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Scientific notation

Writing very large and very small numbers, and comparing them.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you write very large and very small numbers as a number between 1 and 10 times a power of ten. It is not just shorthand: written this way, two numbers can be compared at a glance by their exponents, and multiplying them becomes adding the exponents. This is how every scientific measurement is written, for exactly those reasons.

2. What you already know

You know place value: in $4{,}500$ the $4$ stands for four thousands, and each place to the left is worth ten times as much. You know that multiplying by $10$ moves every digit one place to the left, and dividing by $10$ moves it one place to the right. You have worked with exponents: $10^3$ means $10 \times 10 \times 10 = 1{,}000$. You may also have seen that $10^0 = 1$ and that a negative exponent means dividing, so $10^{-2} = \dfrac{1}{100} = 0.01$. This lesson uses those facts to write numbers that are far too big or too small to write out comfortably.

3. Words this lesson uses

4. A small number times a power of ten

Some numbers are awkward to write. The sun is about $93{,}000{,}000$ miles from Earth. A bacterium might be $0.000002$ meters long. Counting the zeros is slow, and it is easy to drop one.

Scientific notation splits a number into two parts: the digits that matter, and how big the number is. The digits go in a front number that is at least $1$ and less than $10$. The size goes in a power of ten:

$$93{,}000{,}000 = 9.3 \times 10^{7}, \qquad 0.000002 = 2 \times 10^{-6}.$$

Why does this work? Multiplying by $10$ moves the decimal point one place to the right. So $9.3 \times 10^{7}$ is $9.3$ with the point moved $7$ places to the right: $93{,}000{,}000$. A negative exponent means dividing by $10$ instead, which moves the point to the left: $2 \times 10^{-6}$ is $2$ with the point moved $6$ places left, $0.000002$.

So the exponent tells you the size at a glance. A positive exponent means a number of $10$ or more. A negative exponent means a number less than $1$. An exponent of $0$ means the number is between $1$ and $10$ already, since $10^{0} = 1$.

The rule that the front number must be at least $1$ and less than $10$ matters. $93 \times 10^{6}$ has the same value as $9.3 \times 10^{7}$, but it is not in scientific notation. Keeping one digit before the point means every number has exactly one way to be written, which is what makes comparing them quick.

5. Writing a large number

Take $5{,}280{,}000$. Follow three moves.

1. Find the front number. Put the decimal point just after the first digit: $5.28$. Drop zeros at the end, since they do not change the value once the point is there.
2. Count the places. The point was at the end of $5280000$ and moved to just after the $5$. That is $6$ places to the left.
3. Write the power. Moving left $6$ places divided by $10^{6}$, so multiply by $10^{6}$ to keep the value: $5.28 \times 10^{6}$.

A quick way to count: the exponent is one less than the number of digits before the point. $5{,}280{,}000$ has $7$ digits, so the exponent is $6$. This also shows why $10^{6}$ is "one million": a one followed by six zeros.

Check by going backward. $5.28 \times 10^{6}$ means moving the point $6$ places right: $5.28 \to 52.8 \to 528 \to 5280 \to 52800 \to 528000 \to 5280000$. The digits are back where they started.

6. Writing a small number: negative exponents

Take $0.00047$. The moves are the same, but the point travels the other way.

1. Find the front number. Skip the zeros at the start and put the point just after the first digit that is not zero: $4.7$.
2. Count the places. The point moved from just after the first $0$ to just after the $4$: $4$ places to the right.
3. Write the power. Moving right made the number $10^{4}$ times bigger, so multiply by $10^{-4}$ to shrink it back: $4.7 \times 10^{-4}$.

Why is $10^{-4}$ the same as dividing by $10{,}000$? Look at the pattern: $10^{2} = 100$, $10^{1} = 10$, $10^{0} = 1$. Each step down divides by $10$. Keep going: $10^{-1} = 0.1$, $10^{-2} = 0.01$, $10^{-3} = 0.001$, $10^{-4} = 0.0001$. A negative exponent is not a negative number. $4.7 \times 10^{-4}$ is a small positive number, $0.00047$. A minus sign in front of the whole thing, as in $-4.7 \times 10^{-4}$, is what makes a number negative.

7. Comparing two numbers

To compare two positive numbers in scientific notation, look at the exponents first. The one with the larger exponent is larger, whatever the front numbers are. $2.1 \times 10^{9}$ is larger than $9.8 \times 10^{8}$, because $10^{9}$ is ten times $10^{8}$ and no front number reaches $10$. If the exponents are equal, compare the front numbers: $6.4 \times 10^{5}$ is larger than $3.9 \times 10^{5}$.

To say how many times larger, divide. Divide the front numbers and subtract the exponents:

$$\dfrac{8 \times 10^{9}}{2 \times 10^{5}} = \dfrac{8}{2} \times 10^{9 - 5} = 4 \times 10^{4}.$$

So the first number is $40{,}000$ times the second. An estimate like this is often all you need: "about $10{,}000$ times bigger" is a real answer in science, and you can read it off the exponents alone.

8. Multiplying and dividing

Scientific notation makes multiplying easy, because multiplication can be done in any order. Group the front numbers together and the powers of ten together:

$$(3 \times 10^{4}) \times (2 \times 10^{5}) = (3 \times 2) \times (10^{4} \times 10^{5}) = 6 \times 10^{9}.$$

The exponents add because $10^{4}$ is four tens multiplied together and $10^{5}$ is five more, making nine in all. Division works the same way, with the front numbers divided and the exponents subtracted.

Sometimes the new front number is too big or too small. $(5 \times 10^{3}) \times (4 \times 10^{2}) = 20 \times 10^{5}$, and $20$ is not less than $10$. Rewrite $20$ as $2 \times 10^{1}$, then combine: $2 \times 10^{6}$. In the other direction, $(2 \times 10^{7}) \div (8 \times 10^{3}) = 0.25 \times 10^{4}$, and $0.25 = 2.5 \times 10^{-1}$, so the answer is $2.5 \times 10^{3}$. The rule to remember: when the front gets smaller by a factor of $10$, the exponent goes up by one, and the other way around.

9. Adding, subtracting and calculator displays

Adding is different from multiplying. You can only add the front numbers when the powers of ten match, just as you can only add digits in the same place-value column. $(4.2 \times 10^{5}) + (3.1 \times 10^{5}) = 7.3 \times 10^{5}$. If the exponents differ, rewrite one first: $6 \times 10^{4} = 0.6 \times 10^{5}$, so $(2.5 \times 10^{5}) + (6 \times 10^{4}) = (2.5 + 0.6) \times 10^{5} = 3.1 \times 10^{5}$. Never add the exponents when adding the numbers.

Calculators and spreadsheets have their own way to write scientific notation. A screen showing 3.1E5 or 3.1e+05 means $3.1 \times 10^{5}$, and 2.5E-4 means $2.5 \times 10^{-4}$. The letter E stands for "exponent". When you copy such a result into your work, write it properly with the power of ten, and keep an eye on the sign after the E.

10. The method, step by step, and how to check it

To write a number in scientific notation:

1. Put the decimal point just after the first digit that is not zero.
2. Count how many places the point moved.
3. Use that count as the exponent: positive if the original number was $10$ or more, negative if it was less than $1$.
4. Check the front number is at least $1$ and less than $10$.

To multiply or divide: work on the front numbers and the powers of ten separately (exponents add when you multiply, subtract when you divide), then fix the front number if it has left the range from $1$ to $10$.

Why this is allowed. Multiplying by $10^{n}$ and then by $10^{-n}$ is multiplying by $1$, so moving the point one way and writing the matching power the other way never changes the value. That is the whole trick.

How to check. Go backward: move the point as the exponent says and see whether you get the original number. For a calculation, estimate with the exponents alone: $10^{4} \times 10^{5}$ is $10^{9}$, so an answer near $10^{12}$ has gone wrong somewhere.

11. In the world: how old is sunlight?

Light travels about $186{,}000$ miles each second, and the sun is about $93{,}000{,}000$ miles away. How long does sunlight take to reach Earth? In scientific notation the distance is $9.3 \times 10^{7}$ miles and the speed is $1.86 \times 10^{5}$ miles per second. Time is distance divided by speed: $\dfrac{9.3}{1.86} = 5$ and $10^{7-5} = 10^{2}$, so the time is $5 \times 10^{2} = 500$ seconds. That is $8$ minutes and $20$ seconds. The sunlight you see left the sun a little over eight minutes ago.

12. In the world: counting blood cells

A human red blood cell is about $7.5 \times 10^{-6}$ meters across, that is, $0.0000075$ meters. How many would fit side by side across one centimeter, which is $1 \times 10^{-2}$ meters? Divide: $\dfrac{1}{7.5} \approx 0.133$ and $10^{-2 - (-6)} = 10^{4}$, giving about $0.133 \times 10^{4} = 1.33 \times 10^{3}$. So roughly $1{,}300$ red blood cells would line up across the width of a fingernail. The negative exponents did the hard part of the counting.

13. In the world: filling a hard drive

Drive makers count one terabyte as $10^{12}$ bytes. A photo from a phone camera might take $4$ megabytes, which is $4 \times 10^{6}$ bytes. How many such photos fit on a $1$-terabyte drive? Divide: $\dfrac{1 \times 10^{12}}{4 \times 10^{6}} = 0.25 \times 10^{6}$. The front number $0.25$ is less than $1$, so rewrite it as $2.5 \times 10^{-1}$ and combine: $2.5 \times 10^{5}$, or $250{,}000$ photos. At $20$ photos a day, that is over $34$ years of pictures.

14. Mistakes to watch for

A front number outside the range. $47 \times 10^{3}$ has the right value but is not scientific notation. Write $4.7 \times 10^{4}$.

The wrong sign on the exponent. A number less than $1$ has a negative exponent. $0.006$ is $6 \times 10^{-3}$, not $6 \times 10^{3}$.

Reading a negative exponent as a negative number. $10^{-3}$ is $0.001$, a small positive number.

Multiplying the exponents. $10^{4} \times 10^{5} = 10^{9}$, not $10^{20}$.

Adding the exponents when adding numbers. Line up the powers of ten first, then add only the front numbers.

15. Writing a distance in scientific notation

1. Write $93{,}000{,}000$ in scientific notation. Find the decimal point.

   $93000000.$

   A whole number has its decimal point at the end.

2. Move the point to just after the first digit.

   $9.3000000$

   That makes the front number between $1$ and $10$.

3. Count the places it moved.

   $7 \text{ places to the left}$

   There were $8$ digits, and the point now sits after the first one.

4. Write the number with a power of ten.

   $9.3 \times 10^{7}$

   Moving left $7$ places divided by $10^{7}$, so multiply by $10^{7}$ to keep the value.

5. Check by moving the point back.

   $9.3 \times 10{,}000{,}000 = 93{,}000{,}000$

   The original number comes back, so the answer is right.

16. Writing a small number

1. Write $0.000052$ in scientific notation. Find the first digit that is not zero.

   $0.0000\underline{5}2$

   The zeros in front only hold places.

2. Move the point to just after that digit.

   $5.2$

   The front number must be at least $1$ and less than $10$.

3. Count the places the point moved.

   $5 \text{ places to the right}$

   It passed four zeros and the $5$.

4. Decide the sign of the exponent.

   $\text{exponent} = -5$

   Moving right made the number bigger, so the power of ten must make it smaller.

5. Write the number in scientific notation.

   $5.2 \times 10^{-5}$

   $10^{-5}$ divides by $100{,}000$.

6. Check by dividing.

   $5.2 \div 100{,}000 = 0.000052$

   The original number comes back.

17. A product that needs fixing

1. Multiply $(8 \times 10^{6}) \times (3.5 \times 10^{4})$. Group the parts.

   $(8 \times 3.5) \times (10^{6} \times 10^{4})$

   Multiplication can be done in any order.

2. Multiply the front numbers.

   $8 \times 3.5 = 28$

   $8 \times 3 = 24$ and $8 \times 0.5 = 4$.

3. Add the exponents.

   $10^{6} \times 10^{4} = 10^{10}$

   Six tens times four more tens is ten tens.

4. Write the product so far.

   $28 \times 10^{10}$

   The value is right, but $28$ is not less than $10$.

5. Rewrite the front number.

   $28 = 2.8 \times 10^{1}$

   Moving the point one place left takes out one factor of $10$.

6. Substitute it back in.

   $2.8 \times 10^{1} \times 10^{10}$

   The value has not changed.

7. Combine the powers of ten.

   $2.8 \times 10^{11}$

   $1 + 10 = 11$.

8. Check the size with a rough estimate.

   $10^{7} \times 10^{4} = 10^{11}$

   Rounding $8 \times 10^{6}$ up to $10^{7}$ and $3.5 \times 10^{4}$ to $10^{4}$ gives the same power.

18. Your turn: work out $(9.6 \times 10^{8}) \div (3 \times 10^{3})$.

1. Split into the front numbers and the powers of ten.

   $\dfrac{9.6}{3} \times \dfrac{10^{8}}{10^{3}}$

   Dividing a product can be done one factor at a time.

2. Divide the front numbers.

   $9.6 \div 3 = 3.2$

   $9 \div 3 = 3$ and $0.6 \div 3 = 0.2$.

3. Your turn: work this step out. Its working is at the end of the packet.

   Subtract the exponents.

4. Your turn: work this step out. Its working is at the end of the packet.

   Write the answer.

19. Guided practice

How many times larger is $6 \times 10^{6}$ than $3 \times 10^{3}$?

• $2 \times 10^{3}$
• $2 \times 10^{9}$
• $3 \times 10^{2}$
• $5$

20. Guided practice

Complete the worked solution of $(2 \times 10^{6}) \times (7 \times 10^{6})$.

1. Multiply the front numbers.

   $2 \times 7 =$ _u

   The fronts and the powers of ten are multiplied separately.

2. Add the exponents.

   $10^{6} \times 10^{6} = 10$ to the power _v

   Multiplying powers of ten adds the exponents.

3. The front is $10$ or more, so divide it by $10$.

   front $=$ _w

   Scientific notation needs a front number less than $10$.

4. Add $1$ to the exponent to make up for the $10$ taken out.

   exponent $=$ _z

   Dividing the front by $10$ and multiplying the power by $10$ keeps the value the same.

21. Guided practice

Write $450000$ in scientific notation.

_front × 10^_exp

22. Practice

Write $0.00059$ in scientific notation.

_front × 10^_exp

23. Practice

Write $6.2 \times 10^{5}$ as an ordinary number.

Answer:

24. Practice

Multiply $(3 \times 10^{9}) \times (3 \times 10^{-6})$ and write the answer in scientific notation.

_front × 10^_exp

25. Somewhere new

A school district is backing up $8$ million photographs. One backup drive holds $4 \times 10^{3}$ photographs. How many drives does the whole collection fill? Write the answer in scientific notation.

_front × 10^_exp drives

26. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

27. Test question

Work out $(9 \times 10^{7}) \div (3 \times 10^{4})$ and write the answer in scientific notation.

_front × 10^_exp

28. What you can do now

You can write a number in scientific notation and compare two written that way. Without looking: write 0.00042 in scientific notation, and say which is bigger, $3 \times 10^8$ or $9 \times 10^7$.

Working for the steps left to you

18. Your turn: work out $(9.6 \times 10^{8}) \div (3 \times 10^{3})$., step 3

18. Your turn: work out $(9.6 \times 10^{8}) \div (3 \times 10^{3})$., step 4