On This Page
- The a × 10n rule
- What the calculator does
- Converting in both directions
- The four operations at a glance
- Significant figures
- Five problems, worked
- How the calculator is verified
- FAQ
- References
The a × 10n Rule
A number in scientific notation has a coefficient a with 1 ≤ |a| < 10, times 10 raised to a whole-number exponent n. 450,000,000 becomes 4.5 × 108 and 0.0000045 becomes 4.5 × 10−6: easier to read, compare and type. 45 × 103 breaks the rule and normalizes to 4.5 × 104. Physics and astronomy depend on it. Light travels at 3 × 108 m/s, a proton weighs 1.67 × 10−27 kg, and Andromeda is about 2.4 × 1022 m away.
What the Calculator Does
Convert number accepts a plain decimal (0.00045), e-notation (4.5e-4) or the written form (4.5 × 10^-4). It returns scientific notation, standard form, engineering notation, e-notation and the significant-figure count, all updating as you type. Add / Subtract, Multiply and Divide work on two numbers and show every step, including any renormalizing.
Converting in Both Directions
To standard form, move the decimal point by the exponent: right for a positive exponent (6.02 × 103 = 6,020), left for a negative one (6.02 × 10−3 = 0.00602).
To scientific notation, move the decimal until one non-zero digit sits to its left and count the moves. Moving left gives a positive exponent and moving right gives a negative one. 0.0000735 needs five moves right, so it is 7.35 × 10−5. Scientific notation practice problems has more of these, worked step by step.
Engineering notation keeps the exponent to multiples of three, so it lines up with SI prefixes: kilo (103), mega (106), milli (10−3), micro (10−6). 0.00045 becomes 450 × 10−6, that is 450 micro-units. E-notation (4.5e-4) is how code and calculator screens show it.
The Four Operations at a Glance
| Operation | Rule | Example |
|---|---|---|
| Multiply | Multiply coefficients, add exponents | (3.0 × 104)(4.5 × 102) = 13.5 × 106 = 1.35 × 107 |
| Divide | Divide coefficients, subtract exponents | (7.2 × 108) ÷ (1.8 × 103) = 4.0 × 105 |
| Add | Rewrite at the larger exponent, then add coefficients | 5.4 × 103 + 0.30 × 103 = 5.70 × 103 |
| Subtract | Same alignment, then subtract | 5.0 × 106 − 0.2 × 106 = 4.8 × 106 |
After any operation, check the coefficient is still between 1 and 10. A product like 13.5 moves up an exponent, and a quotient like 0.5 × 104 becomes 5.0 × 103. The same align-then-combine habit applies when adding large distances.
Significant Figures
In scientific notation every digit of the coefficient is significant, which clears up ambiguity. 4500 could have two, three or four significant figures, but 4.5 × 103 has exactly two.
- Non-zero digits always count (3.45 has 3). Leading zeros never count (0.0045 has 2). Trailing zeros after a decimal point do count (3.450 has 4).
- When multiplying or dividing, keep the fewest significant figures of any input. When adding or subtracting, round to the least precise decimal place.
- Subtracting close numbers throws precision away. 1.000 × 104 − 1.001 × 104 = −1 × 101 (−10), a single significant figure from two four-figure inputs.
Reporting 1.234567 × 10−3 mol from a 3-figure instrument overstates your precision; 1.23 × 10−3 is correct. Round the same way before working out a percentage change between two measurements, so errors do not compound. For rates such as rise over run in lab data, the Slope Calculator takes the numbers once they are in standard form.
Five Problems, Worked
- 93,000,000 (miles from Earth to the Sun): 7 moves left gives 9.3 × 107.
- 0.00000602: 6 moves right gives 6.02 × 10−6, 3 significant figures.
- (3.0 × 104)(4.5 × 102): 13.5 × 106 normalizes to 1.35 × 107 = 13,500,000.
- (8.4 × 106) ÷ (2.0 × 102): 4.2 × 104 = 42,000.
- 3.0 × 105 + 4.0 × 103: rewrite the second as 0.04 × 105, so the sum is 3.04 × 105.
How the Calculator Is Verified
The operation dropdown itself is rebuilt by syncOpToMode() every time the mode changes: selecting "add" hides every option except add and subtract, selecting "multiply" or "divide" hides everything but that single operation, so there is no dropdown state where an operation exists that the current mode does not run. Addition and subtraction align both numbers to the larger exponent before combining coefficients; multiplication and division add or subtract the exponents directly and multiply or divide the coefficients, two different mechanisms for the two operation types. After each operation the code checks whether the result's exponent still needs normalizing: a coefficient landing outside 1 to 10 triggers an explicit renormalization step in the displayed output, and dividing by zero returns its own error step instead of Infinity.
Frequently Asked Questions
How to do scientific notation on a calculator, and how do you enter it?
On a physical calculator, the key is usually labelled EE, EXP, or ×10x. Press the coefficient digits, then EE (or EXP), then the exponent. Do not type "×10" separately. For example, to enter 3.2 × 105, press 3 . 2 EE 5. On this online calculator, type the number in any accepted format: plain decimal, e-notation (3.2e5), or the written form (3.2 × 10^5).
How do you do scientific notation on a calculator when the exponent is negative?
After pressing EE or EXP, press the +/− or (-) key before typing the exponent digits. On this calculator, type a minus sign directly: 3.2e-5 or 3.2 × 10^-5 both parse correctly.
How to put scientific notation into a calculator when the number has many zeros
Count the zeros to determine the exponent rather than typing all the zeros. 0.000000035 has seven leading zeros after the decimal, so the exponent is −8 (the digit 3 is the eighth decimal place). Type 3.5e-8 instead of 0.000000035. This avoids keying errors and the calculator handles the rest.
References
- NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities: the authoritative US federal reference for scientific notation and SI unit conventions, including guidance on the use of powers of ten.
- Khan Academy: Scientific Notation Review: a clear walkthrough of writing, reading, and operating with scientific notation, covering positive and negative exponents.
- BIPM SI Brochure, 9th Edition: the international reference for the International System of Units, including the SI prefixes and their correspondence with powers of ten used in engineering notation.