Standard Form Converter and Calculator

Using the three modes

The first mode takes an ordinary number — 384000, or 0.00045 — and rewrites it as a × 10ⁿ. The second goes the other way: give it the mantissa and the index and it writes the digits out in full, which is the quickest way to see what 6.02 × 10²³ actually means. The third does arithmetic on two numbers in standard form with any of the four operations.

The significant-figures selector rounds the mantissa only, never the index. Where the exact value and the rounded value differ, both are shown, so you can see what the rounding has cost before you copy an answer down.

Every mode also shows the two forms you will meet outside a maths lesson: the keys you press on a calculator, and the E notation a spreadsheet or a piece of lab software uses to display the same number.

How the conversion is done

The index is read straight off the digit positions rather than calculated with a logarithm. That is the safer method for a computer as well as for a student: log₁₀(1000) can come back from floating-point arithmetic as 2.9999999999999996, and rounding that to 3 hides the problem instead of avoiding it. Counting places is exact.

For arithmetic the two halves are handled separately. The powers of ten follow the index laws — 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ for multiplication, 10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ for division — and the mantissas are multiplied or divided as ordinary decimals. Addition and subtraction cannot work that way at all, so both numbers are first rewritten with the same index, in the same spirit as putting two fractions over a common denominator.

Whatever the operation, the mantissa can land outside 1 ≤ a < 10, and the answer is then re-normalised in a step of its own. 5 × 10³ multiplied by 4 × 10⁵ gives 20 × 10⁸, which is a correct value and an unacceptable answer; the tool shows it becoming 2 × 10⁹ rather than tidying it away in silence.

What it will not do

It will not write zero in standard form. There is no mantissa between 1 and 10 and no index that produces zero, so the tool says so rather than returning 0 × 10⁰, which is not standard form and would teach the wrong thing.

It will not accept E notation as an input in the conversion mode. Typing 3E8 into a box labelled "ordinary number" is a sign that the number is already in standard form, and the tool points you at the mode that expands it instead of guessing what you meant.

It will not silently correct a mantissa outside 1 ≤ a < 10. Enter 12 × 10⁵ and the working states plainly that this is a correct value but not standard form, then shows the point moving one place and the index rising by one. That single step is where the accuracy mark usually goes.

Where marks are lost

The mantissa outside 1 to 10 is the classic. 12 × 10⁵ is exactly the same quantity as 1.2 × 10⁶, but only the second is standard form, and examiners mark the notation as well as the value. The same goes for 0.4 × 10⁻³, which needs the point moved the other way to become 4 × 10⁻⁴.

The second is the calculator. The ×10ˣ key — EXP on older models — already contains the times ten. Pressing 3, then ×, then 10, then ×10ˣ, then 8 enters 3 × 10 × 10⁸, which is ten times too big and looks perfectly reasonable on the screen. Press the mantissa, the ×10ˣ key, then the index, and nothing else.

The third is adding indices when the numbers are being added. 3 × 10⁸ plus 4 × 10⁶ is not 7 × 10¹⁴ and not 7 × 10⁸; the second number has to be rewritten as 0.04 × 10⁸ first. And a negative index means a small positive number, not a negative one: 1.6 × 10⁻¹⁹ is a very small positive charge, not a debt.

• Leaving the mantissa outside 1 ≤ a < 10, as in 12 × 10⁵
• Pressing × 10 as well as the ×10ˣ or EXP key
• Adding or subtracting indices when the numbers themselves are being added
• Reading a negative index as a negative number
• Rounding the index instead of the mantissa when asked for significant figures