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Scientific Notation Examples & Addition Calculator

11 months ago
Last Update on October 3, 2026
7 mins
Learn scientific notation with worked conversions, real-life examples and 12 practice answers. Add numbers exactly with a step-by-step calculator.
Scientific notation examples: 58,000 equals 5.8 times 10 to the fourth power, and 0.000037 equals 3.7 times 10 to the negative fifth power.
Scientific notation writes a number as a × 10n. For example, 58,000 = 5.8 × 104 and 0.000037 = 3.7 × 10−5. For a nonzero number, the coefficient has magnitude at least 1 and less than 10, and the exponent is an integer.

Use the examples to see what the exponent means, then try the addition calculator to check your own work. The central idea is keeping the value unchanged: when the coefficient becomes ten times smaller, the power of ten must become ten times larger.

Adding scientific notation calculator

Enter the coefficient and exponent separately for each number. Negative coefficients allow subtraction: to calculate A − B, add A and −B.

First number
Second number

Use ordinary decimal coefficients with at most 30 digits each and whole-number exponents from −300 to 300. A leading minus is allowed. Do not enter commas or E notation in these fields. Zero and coefficients outside the usual scientific-notation range are accepted; every nonzero result is normalized.

The calculator uses exact decimal arithmetic within these input limits. It does not round to a measurement’s significant figures, retain trailing zeros as precision labels, or convert units. For measured quantities, decide the appropriate rounding separately.

Scientific notation examples: big, small and negative

The coefficient, a, supplies the significant digits. The power, 10n, supplies the scale. The condition 1 ≤ |a| < 10 means that 1 is allowed but 10 is not; the vertical bars mean “absolute value.” A negative number keeps its minus sign on the coefficient.

Exact numerical conversions
Ordinary decimalScientific notationCheck the scale
58,0005.8 × 1045.8 × 10,000
5,800,000,0005.8 × 1095.8 billion
0.0000373.7 × 10−53.7 ÷ 100,000
−0.00409−4.09 × 10−3−4.09 ÷ 1,000
77 × 100100 = 1
11 × 100The lower boundary is included

From decimal to scientific notation

  1. Keep the sign. Place the decimal point after the first nonzero digit to form the coefficient.
  2. Count how many places the point moved. For a value whose magnitude is at least 10, moving left gives a positive exponent. For a nonzero value whose magnitude is below 1, moving right gives a negative exponent.
  3. Multiply back to check. The power of ten must restore the original value.

For 58,000, moving four places left gives 5.8; multiplying by 104 restores 58,000. For 0.000037, moving five places right gives 3.7; dividing by 105, or multiplying by 10−5, restores 0.000037.

58,000 becomes 5.8 times 10 to the fourth power; 0.000037 becomes 3.7 times 10 to the negative fifth power. The power of ten reverses the coefficient’s decimal shift.
The coefficient and the power of ten change in opposite ways so that the number stays equal.

From scientific notation to ordinary decimal notation

Now read the multiplication directly: a positive exponent shifts the coefficient’s decimal point right; a negative exponent shifts it left. Add placeholder zeros when needed.

  • 3.45 × 104 = 34,500
  • 5.67 × 103 = 5,670
  • 6.12 × 10−6 = 0.00000612

For more conversion practice, use the scientific notation to decimal converter. If the zeros are confusing, review the place-value chart and calculator.

Zero is the exception. No coefficient with 1 ≤ |a| < 10 represents zero. Write 0. Although 0 × 10n equals zero for every integer n, it is not normalized scientific notation.

Adding scientific notation: five worked examples

Match powers, add coefficients, then normalize. You can choose either exponent as the common exponent; using the larger one often keeps the arithmetic compact. When quantities have units, make the units match first too.

1. Add numbers with the same exponent

(4.5 × 106) + (2.3 × 106)
= (4.5 + 2.3) × 106
= 6.8 × 106

Both coefficients count millions, so add 4.5 and 2.3. The exponent remains 6. A decimal check gives 4,500,000 + 2,300,000 = 6,800,000.

2. Add numbers with different exponents

(5 × 107) + (3 × 105)
= (5 + 0.03) × 107
= 5.03 × 107

Changing 105 to 107 makes the power 100 times larger. Divide its coefficient by 100 to compensate: 3 becomes 0.03. Check: 50,000,000 + 300,000 = 50,300,000.

Align both addends to 10 to the seventh power: 3 times 10 to the fifth equals 0.03 times 10 to the seventh. Add 5 and 0.03 to get 5.03 times 10 to the seventh.
An intermediate coefficient such as 0.03 is useful for aligning terms, even though it is not normalized.

3. Add small numbers with negative exponents

(7.5 × 10−3) + (2.1 × 10−3)
= 9.6 × 10−3 = 0.0096

A negative exponent does not make the number negative. These are 0.0075 and 0.0021, both positive. With different exponents, the same alignment rule applies: (3.6 × 10−4) + (8.7 × 10−5) = (3.6 + 0.87) × 10−4 = 4.47 × 10−4.

4. Normalize a coefficient of 10 or more

(8.4 × 106) + (7.9 × 106)
= 16.3 × 106
= 1.63 × 107

The intermediate value is correct, but 16.3 is too large for normalized scientific notation. Divide the coefficient by 10 and increase the exponent by 1. Keep 1.63 rather than rounding it to 1.6 unless the question asks for that rounding.

5. Subtract or cancel terms

(−6.4 × 105) + (2.4 × 105)
= −4 × 105

Keep track of the signed coefficients. Equal and opposite terms, such as 7.2 × 104 and −7.2 × 104, sum to zero. For a refresher on signed arithmetic, see the negative numbers guide.

Common mistakes, and how to catch them

Adding the exponents

(2 × 103) + (3 × 103) is 5 × 103, not 5 × 106. Adding exponents belongs to multiplying powers: 103 × 103 = 106.

Changing only the exponent

3 × 105 is 0.03 × 107, not 3 × 107. The latter is 100 times too big. Check by expanding both values.

Confusing the two minus signs

3 × 10−4 is positive 0.0003. −3 × 104 is negative 30,000. The coefficient controls the sign; the exponent controls the scale.

Forgetting the final range

0.58 × 105 and 58 × 103 both equal 58,000, but the normalized answer is 5.8 × 104. The coefficient must have exactly one digit before its decimal point, and that digit must be nonzero.

Scientific, standard, decimal and exponential notation

Ordinary decimal notation writes out a value such as 456.789, 0.00025 or 12,500,000. In many US classrooms, “standard notation” means this ordinary form. In UK school mathematics, “standard form” commonly means scientific notation instead. Read the example or definition in your course rather than assuming these phrases always mean the same thing.

Exponential notation is broader: 28 = 256 and 105 = 100,000 are powers. Normalized scientific notation uses a power of 10 and the coefficient rule. A calculator display such as 3.7E−5 is shorthand for 3.7 × 10−5; E here is not a separate multiplier or Euler’s number. The calculator on this page asks for the two parts in separate fields.

Scientific notation can represent small everyday values such as 7 × 100 too. There is no mandatory “large enough” threshold. It is especially useful when a long line of zeros hides the size or precision of a number.

Real-life examples with units and precision

Changing notation does not change a quantity’s unit. These examples separate exact definitions from rounded representations, so the equals sign does not promise more accuracy than the value has.

QuantityScientific notationWhat it means
Speed of light in vacuum2.99792458 × 108 m/sExactly 299,792,458 m/s in the SI. The familiar 3.0 × 108 m/s is a rounded approximation.
Avogadro constant6.02214076 × 1023 mol−1Exact SI value. One mole contains exactly this number of specified entities. 6.02 × 1023 is a rounded count.
Planck constant6.62607015 × 10−34 J·sExact SI value; 6.626 × 10−34 J·s is a rounded form.
One nanometre1 × 10−9 mThe nano- prefix means one billionth. Thus 100 nm = 1 × 10−7 m.
One terabyte (decimal convention)1 × 1012 bytesOne trillion bytes. The binary unit tebibyte (TiB) is different: 240 bytes.

The three physical constants above follow the SI definitions; the prefix and byte conventions are identified in the final references. For measured distances, populations or biological counts, record the source date, conditions and uncertainty before converting the number. A count of cells per microlitre is not the same quantity as a total cell count.

A practical addition example

Suppose one data file contains 2.4 × 106 bytes and another contains 7.5 × 105 bytes. Both quantities are already in bytes. Rewrite the second as 0.75 × 106 bytes, then add:

(2.4 + 0.75) × 106 = 3.15 × 106 bytes

This hypothetical total is 3,150,000 bytes, or 3.15 MB in the decimal convention.

Value and precision are different. As numbers, 5.0 and 5 are equal. For measurements, 5.0 can communicate a finer stated precision. In a textbook measurement calculation, (5.0 × 107) + (3.0 × 105) has an unrounded sum of 5.03 × 107, but the usual addition rule rounds to the least precise decimal place, giving 5.0 × 107. The calculator returns the unrounded numerical sum; follow your measurement or exam instructions.

Practice: 12 questions with worked answers

Treat these numbers as exact. Try the questions before opening the answers.

  1. Write 58,400,000 in scientific notation.
  2. Write 0.0000726 in scientific notation.
  3. Write −0.00409 in scientific notation.
  4. Write 7.03 × 104 as an ordinary decimal.
  5. Write 6.12 × 10−6 as an ordinary decimal.
  6. Write 9 in scientific notation.
  7. Add (4.6 × 105) + (2.7 × 105).
  8. Add (8.4 × 106) + (7.9 × 106).
  9. Add (6.2 × 107) + (4.5 × 105).
  10. Add (3.6 × 10−4) + (8.7 × 10−5).
  11. Add (−6.4 × 105) + (2.4 × 105).
  12. Add (7.2 × 104) + (−7.2 × 104).
Show answers 1–6: conversions
  1. 5.84 × 107. Moving seven places left makes 5.84; multiply by 107 to recover the original.
  2. 7.26 × 10−5. Moving five places right makes 7.26; compensate with 10−5.
  3. −4.09 × 10−3. Keep the negative sign and use a power of 10−3 for the scale.
  4. 70,300. Multiply 7.03 by 10,000.
  5. 0.00000612. Divide 6.12 by 1,000,000.
  6. 9 × 100. Since 100 = 1, the value stays 9.
Show answers 7–12: addition
  1. 7.3 × 105. Add 4.6 + 2.7 = 7.3 and keep the common exponent.
  2. 1.63 × 107. The sum 16.3 × 106 must be normalized.
  3. 6.245 × 107. Rewrite 4.5 × 105 as 0.045 × 107, then add 6.2 + 0.045.
  4. 4.47 × 10−4. Rewrite 8.7 × 10−5 as 0.87 × 10−4; 3.6 + 0.87 = 4.47.
  5. −4 × 105. The coefficients sum to −6.4 + 2.4 = −4.
  6. 0. Equal and opposite terms cancel.

Five scientific notation project ideas

  1. Powers-of-ten poster. Choose six sourced quantities from microscopic to astronomical scales. Give each a unit, a scientific-notation label and a citation. Do not mix diameters, masses and distances on a single unlabelled size axis.
  2. Number scavenger hunt. Collect 10–20 large or small values from textbooks or reliable sources. Convert each both ways and have a partner check the zeros. Mark estimates clearly.
  3. Solar-system scale model. Use one consistent astronomical dataset and specify whether each distance is an average or an instantaneous distance. Choose a scale in metres per astronomical unit, convert every value to the same unit and explain why the model cannot show planet sizes at the same convenient scale.
  4. Microscopic-to-cosmic booklet. Research a representative size for each chosen object. Describe the specific object and measurement, rather than assuming all viruses, cells or galaxies have one fixed size. Use a log-scale layout so each equal step represents a factor of ten.
  5. Numbers in the news. Compare dated values from the same category, currency and measurement period. Write out one example in decimal form to prove your conversion. If comparing budgets, distinguish annual spending from accumulated debt.

Frequently asked questions

Can scientific notation have a negative exponent?

Yes. For example, 4.2 × 10−3 = 0.0042. The exponent says to divide by 1,000; it does not make the result negative.

What happens when the coefficient equals 10?

Renormalize it: 10 × 104 = 1 × 105. The scientific-notation upper boundary is strictly less than 10.

Why can’t I add the coefficients immediately?

The coefficients must describe equal-sized units. Five lots of 107 and three lots of 105 are different-sized lots. Rewrite them using a common power before adding.

How is multiplication different from addition?

Multiply the coefficients and add the exponents, then normalize: (3.2 × 105)(4 × 10−3) = 12.8 × 102 = 1.28 × 103. For division, divide coefficients and subtract exponents. See the exponents and scientific notation guide.

Sources & References

Definitions and factual quantities were checked against the sources below. Worked exercises, calculator implementation and diagrams on this page are original. Reference links open in a new tab.

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