Convert numbers between scientific notation, standard form, e-notation, engineering notation, and word form
enter a number or scientific notation
Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in standard decimal form. It's widely used in science, engineering, and mathematics to represent very large or very small quantities in a compact, standardized format.
A number in scientific notation is written as the product of two factors:
a × 10ⁿ
where a is the coefficient (1 ≤ |a| < 10)
and n is the exponent (an integer)
For example, 345,600,000,000 can be written as 3.456 × 10¹¹ in scientific notation. This makes it much easier to read, write, and calculate with extremely large or small numbers.
The coefficient is the number that appears before the multiplication sign. It must be greater than or equal to 1 and less than 10.
In 3.456 × 10¹¹
Coefficient = 3.456
The exponent indicates how many places to move the decimal point. Positive exponents mean large numbers, negative mean small numbers.
In 3.456 × 10¹¹
Exponent = 11
Example:
345,600,000,000
→ Move decimal 11 places left: 3.456
= 3.456 × 10¹¹
Example:
0.00000456
→ Move decimal 6 places right: 4.56
= 4.56 × 10⁻⁶
Uses the multiplication symbol (×) and superscript for exponent.
3.456 × 10¹¹
Computer/calculator format using 'e' or 'E' to represent "× 10^".
3.456e11 or 3.456E+11
Similar to scientific notation but exponents are always multiples of 3, aligning with metric prefixes.
345.6 × 10⁹ (giga-)
The regular decimal representation of the number.
345,600,000,000
Engineering notation uses metric prefixes for common powers of 10:
| Prefix | Symbol | Power of 10 | Decimal |
|---|---|---|---|
| tera- | T | 10¹² | 1,000,000,000,000 |
| giga- | G | 10⁹ | 1,000,000,000 |
| mega- | M | 10⁶ | 1,000,000 |
| kilo- | k | 10³ | 1,000 |
| — | — | 10⁰ | 1 |
| milli- | m | 10⁻³ | 0.001 |
| micro- | μ | 10⁻⁶ | 0.000001 |
| nano- | n | 10⁻⁹ | 0.000000001 |
| pico- | p | 10⁻¹² | 0.000000000001 |
Convert: 5,870,000,000 to scientific notation
Step 1: Move decimal left until between 1 and 10
5,870,000,000 → 5.87
Step 2: Count decimal places moved
Moved 9 places left
Step 3: Write in scientific notation
= 5.87 × 10⁹
Convert: 0.00000234 to scientific notation
Step 1: Move decimal right until between 1 and 10
0.00000234 → 2.34
Step 2: Count decimal places moved
Moved 6 places right (negative exponent)
Step 3: Write in scientific notation
= 2.34 × 10⁻⁶
Convert: 7.89 to scientific notation
Step 1: Already between 1 and 10
7.89 (no movement needed)
Step 2: Decimal moved 0 places
Exponent = 0
Step 3: Write in scientific notation
= 7.89 × 10⁰ = 7.89
Multiply the coefficients and add the exponents:
(2 × 10³) × (3 × 10⁵)
= (2 × 3) × 10⁽³⁺⁵⁾
= 6 × 10⁸
Divide the coefficients and subtract the exponents:
(8 × 10⁶) ÷ (2 × 10²)
= (8 ÷ 2) × 10⁽⁶⁻²⁾
= 4 × 10⁴
First make the exponents the same, then add/subtract coefficients:
(3 × 10⁴) + (5 × 10³)
= (3 × 10⁴) + (0.5 × 10⁴)
= (3 + 0.5) × 10⁴
= 3.5 × 10⁴
Scientific notation (a × 10ⁿ) expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10, while standard form is the regular decimal representation. For example, 5,000 is standard form, while 5 × 10³ is scientific notation.
Use scientific notation when dealing with very large numbers (like astronomical distances), very small numbers (like atomic measurements), when precision matters, in scientific calculations, or when comparing numbers of vastly different magnitudes.
E-notation (like 3.456e11) is the computer/calculator format for scientific notation. The "e" stands for "exponent" and means "times 10 to the power of." So 3.456e11 is the same as 3.456 × 10¹¹. It's just a different way to write the same thing.
Engineering notation is similar to scientific notation, but the exponent is always a multiple of 3 (like 10³, 10⁶, 10⁹). This aligns with metric prefixes (kilo, mega, giga) and is commonly used in engineering and electronics.
Most calculators have an "EE" or "EXP" button for scientific notation. To enter 3.456 × 10¹¹, type: 3.456, press EE or EXP, then type 11. Don't manually type "× 10^" as this will give incorrect results.
Yes, both the coefficient and the exponent can be negative. A negative coefficient means the number itself is negative (like -3.5 × 10⁴ = -35,000), while a negative exponent indicates a small number less than 1 (like 3.5 × 10⁻⁴ = 0.00035).
The order of magnitude is the exponent in scientific notation, representing the power of 10. It gives a rough sense of the scale of a number. For example, 10⁶ (million) and 10⁹ (billion) differ by 3 orders of magnitude, meaning one is 1,000 times larger than the other.
The number of significant figures depends on the precision of your measurement or calculation. In scientific notation, all digits in the coefficient are significant. For example, 3.456 × 10¹¹ has 4 significant figures, while 3.5 × 10¹¹ has 2 significant figures.