 36.4 x 104 and 16 x 104.
Solution
Step 1: Put a subtraction sign between the numbers.
36.4 x 104 – 16 x 104
Step 2: As the exponents are same, subtract the coefficients.
36.4 – 16 = 20.4
Step 3: Write the result with the exponent.
20.4 x 104
It can also be written as.
2.04 x 10
Example 2
Subtract the scientific notations 59.5 x 106 and 19 x 105.
Solution
Step 1: Put a subtraction sign between the numbers.
59.5 x 106 – 19 x 105
Step 2: Exponents are not same, make them same.
We have to make any exponent the same as the other.
59.5 x 106 – 1.9 x 106
Or
595 x 105 – 19 x 105
Take any one form above. Let us take,
595 x 105 – 19 x 105
Step 3: As the exponents are same, subtract the coefficients.
595 – 19 = 576
Step 4: Write the result with the exponent.
576 x 105
It can also be written as.
5.76 x 107
Multiplication of Scientific Notations
For the multiplication of two or more scientific notations simply multiply the coefficients and add the exponents.
Let us take an example in order to understand this concept.
Example
Multiply the scientific notations 6 x 104 and 15 x 105.
Solution
Step 1: Put multiplication sign between the numbers.
6 x 104 x 15 x 105
Step 2: multiply the coefficients.
6 x 15 = 90
Step 3: Add the exponents.
104 x 105 = 104+5 = 109
Step 4: Write the result with the exponent.
90 x 109
It can also be written as.
9.0 x 1010
Division of Scientific Notations
For the division of two or more scientific notations simply divide the coefficients and subtract the exponents.
Let us take an example in order to understand this concept.
Example
Divide the scientific notations 600 x 109 and 15 x 104.
Solution
Step 1: Put a division sign between the numbers.
600 x 109 / 15 x 104
Step 2: divide the coefficients.
600 / 15 = 40
Step 3: subtract the exponents.
109 / 104 = 109-4 = 105
Step 4: Write the result with the exponent.
40 x 105
It can also be written as.
4.0 x 106
Summary
Scientific notation is also referred to as standard form. The work of both terms is the same as providing the easiest way to write a number. We can convert a larger number or a smaller number into standard form and the addition, subtraction, multiplication, or division can be done easily of two or more scientific notations.
Scientific notation can be converted into standard form by placing the decimal point after the first non-zero digit.
