Converting from Scientific Notation

Learn to turn a number like 3.2 × 10⁻⁴ back into an ordinary decimal number by moving the decimal point and filling in zeros.

Intermediate25 min18 problems
Practice Problems

Definition

In the previous lesson you wrote numbers in scientific notation. Now we go the other way: from a×10na \times 10^n back to an ordinary number written out in full, its decimal form. (Careful: "standard form" is another name for scientific notation in the UK, so in this lesson we say "ordinary number" or "decimal form".)
The exponent nn tells you how many places to move the decimal point of the coefficient aa:
  • n>0n > 0: move the point nn places to the right (the number gets bigger)
  • n<0n < 0: move the point ∣n∣|n| places to the left (the number gets smaller, but stays positive)
  • n=0n = 0: do not move it, because 100=110^0 = 1
When you run out of digits, fill the empty places with placeholder zeros:
4.5×106=4,500,0003.2×10−4=0.000324.5 \times 10^6 = 4{,}500{,}000 \qquad 3.2 \times 10^{-4} = 0.00032

Try it now

What is 5×1035 \times 10^3 as an ordinary number?

Worked Examples

Write 4.5×1064.5 \times 10^6 as an ordinary number.
1

Read the exponent

n=6n = 6 is positive, so the point moves 66 places to the right → 6 places right
2

Use the digits you have

4.54.5 has one digit after the point, so the first hop uses the 55: 4545 → 1 place used, 5 left
3

Fill in placeholder zeros

The other 55 places become zeros: 45 00000‾45\,\underline{00000} → 4,500,0004{,}500{,}000
4

Check the size

10610^6 is one million, and 4.54.5 million is 4,500,0004{,}500{,}000 → Makes sense

Common Mistakes

Moving the point to the right for a negative exponent: 3.2×10−4=32,0003.2 \times 10^{-4} = 32{,}000

Why it's wrong: A negative exponent means dividing by 1010 again and again, so the number gets smaller, not bigger.

Correct: Negative exponent, move left: 3.2×10−4=0.000323.2 \times 10^{-4} = 0.00032

Adding nn zeros to the end: 4.5×106=45,000,0004.5 \times 10^6 = 45{,}000{,}000

Why it's wrong: The exponent counts places the point moves, not zeros to write. The digit 55 already uses one of the 66 places.

Correct: Only 6−1=56 - 1 = 5 zeros are needed: 4.5×106=4,500,0004.5 \times 10^6 = 4{,}500{,}000

Writing nn zeros after the point: 3.2×10−4=0.0000323.2 \times 10^{-4} = 0.000032

Why it's wrong: The first hop to the left puts the point in front of the 33 without adding a zero. That is one zero too many.

Correct: With a one-digit whole part, there are ∣n∣−1|n| - 1 zeros after the point: 0.000320.00032 (3 zeros)

Thinking a negative exponent makes the number negative: 5×10−3=−50005 \times 10^{-3} = -5000

Why it's wrong: The sign of the exponent controls the size. Only a minus sign in front of the coefficient makes the number negative.

Correct: 5×10−3=0.0055 \times 10^{-3} = 0.005 (positive), while −5×103=−5000-5 \times 10^{3} = -5000

Interactive Visual

Move the decimal point: scientific notation to ordinary number

Scientific notation to standard form

Move the decimal point with the buttons or drag it. The digits stay where they are.

45.
4.5×1064.5 \times 10^{6}

Each hop of the point changes the exponent by 1. The value stays the same.

The exponent is above 0: move the point right until it is 0.

Orange zeros are placeholders: they keep the digits in their places.

For scale

10⁻²10⁰10²10⁴10⁶10⁸10¹⁰10¹²45321
Powers of 10: each step to the right is 10 times bigger.
1Earth to Sun (m):1.496×10111.496 \times 10^{11}= 149600000000
2people on Earth (about):8×1098 \times 10^{9}= 8000000000
3Mount Everest (m):8.849×1038.849 \times 10^{3}= 8849
4red blood cell (m):7×10−67 \times 10^{-6}= 0.000007
5hydrogen atom (m):1.06×10−101.06 \times 10^{-10}= 0.000000000106

Place Value Chart

Enter a number:

0.00032

Scientific notation: 3.2 × 10⁻⁴

Ones
1
Tenths
0.1
Hundredths
0.01
Thousandths
0.001
Ten-Thousandths
0.0001
Hundred-Thousandths
0.00001
000032

Expanded Form:

3 × 0.0001 + 2 × 0.00001

Click on any place value to highlight it and see its value.

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is 5×1035 \times 10^3 as an ordinary number?

Why It Matters

Scientific notation is compact, but people usually need the ordinary number to picture a size or to use it:
  • Space: the Sun is 1.496×1081.496 \times 10^8 km away, which is 149,600,000149{,}600{,}000 km
  • Biology: a red blood cell is 7×10−67 \times 10^{-6} m wide, which is 0.0000070.000007 m
  • Calculators and spreadsheets show very big or very small results like 6.02E23 or 2.5E-7. The E means "times ten to the power", so you must be able to read it back
  • Science reports and news mix both forms, and you will often need to compare a value in scientific notation with a value written out in full
Once you can convert in both directions, no number is too big or too small to understand.

Real World Applications

Astronomy

Astronomers write distances in scientific notation, but maps, books and news reports often give them in full.

Example:

Light travels about 3×1053 \times 10^5 km every second, which is 300,000300{,}000 km per second.

1Try It Yourself

The average distance from Earth to the Moon is 3.844×1053.844 \times 10^5 km.

Write this distance as an ordinary number of kilometers.

Step 1: Write the mathematical expression

Type the distance in km as an ordinary number:

Biology and Medicine

Cells, bacteria and viruses are measured in tiny parts of a meter, so lab results often use negative exponents.

Example:

A bacterium about 2×10−62 \times 10^{-6} m long is 0.0000020.000002 m, two millionths of a meter.

2Try It Yourself

A flu virus has a diameter of about 1.2×10−71.2 \times 10^{-7} m.

Write the diameter as an ordinary decimal number of meters.

Step 1: Write the mathematical expression

Type the diameter in meters:

Calculators and Spreadsheets

When a result has too many digits for the screen, calculators and spreadsheets switch to E notation: 4.54E9 means $4.54 \times 10^9$.

Example:

A spreadsheet cell showing 2.5E-4 holds the number 0.000250.00025.

3Try It Yourself

A calculator shows 4.54E9 for the age of the Earth in years.

How many years is that, written as an ordinary number?

Step 1: Write the mathematical expression

Type the number of years:

Key Takeaways

  • 1To convert a×10na \times 10^n to an ordinary number, move the decimal point of aa by ∣n∣|n| places
  • 2Positive exponent: move the point to the right, the number is 1010 or more
  • 3Negative exponent: move the point to the left, the number is between 00 and 11 (still positive)
  • 4Exponent 00: the number stays the same, because 100=110^0 = 1
  • 5Fill every empty place with a placeholder zero: 3.2×10−4=0.000323.2 \times 10^{-4} = 0.00032
  • 6Calculator E notation is scientific notation: 6.02E23 =6.02×1023= 6.02 \times 10^{23}

Frequently Asked Questions

Subtract the number of digits after the point in the coefficient from the exponent. For 1.496×1081.496 \times 10^8: 8−3=58 - 3 = 5 zeros, so 149,600,000149{,}600{,}000.
Subtract the number of digits after the point in the coefficient from the exponent. For 1.496×1081.496 \times 10^8: 8−3=58 - 3 = 5 zeros, so 149,600,000149{,}600{,}000.
When the coefficient is between 11 and 1010, there are ∣n∣−1|n| - 1 zeros between the point and the first digit. For 7×10−67 \times 10^{-6}: 55 zeros, so 0.0000070.000007.
No. 10−n10^{-n} is always positive. A number is negative only if the coefficient has a minus sign, like −2.5×10−3=−0.0025-2.5 \times 10^{-3} = -0.0025.
E stands for "exponent" and means ×10□\times 10^{\square}. So 3.1E-5 is 3.1×10−5=0.0000313.1 \times 10^{-5} = 0.000031. It has nothing to do with the number e≈2.718e \approx 2.718.

Glossary

Scientific notation
A number written as a×10na \times 10^n with 1≤a<101 \le a < 10 and nn a whole number
Ordinary (decimal) number
A number written out with all its digits, like 4,500,0004{,}500{,}000 or 0.000320.00032 (not "standard form", which in the UK means scientific notation)
Coefficient
The number aa in a×10na \times 10^n, between 11 and 1010
Exponent
The power nn in 10n10^n: how many places the decimal point moves, and in which direction
Placeholder zero
A zero written only to fill an empty place value, like the zeros in 0.000320.00032 or 4,500,0004{,}500{,}000
E notation
The calculator and spreadsheet way to show scientific notation: 2.5E-7 means 2.5×10−72.5 \times 10^{-7}

Formula Card

Positive exponent

a×10n, n>0a \times 10^{n},\ n > 0: move the point nn places right

Example: $2.75 \times 10^5 = 275{,}000$

Negative exponent

a×10−na \times 10^{-n}: move the point nn places left

Example: $8.4 \times 10^{-5} = 0.000084$ ($n - 1 = 4$ zeros after the point)

Zero exponent

a×100=aa \times 10^{0} = a

Because $10^0 = 1$, the point does not move

Today's challengeOne problem, three tries

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