Scientific Notation Fundamentals

What is scientific notation?

Scientific notation provides a compact way to write very large or very small numbers. Understanding scientific notation helps you grasp how floating-point numbers work - and crucially, their limitations.

Scientific notation follows this pattern: significand x 10^exponent. For example, 2.5 x 10^3 equals 2,500.

In C++, we use 'e' to represent "times 10 to the power of":

• 2.5e3 means 2.5 x 10^3 = 2,500
• 1.898e27 means 1.898 x 10^27
• 3e-2 means 3 x 10^-2 = 0.03

Significant digits

The digits in the significand (the part before 'e') are called significant digits (or significant figures). More significant digits means greater precision.

• 3.14 has 3 significant digits
• 3.14159 has 6 significant digits

Converting to scientific notation

Follow this procedure:

1. Start with exponent zero
2. Slide the decimal point until exactly one non-zero digit appears left of it
   • Each leftward movement increases the exponent by 1
   • Each rightward movement decreases the exponent by 1
3. Remove leading zeros
4. For trailing zeros:
   • Keep them if the original number had a decimal point
   • Remove them if the original number had no decimal point

Examples:

750.820 -> 7.50820e2 (6 significant digits)
0.0056300 -> 5.6300e-3 (5 significant digits)
85070 -> 8.507e4 (4 significant digits, trailing zero dropped)
85070. -> 8.5070e4 (5 significant digits, trailing zero kept due to decimal)

Why trailing zeros matter

When a number has a decimal point, trailing zeros indicate precision:

• 125.0g means the measurement is precise to 0.1g
• 125.000g means the measurement is precise to 0.001g

In scientific notation:

• 125.0g = 1.250e2 (4 significant digits)
• 125.000g = 1.25000e2 (6 significant digits)

Now that we've covered scientific notation, we're ready to explore floating-point numbers.