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Scientific Notation Converter

Convert both ways and calculate with powers of ten, step by step
Write 0.00047 in scientific notation
(3 x 10^5)(4 x 10^-2)
5.2 x 10^4 + 6.0 x 10^3
Convert 2.9979 x 10^8 to standard notation

The Rule: One Digit, Then a Power of Ten

A number is in scientific notation when it is written as

a×10n,1≤∣a∣<10,  n an integera \times 10^{n}, \qquad 1 \le |a| < 10, \; n \text{ an integer}

The coefficient aa must have exactly one non-zero digit before the decimal point. That single condition is what makes the form unique: 47,00047{,}000 can be written as 4.7×1044.7 \times 10^4 and no other way, so two numbers can be compared just by looking at their exponents.

4.7×104  ✓47×103  ×0.47×105  ×4.7 \times 10^4 \;\checkmark \qquad 47 \times 10^3 \;\times \qquad 0.47 \times 10^5 \;\times

The last two are equal in value but not in form; a question asking for scientific notation will mark them wrong.

The exponent counts places, not zeros. A positive nn means a large number, a negative nn a small one — never a negative number. −4.7×104-4.7 \times 10^{4} is negative; 4.7×10−44.7 \times 10^{-4} is small.

Converting, Reading and Calculating

To scientific notation: put the decimal point after the first non-zero digit and count how many places it moved. Moved left (the number was big) → positive exponent. Moved right (the number was small) → negative exponent. So 0.00047→4.7×10−40.00047 \to 4.7 \times 10^{-4}, four places right.

Back to standard notation: move the decimal point nn places right for positive nn, left for negative nn, padding with zeros.

Reading E: calculators and spreadsheets print 6.02×10236.02 \times 10^{23} as 6.02E23. The E means "times ten to the", nothing else.

Multiply and divide: handle the coefficients and the powers separately, using 10a×10b=10a+b10^a \times 10^b = 10^{a+b} and 10a÷10b=10a−b10^a \div 10^b = 10^{a-b}. If the new coefficient falls outside [1,10)[1, 10), renormalise.

Add and subtract: the exponents must match first. Rewrite the smaller number with the larger exponent, then add the coefficients.

Common Mistakes to Avoid

  • Leaving the coefficient out of range. 12×10312 \times 10^3 is not scientific notation; renormalise to 1.2×1041.2 \times 10^4.
  • Counting zeros instead of decimal places. In 0.000470.00047 the point moves 4 places, even though there are only 3 zeros after it.
  • Adding the exponents when adding numbers. Exponents add only for multiplication. For a sum, match the exponents and add the coefficients.
  • Flipping the sign of the exponent. A number smaller than 1 always gets a negative exponent. Sanity check: 10−4=0.000110^{-4} = 0.0001.
  • Renormalising without fixing the exponent. Turning 12×10312 \times 10^3 into 1.21.2 means the power must rise to 10410^4; the two changes always move in opposite directions.
  • Rounding the coefficient too early in a multi-step calculation, which can shift the final significant digits.

Examples

Step 1: The first non-zero digit is 4, so the decimal point belongs right after it: 4.74.7.
Step 2: Count how far the point moved: from 0.000470.00047 to 4.74.7 is 4 places to the right.
Step 3: Moving right means a negative exponent: n=−4n = -4.
Step 4: Result: 4.7×10−44.7 \times 10^{-4}.
Step 5: Check: 4.7×0.0001=0.000474.7 \times 0.0001 = 0.00047 ✓
Answer: 4.7×10−44.7 \times 10^{-4}

Step 1: Multiply the coefficients: 3×4=123 \times 4 = 12.
Step 2: Add the exponents: 105×10−2=105+(−2)=10310^{5} \times 10^{-2} = 10^{5 + (-2)} = 10^{3}.
Step 3: So far: 12×10312 \times 10^{3} — the coefficient is 12, which is not below 10.
Step 4: Renormalise: 12=1.2×10112 = 1.2 \times 10^{1}, so 12×103=1.2×10412 \times 10^3 = 1.2 \times 10^{4}.
Step 5: Check in ordinary numbers: 300,000×0.04=12,000300{,}000 \times 0.04 = 12{,}000 ✓
Answer: 1.2×1041.2 \times 10^{4}

Step 1: The exponents differ, so they must be matched before adding.
Step 2: Rewrite the smaller number with exponent 4: 6.0×103=0.60×1046.0 \times 10^{3} = 0.60 \times 10^{4}.
Step 3: Add the coefficients: 5.2+0.60=5.85.2 + 0.60 = 5.8.
Step 4: Result: 5.8×1045.8 \times 10^{4}, already normalised.
Step 5: Check: 52,000+6,000=58,00052{,}000 + 6{,}000 = 58{,}000 ✓
Answer: 5.8×1045.8 \times 10^{4}

Frequently Asked Questions

It means the number is smaller than 1, not that the number is negative. Ten to the power minus four is 0.0001, so 4.7 times ten to the minus four equals 0.00047. A negative number has a minus sign on the coefficient instead.

E stands for exponent and reads as times ten to the power of. So 6.02E23 is 6.02 times ten to the twenty-third, and 1.2E-5 is 0.000012. It is a display convention, not an error message.

It makes the representation unique, so every number has one correct form. That uniqueness is what lets you compare two quantities by their exponents alone and read the significant digits straight off the coefficient.

Rewrite one of them so both share the same power of ten, usually by raising the smaller exponent to match the larger. Then add the coefficients and keep the shared power. If the sum reaches 10 or more, renormalise by shifting the point one place and raising the exponent by one.

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