How this instrument works
Multiplying two numbers already in mantissa-and-exponent form is a two-step process: multiply the mantissas together, and separately add the exponents together. The powers of 10 combine by simple addition because multiplying powers of the same base adds their exponents — the same rule behind 10²×10³=10⁵.
There is one wrinkle: multiplying two mantissas that are each between 1 and 10 can produce a result of 10 or more, which breaks the rule that a proper mantissa must stay under 10. Whenever that happens, this calculator renormalizes automatically — dividing the oversized mantissa by 10 and bumping the exponent up by one to absorb that extra factor, restoring the standard one-mantissa-under-10 form.
The companion Scientific Notation Converter page on this site handles the earlier step in the pipeline: turning a single ordinary number into this mantissa-and-exponent form in the first place. This page instead assumes both numbers already arrive in that form and focuses purely on the arithmetic of combining them.
- Enter the first number's mantissa and exponent into their fields.
- Enter the second number's mantissa and exponent the same way.
- Read Product mantissa and Product exponent — if the raw mantissa product reached 10 or more, it has already been renormalized for you.
- Multiply the reported mantissa by 10 raised to the reported exponent to confirm it reproduces the correct total.
Worked example — (2.5×10³) times (4×10²)
Multiplying the mantissas gives 2.5×4=10, and adding the exponents gives 3+2=5 — but a mantissa of exactly 10 sits one step too high for the standard form, so this calculator renormalizes it: dividing by 10 gives a mantissa of 1.0, and the exponent absorbs that extra factor of 10, becoming 5+1=6. The final answer is 1.0×10⁶, which checks out: 2,500×400=1,000,000, and 1.0×10⁶ is exactly one million.
A case that needs no renormalizing: (1.2×10²) times (3×10¹) gives a mantissa of 1.2×3=3.6, already comfortably under 10, and an exponent of 2+1=3 — landing directly on 3.6×10³ with no adjustment required.
Questions
Why do the exponents just add together?
Because multiplying powers of the same base adds their exponents — 10^a times 10^b equals 10^(a+b) by the basic rules of exponents, and that rule applies here exactly as it would to any other power of 10.
Why does the mantissa sometimes need renormalizing?
Two numbers each between 1 and 10 can multiply to something as large as just under 100, which no longer fits the rule that a mantissa must stay under 10 — dividing by 10 and shifting that factor into the exponent restores the standard form without changing the actual value represented.
Does this work for dividing scientific-notation numbers too?
This calculator specifically performs multiplication; division follows the same idea in reverse (dividing mantissas, subtracting exponents), but that direction isn't what this particular page computes.
What if one of the exponents is negative?
Addition works identically regardless of sign — a negative exponent added to a positive one can move the result's exponent either up or down, exactly as ordinary signed addition would suggest.
How is this different from the Scientific Notation Converter page on this site?
That page converts a single plain number into mantissa-and-exponent form to begin with; this page instead assumes two numbers already arrive in that form and multiplies them together, handling the renormalization step that plain multiplication alone wouldn't take care of.