What is scientific notation on a calculator?
Short answer. Scientific notation writes any number as a coefficient of at least 1 and less than 10, times a power of ten. Calculators print that power as E, so 6.42E6 is 6.42 × 10⁶, or 6,420,000. The exponent counts decimal places: positive moves the point right, negative moves it left.
Every number, however long, is a short number and a size. Scientific notation splits it into exactly those two parts: the digits that carry the meaning, and a power of ten that says where the decimal point belongs. Calculators use it because they run out of screen, and scientists use it because 6.42 is easier to think about than 6,420,000.
The one rule: the coefficient sits between 1 and 10
The number in front must be at least 1 and less than 10. Exactly one non-zero digit before the decimal point, no more and no fewer. Everything else is bookkeeping.
6420000 → 6.42 × 10⁶ one digit before the point ✓ 0.00081 → 8.1 × 10⁻⁴ one digit before the point ✓ 64.2 × 10⁵ right value, wrong form (two digits in front) 0.642 × 10⁷ right value, wrong form (a zero in front)
All four lines above are the same quantity, 6,420,000, worn four ways. Only the first is in scientific notation. The other two are what a half finished conversion looks like, and fixing them is one nudge: move the point one place and change the exponent by one to compensate.
Turning a number into scientific notation
Put the decimal point after the first non-zero digit, then count how far it travelled. That count is the exponent. Left is positive, right is negative.
6420000. 6.420000 six places left → 6.42 × 10⁶
←←←←←←
0.00081 8.1 four places right → 8.1 × 10⁻⁴
→→→→
The sign trips people up, so anchor it on size rather than on direction: numbers bigger than 10 get a positive exponent, numbers smaller than 1 get a negative one. A number already between 1 and 10 gets an exponent of zero, because 10⁰ is 1 and nothing needs to move.
Going the other way is the same walk in reverse. Read the exponent, move the point that many places, and fill the gaps with zeros. The digits themselves never change. If your answer contains a digit that was not in the original, you did something other than move a decimal point.
| Scientific notation | On the display | Written out |
|---|---|---|
| 6.42 × 10⁶ | 6.42E6 | 6,420,000 |
| 1.6 × 10³ | 1.6E3 | 1,600 |
| 4.07 × 10⁵ | 4.07E5 | 407,000 |
| 7 × 10⁰ | 7E0 | 7 |
| 8.1 × 10⁻⁴ | 8.1E-4 | 0.00081 |
| 2.5 × 10⁻⁷ | 2.5E-7 | 0.00000025 |
That middle column is the whole reason the letter turns up in your results. E is not a variable and not an error, it is just a typed stand-in for "times ten to the power of". There is a longer tour of it in what E means on a calculator, including the unrelated lowercase e.
Multiplying: coefficients multiply, exponents add
This is where the notation earns its keep. You never write the zeros, so you never miscount them.
(3 × 10⁴) × (2 × 10⁵) 3 × 2 = 6, 4 + 5 = 9 → 6 × 10⁹
(6 × 10⁸) ÷ (1.5 × 10³) 6 ÷ 1.5 = 4, 8 − 3 = 5 → 4 × 10⁵
(5 × 10⁶) × (4 × 10³) 5 × 4 = 20, 6 + 3 = 9 → 20 × 10⁹
coefficient too big, shift one
→ 2 × 10¹⁰
The third line is the case worth practising. Nothing is wrong with 20 × 10⁹, it is simply not normalized, and a calculator will hand it back to you as 2E10. Move the point one place left and add one to the exponent.
Adding: the exponents have to match first
Addition has no shortcut, because you cannot add hundreds of thousands to tens of thousands without lining them up. Rewrite the smaller number so it shares the larger exponent, then add the coefficients and leave the power of ten alone.
4.2 × 10⁵ + 3 × 10⁴ rewrite 3 × 10⁴ = 0.3 × 10⁵ add 4.2 + 0.3 = 4.5 result 4.5 × 10⁵ = 450000 check 420000 + 30000 = 450000 ✓
Note that the rewritten 0.3 × 10⁵ breaks the coefficient rule on purpose. That is allowed mid working. Only the answer has to come back into normal form.
Typing one in, and the mistake everybody makes
You do not type the letter E. There is a key for it, marked EE, EXP or ×10ˣ, and it means "and here comes the exponent". For 6.42 × 10⁶ you press 6.42, the exponent key, then 6.
right 6.42 EE 6 = 6420000
wrong 6.42 × EE 6 = 6.42 × 10 × 10⁶ = 64200000
ten times too big
The exponent key already contains the multiplication. Pressing multiply as well multiplies by an extra ten, and the answer is out by a factor of ten in a way that looks perfectly plausible on the screen.
Engineering notation: the same idea in threes
Engineering notation is scientific notation with one extra restriction. The exponent must be a multiple of three, which lets the coefficient run from 1 up to 1000. The point is that the exponents then line up with the metric prefixes, so the display reads straight off as kilo, milli, micro.
47000 ohms scientific 4.7 × 10⁴
engineering 47 × 10³ = 47 kΩ
0.000022 farad scientific 2.2 × 10⁻⁵
engineering 22 × 10⁻⁶ = 22 µF
Same values, different parking. If your calculator has an ENG key, that is what it toggles, and it is the reason an answer can suddenly show a coefficient bigger than 10 without being wrong.
What the notation quietly rounds off
Scientific notation is a display format, and display formats keep a fixed number of significant figures. Six figures is common. That is enough for almost everything and not enough for everything:
true value 123456789 shown 1.23457 × 10⁸ six significant figures written out 123457000 lost 211
Usually that is the correct trade. Nobody needs nine digits of a measurement that was accurate to three. It starts to matter when the rounded number goes back in as the input to the next calculation, because then the error travels. The size of that gap is what the percent error formula puts a number on, and here it is 211 ÷ 123456789 × 100, which is about 0.00017%.
The deeper version of the same problem is that most calculators do not store your number exactly in the first place. They store a binary approximation, which is why 0.1 + 0.2 comes out as 0.30000000000000004 on machine after machine. Prism computes with exact rational arithmetic, so 0.1 + 0.2 is 0.3, and the digits that scientific notation leaves off the screen are still intact underneath it rather than gone.
Scientific mode, with EE where your thumb expects it
Prism has the exponent key, powers and roots, and it does not round the answer before you have seen it. Scientific mode is free forever, no subscription. See the app.