Converting and Reading Scientific Notation | Grade 8 Math | LevelKite

The Sun is about $93{,}000{,}000$ miles from Earth. Green light has a wavelength of about $0.00000055$ meters. Both numbers are written in standard notation, the everyday way of writing a number with digits and place value. They are correct, but they are hard to read. You have to count places to see how large or how small each one is.

Scientific notation writes a nonzero value as two parts multiplied together: a number whose size is at least 1 and less than 10, and a power of 10 such as $10^{3}$ or $10^{-2}$ . The power of 10 shows the size of the value at a glance.

In this lesson you will convert between the two notations. You will also read the scientific notation that calculators and spreadsheets show on their screens.

Scientific notation works because of what a power of 10 does to place value.

Multiplying by 10 makes every digit worth ten times as much, so each digit shifts to the next larger place. On paper this looks like the decimal point moving one place to the right: $4.7 \times 10 = 47$ . Multiplying by $10^{4}$ does that four times, so $4.7 \times 10^{4} = 4.7 \times 10{,}000 = 47{,}000$ .

You already know that $10^{-3} = \frac{1}{10^{3}} = 0.001$ . So multiplying by $10^{-3}$ is the same as dividing by 10 three times. Each division makes every digit worth one tenth as much, so the decimal point moves one place to the left each time: $4.7 \times 10^{-3} = 4.7 \times 0.001 = 0.0047$ .

Since $10^{0} = 1$ , multiplying by $10^{0}$ changes nothing: $4.7 \times 10^{0} = 4.7$ .

In every result the digits 4 and 7 stay in the same order. Only their places change.

Key concept

Scientific notation: a coefficient times a power of 10

A nonzero number is in scientific notation when it is written as $a \times 10^{n}$ , a coefficient times a power of 10.

The coefficient $a$ has a magnitude, written $\lvert a \rvert$ , which is its size without the sign. The magnitude must be at least 1 and less than 10.
The exponent $n$ is an integer, such as $7$ , $0$ or $-4$ .

The coefficient carries the digits and the exponent carries the size. With such a coefficient, a positive $n$ gives a number whose magnitude is 10 or more, a negative $n$ gives a number whose magnitude is less than 1, and $n = 0$ gives the coefficient itself.

Zero has no scientific notation, because a coefficient with magnitude at least 1 times a power of 10 is never 0.

a \times 10^{n} \quad \text{where } 1 \le \lvert a \rvert < 10 \text{ and } n \text{ is an integer}

The same value in standard notation and in scientific notation

Standard notation	Scientific notation	Why the two are equal
$5{,}000{,}000$	$5 \times 10^{6}$	$5 \times 1{,}000{,}000 = 5{,}000{,}000$
$47{,}000$	$4.7 \times 10^{4}$	$4.7 \times 10{,}000 = 47{,}000$
$8.2$	$8.2 \times 10^{0}$	$8.2 \times 1 = 8.2$
$0.36$	$3.6 \times 10^{-1}$	$3.6 \times 0.1 = 0.36$
$0.0009$	$9 \times 10^{-4}$	$9 \times 0.0001 = 0.0009$
$-62{,}000$	$-6.2 \times 10^{4}$	$-6.2 \times 10{,}000 = -62{,}000$

Going from scientific notation to standard notation means doing the multiplication, as the last column of the table does. The coefficient already has a magnitude of at least 1 and less than 10, and $n$ stands for the integer exponent on 10.

If $n$ is positive, the decimal point moves $n$ places to the right, because you multiply by 10 that many times.
If $n$ is negative, the decimal point moves to the left, because you divide by 10 instead. An exponent of $-5$ moves it 5 places.
If $n$ is 0, nothing moves. The value is the coefficient.

Fill any empty places with zeros, and keep the sign of the coefficient. Two examples:

$3.08 \times 10^{6} = 3.08 \times 1{,}000{,}000 = 3{,}080{,}000$

$6.04 \times 10^{-5} = 6.04 \times 0.00001 = 0.0000604$

The zero inside $3.08$ and $6.04$ is a digit of the coefficient, so it stays in the standard notation too.

Going the other way takes two decisions. Start with a nonzero number in standard notation.

Coefficient. Put the decimal point just after the first nonzero digit. Keep the sign, and keep the digits in the same order. Leave off the zeros that only filled places: the ones in front of the first nonzero digit, and the ones at the end of an integer. The coefficient now has a magnitude of at least 1 and less than 10.

Exponent. Count the places the decimal point moved. Then choose the power of 10 that undoes the move, so the value stays the same.

Original magnitude 10 or more: the point moved left, which made the magnitude smaller. A positive exponent brings it back, so the exponent is the count.
Original magnitude less than 1: the point moved right, which made the magnitude larger. A negative exponent brings it back, so the exponent is the negative of the count.
Original magnitude at least 1 and less than 10: the point did not move, so the exponent is 0.

Do not round or add digits. Keep any unit and any word such as about, so an exact value stays exact and an approximate value stays approximate.

Worked example

A large number into scientific notation

The Sun is about $93{,}000{,}000$ miles from Earth. Write this distance in scientific notation.

Find the decimal point. A whole number has its decimal point just after its last digit, even when the point is not written.
Make the coefficient. Put the decimal point just after the first nonzero digit, the 9. Leave off the zeros at the end, because they only filled places. The coefficient is 9.3, and its magnitude is at least 1 and less than 10.
$1 \le 9.3 < 10$
Count the move. The decimal point went from the end of the number to just after the 9. It passed the 3 and six zeros, so it moved 7 places to the left. That divided the number by 10 seven times.
$93{,}000{,}000 \div 10^{7} = 9.3$
Undo the move so the value stays the same. Dividing by 10 seven times is undone by multiplying by 10 seven times, so the exponent is 7. The original number is 10 or more, so a positive exponent is what you expect.
$93{,}000{,}000 = 9.3 \times 10{,}000{,}000 = 9.3 \times 10^{7}$
Check the form and keep the context. Using 93 as the coefficient with a smaller power of 10 gives the same value, but 93 is not less than 10, so that form is not scientific notation. The distance was approximate and measured in miles, so the answer keeps the word about and the unit.
$93 \times 10^{6} = 9.3 \times 10^{7}$

Answer

The Sun is about $9.3 \times 10^{7}$ miles from Earth.

Worked example

A small number into scientific notation

Green light has a wavelength of about $0.00000055$ meters. Write this length in scientific notation.

Make the coefficient. The first nonzero digit is the first 5, so put the decimal point just after it. Leave off the zeros in front, because they only filled places. The coefficient is 5.5, and its magnitude is at least 1 and less than 10.
$1 \le 5.5 < 10$
Count the move. The decimal point passed six zeros and one 5, so it moved 7 places to the right. That multiplied the number by 10 seven times.
$0.00000055 \times 10^{7} = 5.5$
Undo the move so the value stays the same. Multiplying by 10 seven times is undone by dividing by 10 seven times, which is what an exponent of negative 7 does. The original number is less than 1, so a negative exponent is what you expect.
$0.00000055 = 5.5 \times 10^{-7}$
Check by multiplying out. The power of 10 with exponent negative 7 is 0.0000001, and 5.5 times that is the number you started with. The wavelength was approximate and measured in meters, so the answer keeps the word about and the unit.
$5.5 \times 10^{-7} = 5.5 \times 0.0000001 = 0.00000055$

Answer

Green light has a wavelength of about $5.5 \times 10^{-7}$ meters.

Counting zeros is not the same as counting places. Suppose you count the five zeros in $4{,}500{,}000$ and write $4.5 \times 10^{5}$ . Multiply it out to check: $4.5 \times 10^{5} = 4.5 \times 100{,}000 = 450{,}000$ . That is only one tenth of $4{,}500{,}000$ .

The shortcut fails because the exponent counts the places the decimal point moves, and the point also passes the digit 5. It moves 6 places to make the coefficient $4.5$ , which is at least 1 and less than 10. So $4{,}500{,}000 = 4.5 \times 10^{6}$ .

For a nonzero whole number, the count of zeros equals the exponent only when every digit after the first is 0, as in $5{,}000{,}000 = 5 \times 10^{6}$ . Counting places works for every nonzero number.

Calculators and spreadsheets often switch to scientific notation on their own. They do it when a number has too many digits for the screen or is very close to zero. Many screens use the letter E, short for exponent, in place of a raised exponent.

Read E as times 10 to the power of. The number in front of E is the coefficient, and the integer after E is the exponent on 10. So 3.2E8 means $3.2 \times 10^{8}$ , which is $320{,}000{,}000$ .

Three things to watch:

The exponent belongs to the 10, not to the coefficient. 3.2E8 is not $3.2^{8}$ , which is only about $11{,}000$ .
A negative sign has two possible positions with two different meanings. In -2.5E6 the sign is in front, so the number is negative: $-2.5 \times 10^{6} = -2{,}500{,}000$ . In 2.5E-6 the sign is on the exponent, so the number is positive and less than 1: $2.5 \times 10^{-6} = 0.0000025$ .
Some screens show a lowercase e, a plus sign or an extra zero, as in 3.2e+08. The exponent is still 8.

After you rewrite the E, check the form as before: the value is not zero, and the coefficient has a magnitude of at least 1 and less than 10.

What a screen shows and what it means

Screen shows	Scientific notation	Standard notation
3.2E8	$3.2 \times 10^{8}$	$320{,}000{,}000$
7E5	$7 \times 10^{5}$	$700{,}000$
4.5E-7	$4.5 \times 10^{-7}$	$0.00000045$
-6.1E3	$-6.1 \times 10^{3}$	$-6{,}100$
1.08E+12	$1.08 \times 10^{12}$	$1{,}080{,}000{,}000{,}000$

Worked example

Reading a calculator display

Light travels about $300{,}000{,}000$ meters each second. To find how far it travels in one day, you multiply by the $86{,}400$ seconds in a day. The calculator shows 2.592E13. Write the distance in scientific notation and in standard notation.

Read the letter E as the words times 10 to the power of. The number in front of E is the coefficient, and the number after E is the exponent on 10, not an exponent on 2.592.
$2.592 \times 10^{13}$
Check the form. The value is not zero, the coefficient 2.592 has a magnitude of at least 1 and less than 10, and the exponent 13 is an integer. So this is scientific notation.
$1 \le 2.592 < 10$
Convert to standard notation by doing the multiplication. The exponent is positive 13, so the decimal point moves 13 places to the right. The digits 5, 9 and 2 fill 3 of those places, and zeros fill the other 10.
$2.592 \times 10^{13} = 25{,}920{,}000{,}000{,}000$
Add what the screen leaves out. The calculator shows only a number. The unit comes from the problem: meters per second times seconds gives meters. The speed was approximate, so the distance is approximate too.

Answer

Light travels about $2.592 \times 10^{13}$ meters in one day, which is about $25{,}920{,}000{,}000{,}000$ meters.

Here is the idea to keep. Scientific notation splits a nonzero number into two parts: a coefficient that carries the digits, with a magnitude of at least 1 and less than 10, and a power of 10 that carries the size. To convert, move the decimal point to make the coefficient, then choose the exponent that undoes the move. To go back, do the multiplication. On a screen, E means times 10 to the power of, and the coefficient may be rounded. Through every step, keep the units and any about from the problem.

Next, you will use scientific notation to compare numbers, estimate very large and very small quantities, and find how many times as large one quantity is as another.