What is Scientific Notation?
Scientific notation is a method of expressing numbers that are too big or too small to be conveniently written in decimal form. It is also referred to as ‘scientific form’ in Britain, It is commonly used by scientists, mathematicians, and engineers for complex calculations with lengthy numbers. On scientific calculators, it is usually known as "SCI" display mode.
To write in scientific notation, follow the general form
where N is a number between 1 and 10, but not 10 itself, and m is any integer (positive or negative number).
In this article let us discuss what is the scientific notation, the definition of scientific notation, a scientific notation to standard form, and scientific notation examples.
Scientific Notation Definition
Scientific notation is a method of expressing numbers in terms of a decimal number between 1 and 10, but not 10 itself multiplied by a power of 10.
The general for of scientific notation is
In scientific notation, all numbers are written in the general form as
N × 10m
N times ten raised to the power of m, where the exponent m is an integer, and the coefficient N is any real number. The integer m is called the order of magnitude and the real number N is called the significand.
The digit term in the scientific notation indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,
4660000 = 4.66 x 106
This number only has 3 significant figures. The zeros are not important, they are just placeholders. As another example,
0.00053 = 5.3 x 10-4
This number has 2 significant figures. The zeros are only placeholders.
Scientific Notation Rules:
While writing the numbers in the scientific notation we have to follow certain rules they are as follows:
The scientific notations are written in two parts one is the just the digits, with the decimal point placed after the first digit, followed by multiplication with 10 to a power number of decimal point that puts the decimal point where it should be.
If the given number is greater than 1 and multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive
Example: Scientific notation for 8000 will be 8 × 103.
If the given number is smaller than 1 means in the form of decimal numbers, then the decimal point has to move to the right, and the power of 10 will be negative.
Example: Scientific notation for 0.008 will be 8 × 0.001 or 8 × 10-3.
Standard Form to Scientific Notation
To write 412,000,000,000 in scientific notation:
Use the general form N x 10m
Step1: Move the decimal place to the left to create a new number from 1 upto 10.
412,000,000,000 is a whole number, the decimal point will be given at the end of the number: 412,000,000,000.
So, you get N = 4.12.
Step2: Determine the exponent, it will be the number of times you moved the decimal.
Here, you moved the decimal 11 times and because you moved the decimal to the left, the exponent is positive. Therefore, m = 11, and so you get 1011
Step 3: Substitute the value of N and m in the general form of scientific notation
N x 10m
4.12 x 1011
Hence 4.12 x 1011 is in scientific form
Now write .00000041 in Scientific Notation.
Step 1: Move the decimal place to the right to create a new number from 1 upto 10.
So we get N = 4.1.
Step 2: Determine the exponent,it will be the number of times you moved the decimal.
Here, you moved the decimal 7 times and because you moved the decimal to the right, the exponent is negative. Therefore, m = –7, and so you get 10-7
Step 3: Substitute the value of N and m in the general form of scientific notation
N x 10m
4.1 x 10-7
Hence 4.1 x 10-7 is in scientific form.
Similarly, scientific notations can be converted to standard form.
Let us understand this with help of examples.
Scientific Notation to Standard Form
To write 5.56 × 104 in standard form
Given that 5.56 × 104 is in scientific notation.
Here Exponent m = 4
Since the exponent is positive we need to move the decimal point to 4 places to the right.
Therefore,
5.56 × 104
= 5.56 × 10000
= 55,600.
So, the standard form is 55,600.
Solved Examples
Change scientific notation to standard form of 1.86 × 107
Solution: Given that 1.86 × 107 is in scientific notation.
Here Exponent m = 7
Since the exponent is positive we need to move the decimal point to 7 places to the right.
Therefore,
1.86 × 107
= 1.86 × 10000000
= 1,86,00,000.
So, the standard form is 1,86,00,000.
Convert 0.0000078 into scientific notation.
Solution: Given that 0.0000078 is in standard form
To convert it in scientific notation use the general form
N x 10m
Move the decimal point to the right of 0.0000078 up to 6 places.
We get N = 7.8
Since the numbers are less than 1 we move the decimal point to the right, So we use a negative exponent here.
We get m = -6
Put the value of N and m in general form
Therefore , 0.0000078 = 7.8 × 10-6
7.8 x 10-6 is the scientific notation.
Quiz Time:
Change scientific notation to standard form
1. 6.7 x 106
2. 4.5 x 10-9
Convert into scientific notations
1. 670000000000
2. 0.00000000089
Importance of Scientific Notation
Scientific Notation is a manner in which all scientists easily handle very large numbers or the very small numbers. Any number can be written in scientific notation when it falls between 1 and 10 and is multiplied by a power of 10. It is used globally by engineers, mathematicians and statisticians for important calculations and denotations. It is of great significance for the purpose of representing numbers.
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FAQs on Scientific Notations
1. What is the standard form of scientific notation?
The standard form, or general formula, for scientific notation is a × 10ⁿ. In this expression, 'a' is the coefficient, which must be a number greater than or equal to 1 but less than 10 (1 ≤ a < 10). The term 'n' is an integer that represents the exponent or the power of 10.
2. What are the basic rules for writing any number in scientific notation?
To convert a number into scientific notation, follow these two primary rules:
- Move the decimal point so that it is placed right after the first non-zero digit. This creates the coefficient 'a'.
- Count the number of places the decimal point was moved. This count becomes the exponent 'n'. If the decimal was moved to the left (for large numbers like 58,000), the exponent is positive. If it was moved to the right (for small numbers like 0.0058), the exponent is negative.
3. How do you write a large number like 93,000,000 in scientific notation?
To express 93,000,000 in scientific notation, the decimal point (assumed at the end) is moved 7 places to the left to sit between the 9 and 3. This gives a coefficient of 9.3. Since the decimal was moved 7 places to the left, the exponent is positive 7. Therefore, 93,000,000 is written as 9.3 × 10⁷.
4. How do you write a small decimal like 0.000056 in scientific notation?
To write 0.000056 in scientific notation, you move the decimal point 5 places to the right to place it after the first non-zero digit, 5. This results in a coefficient of 5.6. Because the decimal was moved to the right, the exponent is negative 5. So, 0.000056 is expressed as 5.6 × 10⁻⁵.
5. Why is scientific notation important in subjects like Science and Maths?
Scientific notation is crucial because it provides a compact and standard way to handle extremely large or small numbers. In science, it is used to write values like the distance to a star or the size of a bacterium, which would be cumbersome with long strings of zeros. This standard format simplifies calculations, reduces the risk of errors, and makes it easier to compare the magnitude of different quantities.
6. What is the main difference between scientific notation and standard exponential form?
The key difference is the rule for the coefficient ('a'). In scientific notation, the coefficient must be strictly between 1 and 10 (1 ≤ a < 10). For example, the number 4500 can only be written as 4.5 × 10³ in scientific notation. In general exponential form, you could also write it as 45 × 10² or 0.45 × 10⁴. The strict rule in scientific notation ensures a unique representation for every number, making comparisons straightforward.
7. How do you multiply and divide numbers written in scientific notation?
Operations with scientific notation follow specific rules based on the laws of exponents:
- For multiplication: Multiply the coefficients and add the exponents. For example, (3 × 10⁴) × (2 × 10²) = (3×2) × 10⁴⁺² = 6 × 10⁶.
- For division: Divide the coefficients and subtract the exponents. For example, (8 × 10⁵) ÷ (4 × 10²) = (8÷4) × 10⁵⁻² = 2 × 10³.
8. How does the sign of the exponent (positive or negative) change the value of a number in scientific notation?
The sign of the exponent indicates the number's magnitude. A positive exponent (like in 5 × 10⁶) signifies a large number, as it means multiplying the coefficient by 10 multiple times (5,000,000). A negative exponent (like in 5 × 10⁻⁶) signifies a very small decimal number between 0 and 1, as it means dividing the coefficient by 10 multiple times (0.000005).
9. Why can a number have only one correct form in scientific notation?
A number has only one unique representation in scientific notation because of the strict rule that the coefficient 'a' must be greater than or equal to 1 and less than 10. This constraint prevents multiple valid forms. For example, the number 250 must be 2.5 × 10², not 25 × 10¹ or 0.25 × 10³. This uniqueness creates a universal standard for writing and comparing numbers.
10. How do you convert a number from scientific notation back to its standard form?
To convert from scientific notation (a × 10ⁿ) to standard form, you look at the exponent 'n'. If 'n' is positive, move the decimal point in the coefficient 'a' to the right by 'n' places, adding zeros if necessary. For example, 3.14 × 10⁵ becomes 314,000. If 'n' is negative, move the decimal point to the left by 'n' places. For example, 7.8 × 10⁻⁴ becomes 0.00078.
