Scientific Notation

Learning goals

Write a positive number as $a \times 10^{n}$ with $1 \le a < 10$
Explain why the coefficient range makes each positive number's form unique
Convert between a number and its scientific notation, moving the point left or right as needed
Recognize that a negative power-of-ten exponent gives a small positive value, never a negative one
Compare magnitudes by exponent first, coefficient only to break a tie
Multiply and divide by combining coefficients and exponents, then renormalize

What scientific notation is

A positive number is in scientific notation when it is written in the form

a \times 10^{n},

where the coefficient $a$ satisfies $1 \le a < 10$ and the exponent $n$ is an integer (it may be positive, zero, or negative). The two pieces do two separate jobs. The coefficient $a$ carries the number’s digits, written in order. The power $10^n$ carries the magnitude, the size, telling you where the decimal point really sits. Pulling these apart is the whole point: $4.2 \times 10^9$ and $4.2 \times 10^{-9}$ share the same digits ( $4.2$ ) and differ only in size.

A power of ten with a positive exponent is a $1$ followed by that many zeros, so $10^3 = 1000$ . A power of ten with a negative exponent is the reciprocal of the matching positive power, $10^{-n} = \frac{1}{10^n}$ , so $10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$ .

A single digit sitting in a place-value chart is already a number in this form. In the chart below you choose the digit and the column it occupies. The readout then names that digit’s value as the digit multiplied by a power of ten, which is the shape scientific notation uses.

Pick the digit $3$ and walk it from the ones column out to the thousands. The value climbs $3$ , $30$ , $300$ , $3000$ , which is $3 \times 10^0$ , then $3 \times 10^1$ , $3 \times 10^2$ , $3 \times 10^3$ . The digit is the coefficient and the column supplies the power of ten. Now change the digit without moving it, and watch the power of ten hold still while the value changes. A coefficient such as $1.5$ or $4.83$ does the same job, with more digits to carry.

Where the coefficient stops and the power of ten begins

Digit Column

The 3 sits in the hundreds place. So it is worth 3 times 10², which is 300.

One digit and four columns. Wherever the digit sits, its value is that digit times a power of ten, the shape scientific notation uses; each step left multiplies the value by ten and each step right divides it by ten.

Why the coefficient stays between 1 and 10

The rule $1 \le a < 10$ looks fussy, but it is there for a real reason: without it, the same number could be written many different ways. Take $3000$ . Every line below equals $3000$ :

3000 = 3 \times 10^3 = 30 \times 10^2 = 0.3 \times 10^4 = 300 \times 10^1.

All four are true, yet only the first has a coefficient in the range $1 \le a < 10$ . Restricting the coefficient to a single nonzero digit before the decimal point means $a$ is at least $1$ but less than $10$ . Requiring that range picks out exactly one of these forms as the official one. So two people who convert $3000$ correctly always get the identical expression, $3 \times 10^3$ .

Here is the reason there is only one such form. Take $6200$ and divide by $10$ again and again: $6200,\; 620,\; 62,\; 6.2,\; 0.62,\, \ldots$ , each one a tenth of the number before it. Walking down that list, $62$ is still too big and $0.62$ has already dropped under $1$ . Only $6.2$ lands in between, at or above $1$ but below $10$ . Reaching $6.2$ from $6200$ took three divisions by $10$ , so $6200 = 6.2 \times 10^3$ .

Nothing about that walk was special to $6200$ . Dividing by $10$ always shrinks a number tenfold, and multiplying by $10$ always grows it tenfold. So the candidates step past $1, 10, 100, \ldots$ one at a time and land between $1$ and $10$ exactly once.

So $30 \times 10^2$ is a perfectly correct equation, and it does equal $3000$ . But $30 \times 10^2$ is not proper scientific notation, because the coefficient $30$ is not less than $10$ . The same goes for $0.3 \times 10^4$ , whose coefficient $0.3$ is less than $1$ .

Check your understanding

Which of these writes $4500$ in proper scientific notation?

45 \times 10^{2}

0.45 \times 10^{4}

4.5 \times 10^{3}

4.5 \times 10^{2}

Large numbers: a positive exponent

The decimal point has to end up with exactly one nonzero digit in front of it. The exponent then comes from counting how far the point moved to reach that spot.

Start with $150{,}000{,}000$ . Its decimal point is at the far right (a whole number’s point sits after the last digit). Slide it left until just one digit, the leading $1$ , stands before it:

150{,}000{,}000. \;\longrightarrow\; 1.5 \times 10^{?}

Counting the hops, the point moves $8$ places to the left to get from after the final zero to between the $1$ and the $5$ . Each leftward hop is a division by $10$ , and there were $8$ of them. So to keep the value unchanged you must multiply back by $10$ eight times, that is by $10^8$ :

150{,}000{,}000 = 1.5 \times 10^{8}.

The exponent is positive because the original number is large (at least $10$ ), and it equals the number of places the point moved left. The check is quick: $10^8$ is $1$ followed by eight zeros, and $1.5 \times 100{,}000{,}000 = 150{,}000{,}000$ , the number you started with.

Worked example 1 Write $48{,}300$ in scientific notation

The decimal point starts after the final $0$ , as the form $48{,}300.$ shows. Move the point left until a single nonzero digit sits in front, which means placing it just after the $4$ :

48{,}300. \;\longrightarrow\; 4.8300

Count the hops the point made: from after the last $0$ to between the $4$ and the $8$ is $4$ places. Trailing zeros that fall after the last nonzero digit are dropped from the coefficient, since $4.8300$ and $4.83$ are the same number. So the coefficient is $4.83$ . The point moved $4$ places left, so the exponent is $4$ :

48{,}300 = 4.83 \times 10^{4}.

The coefficient $4.83$ is between $1$ and $10$ , as required, and $4.83 \times 10{,}000 = 48{,}300$ checks out.

Check your understanding

Write $52{,}000$ in scientific notation.

5.2 \times 10^{4}

52 \times 10^{3}

5.2 \times 10^{5}

0.52 \times 10^{5}

Small numbers: a negative exponent

Numbers smaller than $1$ work the same way. But now the decimal point moves the other direction, to the right, and the exponent comes out negative.

Take $0.00042$ . To get one nonzero digit in front of the point, slide the point right until it sits just after the $4$ :

0.00042 \;\longrightarrow\; 4.2 \times 10^{?}

The point moves $4$ places to the right to travel from its start to between the $4$ and the $2$ . Each rightward hop is a multiplication by $10$ , and there were $4$ of them. So to leave the value unchanged you must divide back by $10$ four times, that is multiply by $10^{-4}$ :

0.00042 = 4.2 \times 10^{-4}.

The negative exponent is not a sign that the number itself is negative. The number $0.00042$ is positive; it is simply small, between $0$ and $1$ . The minus sign on the exponent means “reciprocal”: $10^{-4} = \frac{1}{10^4} = \frac{1}{10{,}000} = 0.0001$ , and $4.2 \times 0.0001 = 0.00042$ . In scientific notation, a negative exponent on the power of ten always marks a positive number between $0$ and $1$ , never a number below zero.

The exponent is a signed hop count. Land the point after the first nonzero digit: moving it left to get there gives a positive exponent (large numbers), moving it right gives a negative one (small numbers), and a number already between 1 and 10 needs zero hops.

Worked example 2 Write $0.0307$ in scientific notation

The number is between $0$ and $1$ , so expect a negative exponent. Slide the decimal point right until one nonzero digit stands in front of it, which means placing it just after the $3$ :

0.0307 \;\longrightarrow\; 3.07

The leading zeros before the $3$ are placeholders, and they vanish once the point is repositioned. The zero between the $3$ and the $7$ stays, because it is part of the number’s value, not a placeholder. Count the hops: the point moves $2$ places to the right to get from the start of $0.0307$ to just after the $3$ . Moving right gives a negative exponent of that size:

0.0307 = 3.07 \times 10^{-2}.

Check it: $10^{-2} = \frac{1}{100} = 0.01$ , and $3.07 \times 0.01 = 0.0307$ .

Check your understanding

Write $0.00069$ in scientific notation.

6.9 \times 10^{4}

6.9 \times 10^{-4}

69 \times 10^{-5}

6.9 \times 10^{-3}

Converting back to standard form

Going the other way, from scientific notation to an ordinary number, the point moves in the opposite direction from before. You already have the coefficient, and you read the exponent as a direction and a distance to move its decimal point.

Take $6.02 \times 10^{5}$ . Multiplying by $10^{5}$ makes the number a hundred thousand times bigger, so the point travels $5$ places right. Starting from $6.02$ there are only two digits after the point, so three extra zeros fill the remaining places:

6.02 \times 10^{5} = 602{,}000.

Now take $7.4 \times 10^{-3}$ . Here $10^{-3} = 0.001$ is tiny, so multiplying by it must shrink the coefficient, and the point travels $3$ places left, padding with zeros:

7.4 \times 10^{-3} = 0.0074.

In both cases the direction followed the size. A positive exponent grows the number, so move the point right that many places, filling empty places with zeros. A negative exponent shrinks it, so move the point left that many places.

Worked example 3 Write $9.1 \times 10^{6}$ and $2.5 \times 10^{-4}$ as ordinary numbers

For $9.1 \times 10^{6}$ , the $+6$ moves the point $6$ places right. After the $9.1$ there is one digit past the point, so five zeros fill the rest:

9.1 \times 10^{6} = 9{,}100{,}000.

For $2.5 \times 10^{-4}$ , the $-4$ moves the point $4$ places left, which pushes the $2$ into the fourth decimal place and pads the gap with zeros:

2.5 \times 10^{-4} = 0.00025.

Check your understanding

Write $3.6 \times 10^{-2}$ as an ordinary number.

360

0.036

0.0036

-360

Comparing magnitudes at a glance

Compare $8.1 \times 10^{4}$ and $3.5 \times 10^{6}$ . The exponents are $4$ and $6$ , and $6 > 4$ , so $3.5 \times 10^{6}$ is the larger number, even though its coefficient $3.5$ is the smaller of the two. The coefficient $8.1$ cannot close that gap. It is only about twice $3.5$ , while $10^{6}$ is a hundred times $10^{4}$ .

The power of ten carries the magnitude. So as long as both coefficients sit in the standard range $1 \le a < 10$ , the number with the larger exponent is larger. You only look at the coefficients to break a tie when the exponents are equal.

Here is why. Compare $1.2 \times 10^{3}$ with $9.6 \times 10^{2}$ , where the smaller exponent carries the bigger coefficient and the exponents are as close as they can get. Since $1.2$ is at least $1$ , the first number is at least $1 \times 10^{3} = 1000$ . Since $9.6$ is below $10$ , the second is under $10 \times 10^{2} = 1000$ . The two bounds meet exactly at $1000$ , so $9.6 \times 10^{2} < 1000 \le 1.2 \times 10^{3}$ : by hand, $960 < 1200$ .

That gap can never close, whatever numbers you pick. Swapping coefficients changes a number by less than a factor of $10$ , since every coefficient stays below $10$ . Raising the exponent by even $1$ always multiplies the number by a full factor of $10$ , which no coefficient swap can match. That is why a larger exponent always wins once both coefficients are normalized to $1 \le a < 10$ .

This reasoning needs both coefficients to be normalized. If you allowed $30 \times 10^{2}$ , its inflated coefficient could beat a larger exponent, which is one more reason the rule $1 \le a < 10$ matters.

Worked example 4 Order $4 \times 10^{5}$ , $9 \times 10^{4}$ , and $2 \times 10^{5}$ from smallest to largest

Sort by exponent first. Two numbers share the exponent $5$ and one has exponent $4$ , so the exponent- $4$ number is the smallest of the three outright:

9 \times 10^{4} < 4 \times 10^{5} \quad\text{and}\quad 9 \times 10^{4} < 2 \times 10^{5}.

For the two that tie at $10^{5}$ , the exponents match, so compare the coefficients: $2 < 4$ . That makes $2 \times 10^{5}$ smaller than $4 \times 10^{5}$ . Putting it together:

9 \times 10^{4} < 2 \times 10^{5} < 4 \times 10^{5}.

Check your understanding

Which is larger, $7.8 \times 10^{5}$ or $2.6 \times 10^{6}$ ?

7.8 \times 10^{5}

, because

7.8

is the bigger coefficient B

2.6 \times 10^{6}

, because it has the larger exponent C They are equal D There is not enough information to compare them

Check your understanding

Which is larger, $6.2 \times 10^{5}$ or $4.9 \times 10^{5}$ ?

6.2 \times 10^{5}

, because the exponents are equal so the coefficients decide B

4.9 \times 10^{5}

, because a smaller coefficient always wins C They are equal, since the exponent is the same D There is not enough information to compare them

Multiplying and dividing in scientific notation

Scientific notation also makes multiplication and division of awkward numbers manageable, because the laws of exponents do the heavy lifting. After either operation, the new coefficient might fall outside $1 \le a < 10$ , so you renormalize. If it is $10$ or more, shift one factor of ten back into the power, raising the exponent by $1$ . If it is below $1$ , borrow one factor of ten, lowering the exponent by $1$ .

Multiplying

Take $(2 \times 10^{3}) \times (4 \times 10^{2})$ . Reordering the factors puts the coefficients side by side and the powers of ten side by side: $2 \times 4 = 8$ , and $10^{3} \times 10^{2} = 10^{5}$ , so the product is $8 \times 10^{5}$ . The exponents added because the first factor carried three tens and the second carried two, which is five tens in all. To multiply, multiply the coefficients and add the exponents, since $10^{m} \times 10^{n} = 10^{m+n}$ by the product rule.

Worked example 5 Compute $(5 \times 10^{6}) \times (4 \times 10^{3})$ and fix the coefficient

Multiply the coefficients and add the exponents:

(5 \times 10^{6}) \times (4 \times 10^{3}) = (5 \times 4) \times 10^{6+3} = 20 \times 10^{9}.

The coefficient $20$ is not less than $10$ , so $20 \times 10^{9}$ is not yet proper scientific notation. Rewrite $20$ as $2 \times 10$ , then merge that extra factor of ten into the power using the product rule:

20 \times 10^{9} = (2 \times 10^{1}) \times 10^{9} = 2 \times 10^{1+9} = 2 \times 10^{10}.

Now the coefficient $2$ is in range, so the final answer is $2 \times 10^{10}$ .

Check your understanding

Compute $(6 \times 10^{3}) \times (3 \times 10^{5})$ , written in proper scientific notation.

18 \times 10^{8}

1.8 \times 10^{9}

1.8 \times 10^{8}

9 \times 10^{8}

Dividing

To divide, divide the coefficients and subtract the exponents, since $\frac{10^{m}}{10^{n}} = 10^{m-n}$ by the quotient rule. Dividing can also push the coefficient below $1$ , which renormalizes the other way.

Worked example 6 Compute $\dfrac{2.4 \times 10^{7}}{8 \times 10^{3}}$ and fix the coefficient

Split the quotient into a coefficient part and a power-of-ten part:

\frac{2.4 \times 10^{7}}{8 \times 10^{3}} = \frac{2.4}{8} \times \frac{10^{7}}{10^{3}}.

Divide the coefficients, and subtract the exponents with the quotient rule:

= 0.3 \times 10^{7-3} = 0.3 \times 10^{4}.

The coefficient $0.3$ is below $1$ , so $0.3 \times 10^{4}$ is not yet proper scientific notation. Rewrite $0.3$ as $3 \times 10^{-1}$ , then merge that borrowed factor of ten into the power:

0.3 \times 10^{4} = (3 \times 10^{-1}) \times 10^{4} = 3 \times 10^{-1+4} = 3 \times 10^{3}.

Now the coefficient $3$ is in range, so the final answer is $3 \times 10^{3}$ .

Check your understanding

Compute $\dfrac{1.5 \times 10^{6}}{5 \times 10^{2}}$ , written in proper scientific notation.

0.3 \times 10^{4}

3 \times 10^{3}

3 \times 10^{4}

0.3 \times 10^{3}

Common mistakes

Letting the coefficient leave the range from 1 to 10. Proper scientific notation needs $1 \le a < 10$ , so $30 \times 10^{4}$ and $0.5 \times 10^{3}$ are not in scientific notation, even though they are valid numbers. Shift a factor of ten until the coefficient has exactly one nonzero digit before the point.
Reading a negative exponent as a negative number. $4.2 \times 10^{-4} = 0.00042$ is a small positive number. In scientific notation, the minus sign on the power-of-ten exponent marks a value between $0$ and $1$ (it means a reciprocal power). The exponent’s minus sign never makes the number itself negative.
Moving the decimal point the wrong way. A positive exponent grows the number, so the point moves right; a negative exponent shrinks it, so the point moves left. Mixing these up turns $6.02 \times 10^{5}$ into a tiny decimal instead of $602{,}000$ .
Miscounting the places. The exponent equals the number of places the point moves, so count carefully, including the zeros you pass. In $0.00042$ the point moves $4$ places, not $3$ .
Forgetting to renormalize after multiplying or dividing. $20 \times 10^{9}$ is an intermediate result, not a final answer; rewrite it as $2 \times 10^{10}$ so the coefficient is back in range.
Confusing the power $10^{n}$ with the exponent $n$ . In $7 \times 10^{3}$ the exponent is $3$ , while $10^{3} = 1000$ is the power itself. The exponent is the small count; the power is the big number it stands for.

Key takeaways

Scientific notation writes a positive number as $a \times 10^{n}$ with the coefficient $1 \le a < 10$ and the exponent $n$ an integer. The coefficient holds the digits and the power of ten holds the size.
Keeping $1 \le a < 10$ makes the form unique, so each positive number has exactly one scientific-notation expression.
For a large number (at least $10$ ) the exponent is positive and equals how many places the decimal point moves left. For a small number (between $0$ and $1$ ) the exponent is negative and equals how many places the decimal point moves right.
A negative exponent on the power of ten marks a small positive number, using $10^{-n} = \frac{1}{10^{n}}$ ; it never signals a negative value.
To compare, the larger exponent wins (coefficients break a tie).
To multiply, multiply coefficients and add exponents. To divide, divide coefficients and subtract exponents. Either way, renormalize afterward so $1 \le a < 10$ .

Practice

Multiple Choice Questions (MCQ)

Progressively harder sets of questions. Each opens on its own page.

Level 1 Foundational 12 questions Start → Level 2 Intermediate 12 questions Start → Level 3 Challenge 12 questions Start →

Core practice

Practice problems at the level of the course, to be worked out on paper. Hints one at a time, then the answer or the full worked solution, with your progress kept in this browser.

More practice (optional)

Extra sets, as hard as the Challenge set. Each one opens on its own page.

More resources (optional)

Other explanations of this lesson, if you want a second take.

Scientific notation intro (Khan Academy)
Multiplying and dividing in scientific notation (Khan Academy)
Scientific notation (Math is Fun)
Scientific notation (Purplemath)
Prealgebra: scientific notation (free textbook) (OpenStax)

Go deeper (optional)

You can skip this and keep going. Read it if you want to know more.

Why every positive number has exactly one scientific-notation form

The lesson showed the reasoning on $6200$ : dividing again and again by $10$ steps past every power of ten exactly once. Exactly one of those steps lands the coefficient between $1$ and $10$ . Here it is again, run first on a different number, and then written so it covers every positive number at once.

Why exactly one power of ten makes the coefficient land between 1 and 10#

Run the walk on $47{,}000$ first. Dividing by $10$ repeatedly gives $47{,}000,\; 4700,\; 470,\; 47,\; 4.7,\; 0.47,\, \ldots$ , each a tenth of the one before. $47$ is still ten or more, and $0.47$ has already dropped under $1$ . Only $4.7$ sits at or above $1$ while staying below $10$ , reached after four divisions by $10$ , so $47{,}000 = 4.7 \times 10^{4}$ .

Now take any positive number and ask which powers of ten you could divide or multiply it by. Dividing by $10$ moves the decimal point one place left and shrinks the number tenfold; multiplying by $10$ moves it one place right and grows the number tenfold. So the candidate coefficients you could reach, $\ldots, \; a \times 10,\; a,\; a \div 10, \; \ldots$ , each differ from the next by a single factor of $10$ .

The interval from $1$ up to (but not including) $10$ is exactly one factor of ten wide, since its far end, $10$ , is ten times its near end, $1$ . Because each candidate is ten times the one below it, the candidates step past $1, 10, 100, \ldots$ one at a time. So exactly one of them lands at or above $1$ while staying below $10$ . Land lower and the value is under $1$ ; land higher and it is $10$ or more.

That single landing spot is the coefficient $a$ , and the number of factors of $10$ moved to reach it is the exponent $n$ . Since there is exactly one landing spot, every positive number has exactly one way to be written as $a \times 10^{n}$ with $1 \le a < 10$ .

Why a larger exponent always wins, for any two normalized numbers

The lesson showed the reasoning using $1.2 \times 10^{3}$ and $9.6 \times 10^{2}$ , the closest a smaller exponent can get to catching up. Here it is again, checked first on a different pair, and then written for any two normalized numbers.

Why a larger exponent beats any normalized coefficient#

Check it again on $3.1 \times 10^{4}$ versus $9.4 \times 10^{3}$ . Since $3.1 \ge 1$ , the first number is at least $1 \times 10^{4} = 10{,}000$ . Since $9.4 < 10$ , the second is under $10 \times 10^{3} = 10{,}000$ . So $9.4 \times 10^{3} < 10{,}000 \le 3.1 \times 10^{4}$ , which checks by hand: $9400 < 31{,}000$ .

Now compare $a \times 10^{m}$ and $b \times 10^{n}$ for any coefficients with $1 \le a < 10$ and $1 \le b < 10$ , and suppose $m > n$ , so $m$ is at least $n + 1$ .

The first number is at least its smallest possible value, when $a = 1$ : $a \times 10^{m} \ge 1 \times 10^{m} = 10^{m}$ .

The second number is below its largest possible value: since $b < 10$ , $b \times 10^{n} < 10 \times 10^{n} = 10^{n+1}$ . And because $m \ge n + 1$ , that upper bound $10^{n+1}$ is at most $10^{m}$ .

Chaining these together, the second number is strictly less than $10^{n+1} \le 10^{m}$ , while the first number is at least $10^{m}$ :

b \times 10^{n} < 10^{n+1} \le 10^{m} \le a \times 10^{m}.

So $b \times 10^{n} < a \times 10^{m}$ : whenever $m > n$ , the number with the larger exponent is larger, whatever the coefficients are. Nothing in that chain used the values $1.2$ , $9.6$ , $3.1$ , or $9.4$ : it needed only the two range conditions and $m > n$ . When the exponents are equal, the powers of ten match, and the comparison falls back to the coefficients alone.

A bit of history (optional)

Suppose you want to name a number far larger than anything you meet in everyday life. Picture a one with a hundred zeros behind it. Written out in full it swallows a whole line and tells a reader almost nothing about its size. So what do you call it?

Edward Kasner was a mathematician who enjoyed explaining his subject to ordinary readers. On a walk around 1920 he asked his nine-year-old nephew for a word. The boy, Milton Sirotta, said such a number should be called a googol. The name has stuck to it ever since.

Now look at what the notation does to the same monster. A googol is $1 \times 10^{100}$ . The coefficient holds one digit and the exponent holds the magnitude. The whole quantity fits in a few marks on the page.

Once you pull those two pieces apart, no number is too enormous to compare with another.