How to Read and Write Scientific Notation

In chemistry, physics, astronomy, and computer engineering, scientists routinely handle quantities so vast or so minuscule that writing them in standard decimal form is nearly impossible.

For example, the mass of the Earth is approximately 5,972,000,000,000,000,000,000,000 kilograms, while the mass of an electron is approximately 0.00000000000000000000000000000091093837 kilograms. Writing and typing these long strings of zeros is tedious and practically guarantees transcription errors.

Scientific Notation solves this challenge by leveraging powers of 10 to write any real number compactly and clearly. In this guide, you will learn the anatomy of scientific notation, the exact step-by-step rules to convert to and from decimal form, how to perform basic arithmetic on exponential numbers, and the difference between scientific and engineering notation.

1. The Anatomy of Scientific Notation

Every number in standard scientific notation is written in the exact form:

Where:

• a is the Mantissa (or Significand): A real number whose absolute value must strictly satisfy 1 ≤ |a| < 10. In other words, there must be exactly one non-zero digit to the left of the decimal point.
• 10 is the Base: Representing the decimal base-10 numerical system.
• b is the Exponent: An integer (positive, zero, or negative) indicating how many places the decimal point was shifted.

2. How to Write Numbers Greater Than 10 (Positive Exponents)

When converting a number greater than 10 into scientific notation, the exponent will always be positive.

Step-by-Step Procedure:

1. Identify the starting position of the decimal point. (If the number is an integer, the decimal point sits at the very right).
2. Shift the decimal point to the left until only one non-zero digit remains to its left. This is your mantissa a.
3. Count the number of places you shifted. That count becomes your positive exponent b.
4. Write the result as a × 10^b.

Worked Example: Convert 4,750,000 to Scientific Notation

• Original number: 4,750,000.
• Shift decimal left 6 places: 4.750000 → Mantissa a = 4.75.
• Number of shifts: 6 → Exponent b = 6.
• Scientific notation: 4.75 × 10^6.

3. How to Write Numbers Less Than 1 (Negative Exponents)

When converting a small fractional decimal (between 0 and 1), the exponent will always be negative.

Step-by-Step Procedure:

1. Locate the decimal point.
2. Shift the decimal point to the right until it sits immediately after the first non-zero digit. This is your mantissa a.
3. Count the number of places shifted. That count becomes your negative exponent −b.
4. Write the result as a × 10^-b.

Worked Example: Convert 0.000382 to Scientific Notation

• Original decimal: 0.000382
• Shift decimal right 4 places: 3.82 → Mantissa a = 3.82.
• Number of shifts: 4 → Exponent b = −4.
• Scientific notation: 3.82 × 10^-4.

4. How to Convert Back to Standard Decimal Form

To expand scientific notation back into an everyday decimal:

• If the exponent is positive (+b): Shift the decimal point b places to the right, adding trailing zeros as place holders. (e.g. 2.3 × 10^4 = 23,000).
• If the exponent is negative (−b): Shift the decimal point b places to the left, adding leading zeros. (e.g. 5.1 × 10^-3 = 0.0051).

5. Arithmetic Rules for Scientific Notation

Multiplication Rule: Multiply Mantissas, Add Exponents

To multiply two scientific numbers, multiply their mantissas and add their powers of 10:

(2.0 × 10^3) × (4.0 × 10^5) = (2.0 × 4.0) × 10^(3 + 5) = 8.0 × 10^8

Note: If the resulting mantissa is 10 or greater (e.g. 5 × 4 = 20), re-normalize: 20 × 10^7 = 2.0 × 10^8.

Division Rule: Divide Mantissas, Subtract Exponents

To divide two scientific numbers, divide the mantissas and subtract the divisor exponent from the dividend exponent:

(9.0 × 10^7) ÷ (3.0 × 10^2) = (9.0 ÷ 3.0) × 10^(7 − 2) = 3.0 × 10^5

Addition and Subtraction Rule: Match Exponents First

You cannot directly add or subtract scientific notation numbers unless their powers of 10 are identical! Shift the decimal point of the smaller number until its exponent matches the larger number, then add or subtract mantissas.

6. Scientific Notation vs. Engineering Notation

7. Summary and Related Tools

Mastering scientific notation unlocks easy reading of astronomy, physics, and microscopic biological data. Remember: shift left for positive exponents, shift right for negative exponents, and always ensure exactly one non-zero digit precedes the decimal point.

Use our interactive tools to convert and calculate scientific notation effortlessly:

• Scientific Notation Converter
• Scientific Notation Calculator (+, −, ×, ÷)