Scientific Notation Formula: Definition, Rules & Solved Examples

Scientific Notation is a way of expressing numbers that are either too large or too little to be expressed in decimal form. The usual form when writing in scientific notation is N x 10m . A scientific notation is a method of writing a number, an equation, or an expression in a way that adheres to a set of rules. While expressing the number in Scientific Notation, the exponent is positive if the number is very large and it is negative if the number is very small. When expressing huge numbers in scientific notation, we utilize positive exponents for base 10. We utilize negative exponents for base 10 when expressing any little values in scientific notation.

Key Terms: Scientific Notation, Exponents, Decimals, Negative Exponents, Standard Form, Integers, Power, Scientific Form, Positive Exponents

What is Scientific Notation?

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Scientific Notation can be defined as a method through which we express numbers that are either too large or too little to be expressed in decimal form. In the United Kingdom, it is also known as 'Scientific Form,' and it is often used by scientists, mathematicians, and engineers for difficult computations involving large numbers. It's generally referred to as "SCI" display mode on scientific calculators.

All numbers in scientific notation are written in the generic form as N x 10m

Where N is an integer between 1 and 10, but not 10 itself, and m is any positive integer between 1 and 10. (positive or negative number). The order of magnitude is the integer m, and the significand is the real number N. In scientific notation, the digit term denotes the number of significant figures in the number. Only the exponential term places the decimal point.

Read More: Uses of Exponents to Express Small Numbers in Standard Form

• Example 1: 4.66 x 10⁶ = 4660000

There are only three major figures in this number. The zeros are only placeholders and have no significance.

• Example 2: 5.3 x 10^-4 = 0.00053

There are two key figures in this number. The zeros are just placeholders for the rest of the numbers.

Scientific Notation

Formula for Scientific Notation

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Writing a high number in its number form, such as 8.6 billion, is not only ambiguous but also time-consuming, and there is a probability that we will write a few zeros less or more when doing so. We utilize scientific notation to represent very large or very small numbers in a straightforward manner. The following is a general representation of scientific notation or a scientific notation formula:

a × 10b ; 1 ≤ a < 10

1 x a x 10b;

'a' stands for any number from 0 to 10 (higher than 0 and less than 10)

Read More: Difference Between Power And Exponent

Rules for Scientific Notation

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When writing numbers in scientific notation, we must adhere to the following guidelines:

• The scientific notation is divided into two parts: the first is merely the digits, with the decimal point after the first digit, and the second is multiplication with 10 to a power number of the decimal point, which places the decimal point where it belongs.
• The decimal point must move to the left if the given number is higher than 1 and multiples of 10, and the power of 10 will be positive.

For example, the scientific notation for 8000 is 8 x 10³.

If the provided number is less than one in decimal form, the decimal point must be moved to the right, and the power of ten becomes negative.

For instance, the scientific notation for 0.008 is 8 x 0.001 or 8 x 10^-3.

Read More: Integers as Exponents

Standard Form to Scientific Notation

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Here are some examples that illustrate how to express a number in standard form to scientific notation.

Example 1: Express 412,000,000,000 in scientific notation.

• Step 1: To make a new number from 1 to 10, move the decimal position to the left.

Because 412,000,000,000 is a whole number, the decimal point will be appended at the end: 412,000,000,000. So, N = 3.12 is obtained.

• Step 2: Determine the exponent, which is the number of times the decimal has been moved.

The decimal was shifted 11 times, and the exponent is positive since the decimal was moved to the left. As a result, m = 11 and you get 10¹¹.

• Step 3: Now, you have to substitute the value of N and m in the general form of scientific notation

N x 10m

3.12 x 10¹¹

Hence 3.12 x 10¹¹ is in scientific form

Standard Form & Scientific Notation

Example 2: Write 0.00000041 in scientific notation.

• Step 1: The first step involves moving the decimal place to the right to create a new number from 1 up to 10.

So we get N = 4.1.

• Step 2: Now, determine the exponent, it will be the number of times you moved the decimal.

Here, the decimal is moved 7 times and because you moved the decimal to the right, the exponent is negative. Hence, m = –7, and we get 10^-7

• Step 3: The last step involves substituting the value of N and m in the general form of scientific notation

N x 10m

4.1 x 10^-7

As a result, 4.1 is scientific.

Read More: Powers and Negative Exponents

Positive and Negative Exponents

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When expressing huge numbers in scientific notation, we utilize positive exponents for base 10.

5000000, for example, equals 5 x 10⁶, where 6 is the positive exponent.

We utilize negative exponents for base 10 when expressing any little values in scientific notation.

0.00004 = 4 x 10^-5, where the negative exponent is -5.

Positive and Negative Exponents

Things to Remember

• Scientific Notation is the method of expressing numbers that are either too large or too little to be expressed in decimal form.
• All numbers in scientific notation are written in the generic form as N x 10m
• A general representation of scientific notation, or a scientific notation formula: a × 10b; 1 ≤ a < 10
• The scientific notation is divided into two parts: the first is merely the digits, with the decimal point after the first digit, and the second is multiplication with 10 to a power number of the decimal point, which places the decimal point where it belongs.
• When expressing huge numbers in scientific notation, we utilize positive exponents for base 10. 5000000, for example, equals 5 x 10⁶, where 6 is the positive exponent.
• We utilize negative exponents for base 10 when expressing any little values in scientific notation. 0.00004 = 4 x 10^-5, where the negative exponent is -5