Scientific Notation

Learning goals

Write a number as $a \times 10^{n}$ with $1 \le a < 10$
Explain why the coefficient range makes each positive number's form unique
Move the point left for a large number, right for a small one
Read a negative exponent as small, never as negative
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 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 significant digits, the actual sequence of meaningful figures in the number. 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 significant 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. That is what makes scientific notation a unique way to write each positive number. So two people who convert $3000$ correctly always get the identical expression, $3 \times 10^3$ .

Why exactly one power of ten makes the coefficient land in $[1, 10)$ #

Start with $6200$ . Dividing and multiplying by $10$ generates a ladder of candidate coefficients, $\ldots,\; 6200,\; 620,\; 62,\; 6.2,\; 0.62,\; \ldots$ , each one a tenth of the entry before it. Walk along the list: $62$ is still too big, and $0.62$ has already dropped under $1$ . Only $6.2$ lands at or above $1$ while staying below $10$ . Reaching $6.2$ from $6200$ took three divisions by $10$ , so the exponent is $3$ and $6200 = 6.2 \times 10^3$ .

Take any positive number and ask which powers of ten you could factor out of it. Multiplying or dividing a number by $10$ slides its decimal point one place. Dividing by $10$ moves the point one step left and shrinks the number tenfold. Multiplying by $10$ moves the point one step right and grows the number tenfold. So the candidate coefficients you can reach, $\ldots, \;a \times 10,\; a,\; a \div 10,\; \ldots$ , each differ from the next by a single factor of ten.

Now look at the interval from $1$ up to (but not including) $10$ . It is exactly one factor of ten wide: its right end, $10$ , is ten times its left end, $1$ . Because each candidate coefficient is ten times the one below it, the candidates step right past $1, 10, 100, \ldots$ one at a time. So exactly one of them can land at or above $1$ while still staying below $10$ . Land any lower and the value is under $1$ ; land any higher and it is $10$ or more. Either way you are in a different interval.

That single landing spot is the coefficient $a$ , and the number of factors of ten you moved to reach it is the exponent $n$ . Since there is one and only one such spot, there is one and only one way to write the number as $a \times 10^n$ with $1 \le a < 10$ . Nothing here depended on $6200$ , because every positive number has the same list of candidates and the same single entry inside the interval.

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 $1$ .

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,” exactly as you proved for negative exponents: $10^{-4} = \frac{1}{10^4} = \frac{1}{10{,}000} = 0.0001$ , and $4.2 \times 0.0001 = 0.00042$ . A negative exponent always marks a 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 in [1, 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. They vanish once the point is repositioned, and the zero between the $3$ and the $7$ stays because it sits among the significant digits. 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, you read the exponent as a direction and a distance for the 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.

Why a larger exponent wins when both coefficients are in $[1, 10)$ #

Compare $1.2 \times 10^{3}$ with $9.6 \times 10^{2}$ , where the smaller exponent carries much 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 that reads $960 < 1200$ .

Compare $a \times 10^{m}$ and $b \times 10^{n}$ , where both coefficients satisfy $1 \le a < 10$ 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, which is when $a$ is as small as allowed, $a = 1$ . So $a \times 10^{m} \ge 1 \times 10^{m} = 10^{m}$ .

The second number is below its ceiling: since $b < 10$ , we have $b \times 10^{n} < 10 \times 10^{n} = 10^{n+1}$ . And because $m \ge n + 1$ , that ceiling $10^{n+1}$ is at most $10^{m}$ .

Chain 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 $b \times 10^{n} < a \times 10^{m}$ : the number with the larger exponent is larger, no matter what the coefficients are. That holds because keeping each coefficient under $10$ stops the smaller-exponent number from ever catching up. Nothing in that chain used the values $1.2$ and $9.6$ : it needed only the two range conditions and $m > n$ . When the exponents are equal, the powers of ten match, so the comparison falls back to the coefficients alone.

The proof needs both coefficients to be properly 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}.

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.

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. To divide, divide the coefficients and subtract the exponents, since $\frac{10^{m}}{10^{n}} = 10^{m-n}$ by the quotient rule. 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 $1$ .

Worked example 5 Compute $(3 \times 10^{4}) \times (2 \times 10^{5})$

Group the coefficients together and the powers of ten together, which you may do because multiplication can be reordered freely:

(3 \times 10^{4}) \times (2 \times 10^{5}) = (3 \times 2) \times (10^{4} \times 10^{5}).

Multiply the coefficients, and add the exponents with the product rule:

= 6 \times 10^{4+5} = 6 \times 10^{9}.

The coefficient $6$ already satisfies $1 \le 6 < 10$ , so no renormalizing is needed and the answer is $6 \times 10^{9}$ .

Worked example 6 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 $2 \times 10^{10}$ .

Worked example 7 Compute $\dfrac{8.4 \times 10^{7}}{2 \times 10^{3}}$

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

\frac{8.4 \times 10^{7}}{2 \times 10^{3}} = \frac{8.4}{2} \times \frac{10^{7}}{10^{3}}.

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

= 4.2 \times 10^{7-3} = 4.2 \times 10^{4}.

The coefficient $4.2$ is between $1$ and $10$ , so the answer $4.2 \times 10^{4}$ is already in proper form.

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}

Common mistakes

Letting the coefficient leave $[1, 10)$ . 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. The minus sign on the 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 number as $a \times 10^{n}$ with the coefficient $1 \le a < 10$ and the exponent $n$ an integer. The coefficient holds the significant 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 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, then renormalize so $1 \le a < 10$ .

Scientific Notation

Learning goals

What scientific notation is

Why the coefficient stays between 1 and 10

Large numbers: a positive exponent

Small numbers: a negative exponent

Converting back to standard form

Comparing magnitudes at a glance

Multiplying and dividing in scientific notation

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)