Writing and reading scientific notation
How can a number like 150,000,000 or 0.00007 be written so that its size shows at a glance, with no zeros to count?
Where you are 3 in Scientific notation
When you reach for it
Some quantities are too large or too small for ordinary place value to be comfortable. The distance from Earth to the Sun is about 150{,}000{,}000 km, and a human hair is about 0.00007 m wide. Written out like that, the size of each number lives in its count of zeros, and counting zeros is slow: at a glance, is the first one fifteen million or a hundred and fifty million? Scientific notation moves that count into one small number, an exponent, and keeps the digits that matter at the front:
You reach for it whenever a number has a long run of zeros on either side of the decimal point: writing down a measurement from astronomy or from a microscope, reading a science text, or reading a calculator that has run out of room on its screen. It is also the first step before comparing such numbers or calculating with them. It rests on Powers of ten, and for small numbers on zero and negative exponents.
The lesson runs over two days. Day 1 sets out the form, writes large numbers in it and back again, and explains why the leading number is kept between 1 and 10. Day 2 does the same for small numbers, using negative powers of ten, and reads calculator displays such as \texttt{4.2E-6}.
The procedure
Day 1: the form, and large numbers
The form. A number is written in scientific notation when it is written as
The number a is the leading number (also called the coefficient), and 10^{n} is a power of ten with exponent n. So 2.5 \times 10^{4} and 9 \times 10^{11} are in scientific notation. 25 \times 10^{3} is not, because 25 is not less than 10; 0.25 \times 10^{5} is not, because 0.25 is less than 1; and 2.5 \times 3^{4} is not, because its power is not a power of ten.
What multiplying by ten does. Each place-value column is worth ten times the column to its right. Multiplying by 10 therefore moves every digit one column to the left, which looks the same as the decimal point moving one place to the right: 5.2 \times 10 = 52. Multiplying by 10^{7} is multiplying by ten seven times, so it moves the point seven places right. Dividing by 10^{7} moves it seven places left. Everything in this lesson comes from that one fact.
Writing a large number. Take 52{,}000{,}000. Its decimal point is not written, but it sits at the end, after the last zero. For a leading number between 1 and 10 the point must sit just after the first digit, giving 5.2; the zeros that follow can be dropped, since 5.2000000 = 5.2. To get from the end of the number to just after the 5, the point moves seven places left, which divides by 10^{7}. To keep the value the same, multiply by 10^{7} again:
There is a quick count to check the exponent. The point moves past every digit before it except the first, so a whole number with d digits has exponent d - 1. Here 52{,}000{,}000 has 8 digits and the exponent is 7.
Only zeros at the end may be dropped. In 4{,}080{,}000 the zero between the 4 and the 8 holds the 8 in its place, and 4.08 and 4.8 are different numbers, so 4{,}080{,}000 = 4.08 \times 10^{6}.
Back to a plain number. To write 2.06 \times 10^{5} out in full, multiply 2.06 by ten five times, which moves the point five places right: 2.06, then 20.6, 206, 2060, 20600, 206000. The first two moves use up the digits after the point; the other three reach columns that have no digit, and those are filled with zeros, which hold the place:
Why the leading number is kept between 1 and 10. One number can be written as something times a power of ten in many ways, and all of these are true:
The rule 1 \le a < 10 picks exactly one of them. A positive number sits between two neighboring powers of ten, at or above one and below the next: 10^{7} \le 52{,}000{,}000 < 10^{8}. Dividing through by 10^{7} gives 1 \le 5.2 < 10, so the lower power is the one whose leading number lands between 1 and 10, and no other power does. This buys two things.
- Every positive number has exactly one scientific notation, so two people writing the same number write the same thing.
- The exponent alone tells the size. A number with exponent 7 is at least 10^{7} and less than 10^{8}; a whole number with exponent 7 has 8 digits.
The ends of the range are chosen on purpose. The number 10^{7} itself needs a form, and it is 1 \times 10^{7}, so 1 must be allowed. If 10 were allowed as well, 10^{7} would also be 10 \times 10^{6}, two forms for one number, so 10 is left out.
Putting a number into the form. A value such as 52 \times 10^{6} is correct but not in scientific notation. Write the leading number itself in scientific notation, 52 = 5.2 \times 10, then regroup (the associative property) and combine the tens:
Here 10 \times 10^{6} = 10^{7} because it is seven tens multiplied together. In words: when the leading number is made ten times smaller, the exponent goes up by one, and the value does not change.
Day 2: small numbers and calculator displays
Negative powers of ten. From zero and negative exponents, 10^{0} = 1 and 10^{-n} = \dfrac{1}{10^{n}}. Each row of this table is a tenth of the row above it:
| Power of ten | Value |
|---|---|
| 10^{2} | 100 |
| 10^{1} | 10 |
| 10^{0} | 1 |
| 10^{-1} | \displaystyle \frac{1}{10} = 0.1 |
| 10^{-2} | \displaystyle \frac{1}{100} = 0.01 |
| 10^{-3} | \displaystyle \frac{1}{1000} = 0.001 |
| 10^{-4} | \displaystyle \frac{1}{10000} = 0.0001 |
In 10^{-n} the digit 1 sits in the nth decimal place. Multiplying by 10^{-n} is dividing by 10^{n}, so it moves the decimal point n places to the left.
Writing a small number. Take 0.00063. For a leading number between 1 and 10 the point must sit just after the first digit that is not zero, giving 6.3 (the zeros in front vanish, since 00006.3 = 6.3). To get there the point moves four places right, which multiplies by 10^{4}. To keep the value the same, multiply by 10^{-4}, which undoes it:
Again there is a count to check with. The first digit that is not zero, the 6, sits in the fourth decimal place, and the exponent is -4. For any number between 0 and 1, the exponent is minus the decimal place of its first digit that is not zero.
Which sign the exponent has. A number of 10 or more has a positive exponent. A number from 1 up to but not including 10 is already a leading number, and its exponent is 0, since 10^{0} = 1: 7.5 = 7.5 \times 10^{0}. A number between 0 and 1 has a negative exponent: its leading number is at least 1, so the power of ten has to make it smaller, and only a negative power does that.
Back to a decimal. To write 8.4 \times 10^{-3} out in full, move the point three places left: 8.4, then 0.84, 0.084, 0.0084. Columns with no digit are filled with zeros:
Calculator displays. When an answer has too many digits for the screen, a calculator switches to scientific notation and writes it in its own shorthand. The display \texttt{4.2E-6} means 4.2 \times 10^{-6}: the letter E stands for “exponent”, the number after it is the power of ten, and the \times 10 is left out to save room. Some calculators show the exponent smaller and raised instead, but the meaning is the same. The E has nothing to do with the number e met later in mathematics, and \texttt{4.2E-6} is not 4.2 raised to a power.
Going the other way, many calculators have a key marked \texttt{EE}, \texttt{EXP} or \times 10^{x} that enters “times ten to the” in a single press. Pressing 4.2, \texttt{EE}, (-), 6 enters 4.2 \times 10^{-6}.
Worked examples
Day 1: large numbers into scientific notation
Write each number in scientific notation: (a) the distance from Earth to the Sun, about 150{,}000{,}000 km; (b) the number of seconds in a day, 86{,}400; (c) 4{,}070{,}000; (d) 600.
Solution(a) The leading number must be between 1 and 10, so the point goes just after the first digit: 1.5, with the trailing zeros dropped. From the end of 150{,}000{,}000 to just after the 1 the point moves eight places left, so the power is 10^{8}. The distance is about 1.5 \times 10^{8} km. Check: 150{,}000{,}000 has 9 digits and 9 - 1 = 8.
(b) The leading number is 8.64. The point moves four places left, past 6, 4, 0 and 0, so 86{,}400 = 8.64 \times 10^{4}. Check: 5 digits, exponent 4.
(c) The leading number is 4.07. The zero between 4 and 7 stays, because without it the 7 would move into the wrong column. The point moves six places, so 4{,}070{,}000 = 4.07 \times 10^{6}. Check: 7 digits, exponent 6.
(d) The leading number is 6; a whole number between 1 and 10 is allowed, since 6 = 6.0. The point moves two places, so 600 = 6 \times 10^{2}, and indeed 6 \times 100 = 600.
Day 1: back to plain numbers
Write each as an ordinary number: (a) the population of the world, about 8 \times 10^{9}; (b) 3.75 \times 10^{6}; (c) 7.125 \times 10^{3}.
Solution(a) Multiplying by 10^{9} moves the point nine places right. There are no digits after the 8 to use up, so all nine places are filled with zeros: about 8{,}000{,}000{,}000, or eight billion people. Check: the plain number has 10 digits, one more than the exponent.
(b) The point moves six places right. The first two moves use up the 7 and the 5, giving 375, and the remaining four places are filled with zeros: 3.75 \times 10^{6} = 3{,}750{,}000.
(c) The point moves three places right, and the three digits after it, 1, 2 and 5, use up all three moves exactly. No zeros are needed: 7.125 \times 10^{3} = 7{,}125. Scientific notation is not only for numbers full of zeros; it is simply less useful for small whole numbers like this one.
Day 1: two correct values, neither in the form
Torvald and Wiebke copy down the population of a made-up island nation, 46{,}300{,}000. Torvald writes 46.3 \times 10^{6} and Wiebke writes 0.463 \times 10^{8}. Is either value wrong? Rewrite both in scientific notation.
SolutionNeither value is wrong. Moving the point of 46.3 six places right gives 46{,}300{,}000, and moving the point of 0.463 eight places right gives 46{,}300{,}000 too. But neither is in scientific notation: Torvald’s leading number 46.3 is not less than 10, and Wiebke’s 0.463 is less than 1.
For Torvald’s, write 46.3 = 4.63 \times 10 and regroup (the associative property):
For Wiebke’s, 0.463 = 4.63 \div 10, and the division by ten can be moved on to the power of ten:
Making the leading number ten times bigger makes the exponent one smaller.
Both arrive at the same form, which is exactly what the rule 1 \le a < 10 is for. A direct check: 46{,}300{,}000 has 8 digits, so the exponent is 7.
Day 1: what the exponent says about size
(a) Written out in full, how many digits does 3.2 \times 10^{11} have? (b) Which whole numbers have exponent 4 in scientific notation? Give the smallest and the largest, in scientific notation.
Solution(a) Multiplying by 10^{11} moves the point eleven places right. The 3 starts in the ones column and moves eleven columns left, and every column from there down to the ones column is filled, so there are 11 + 1 = 12 digits. Written out, 3.2 \times 10^{11} = 320{,}000{,}000{,}000, and counting confirms 12 digits.
(b) Exponent 4 means the number is at least 10^{4} and less than 10^{5}:
These are exactly the whole numbers with five digits. The smallest is 10{,}000 = 1 \times 10^{4} and the largest is 99{,}999 = 9.9999 \times 10^{4}. The next whole number, 100{,}000, is 1 \times 10^{5}, and its exponent has moved up to 5.
Day 2: small numbers into scientific notation
Write each number in scientific notation: (a) the width of a human hair, about 0.00007 m; (b) Vasilis measures the thickness of a plastic bag as 0.000045 m; (c) 0.000306; (d) 0.5.
Solution(a) The first digit that is not zero is the 7, in the fifth decimal place. Moving the point five places right gives the leading number 7, which multiplies by 10^{5}, so 10^{-5} is written on to undo it: about 7 \times 10^{-5} m.
(b) The 4 is in the fifth decimal place, so the point moves five places to give 4.5: 0.000045 = 4.5 \times 10^{-5} m. The bag and the hair have the same exponent because both lie between 10^{-5} and 10^{-4}; the leading numbers carry the rest.
(c) The 3 is in the fourth decimal place, so 0.000306 = 3.06 \times 10^{-4}. The zero between 3 and 6 stays, for the same reason as on Day 1.
(d) The 5 is in the first decimal place: 0.5 = 5 \times 10^{-1}. Check: \displaystyle 5 \times \frac{1}{10} = 0.5.
Day 2: small numbers back to decimals
Write each as an ordinary decimal: (a) Wiebke’s notes give the mass of an orchid seed as 6.1 \times 10^{-10} kg; (b) 1.04 \times 10^{-3}; (c) 9 \times 10^{0}.
Solution(a) Multiplying by 10^{-10} divides by 10^{10}, so the point moves ten places left. That puts the 6 in the tenth decimal place, with nine zeros between the point and the 6:
Check by counting decimal places in the answer: the 6 is the tenth digit after the point and the 1 the eleventh.
(b) The point moves three places left, so the 1 lands in the third decimal place: 1.04 \times 10^{-3} = 0.00104.
(c) 10^{0} = 1, so the point does not move: 9 \times 10^{0} = 9.
Day 2: reading and typing a calculator display
Yaw’s calculator shows two answers, \texttt{4.2E-6} and \texttt{3.8E10}. (a) Write each in scientific notation and as a plain number. (b) Yaw then wants to enter Wiebke’s 6.1 \times 10^{-10}, and presses 6.1, \times, 10, \texttt{EE}, (-), 10. What number has he entered, and what should he press instead?
Solution(a) The number after E is the exponent of ten. So \texttt{4.2E-6} is 4.2 \times 10^{-6}; the point moves six places left, putting the 4 in the sixth decimal place: 0.0000042. And \texttt{3.8E10} is 3.8 \times 10^{10}; the point moves ten places right, using up the 8 and filling nine more places with zeros: 38{,}000{,}000{,}000.
(b) The \texttt{EE} key already means “times ten to the”. So the keys Yaw pressed mean 6.1 \times 10 \times 10^{-10}, and since 10 \times 10^{-10} = 10^{-9} he has entered 6.1 \times 10^{-9}, ten times too big. The right keys are 6.1, \texttt{EE}, (-), 10, and the screen then shows \texttt{6.1E-10}.
Boundaries
- Zero has no scientific notation. A power of ten is never zero, and the leading number is at least 1, so a \times 10^{n} is never 0. Zero is simply written 0.
- Negative numbers carry their sign in front. The rule 1 \le a < 10 is about the size of the leading number, and the minus sign rides outside: -3{,}200 = -3.2 \times 10^{3} and -0.0041 = -4.1 \times 10^{-3}. The sign of the exponent says whether the size is above or below 1; it says nothing about whether the number is negative.
- The exponent must be an integer. 10^{2.5} is a real number, but it is not a place-value column, so a form such as 3 \times 10^{2.5} is not scientific notation.
- Trailing zeros. As numbers, 1.5 \times 10^{8} and 1.50 \times 10^{8} are equal. In science the extra zero is sometimes kept to show that a measurement is known to more digits, and a value given as “about 1.5 \times 10^{8} km” does not become more exact by being written out as 150{,}000{,}000: those zeros only hold places.
- Calculator shorthand is not mathematics notation. \texttt{4.2E-6} is a display, and the exact look differs from one calculator to another. In written work it becomes 4.2 \times 10^{-6}.
Where this goes next
Because every positive number has exactly one scientific notation, comparing two of them is mostly a matter of comparing exponents, and the leading numbers only settle a tie; that is Comparing numbers in scientific notation. Multiplying and dividing come next in Calculating in scientific notation: the leading numbers combine, the exponents add or subtract, and the Day 1 step of putting a value back into the form, as with 46.3 \times 10^{6}, is how such a calculation ends. Both sit with this lesson under Scientific notation. Further on, the same habit of separating a number’s digits from its size by a power of ten is how scientists write measurements, and how computers store very large and very small numbers.
Exercises
Day 1: large numbers into scientific notation
Write each number in scientific notation.
- 93{,}000
- 6{,}200{,}000
- 405{,}000{,}000
- 70
- 12{,}345
Day 1: back to plain numbers
Write each as an ordinary number.
- 2.8 \times 10^{5}
- 9.01 \times 10^{7}
- 1 \times 10^{9}
- 6.734 \times 10^{2}
Day 1: into the form
Each value is correct but not in scientific notation. Rewrite it in scientific notation.
- 36 \times 10^{4}
- 0.7 \times 10^{6}
- Sanaa estimates the number of grains of sand on a small beach and writes 20 \times 10^{12}. Write her estimate in scientific notation and as a plain number.
Day 2: small numbers
Write each number in scientific notation.
- 0.0081
- 0.000000502
- 0.2
- 9.5
Write each as an ordinary decimal.
- 3.9 \times 10^{-4}
- 7.07 \times 10^{-6}
Day 2: calculator displays and measurements
- A calculator shows \texttt{6.5E-8}. Write the number in scientific notation and as a decimal.
- A calculator shows \texttt{2.47E11}. Write the number in scientific notation and as a plain number.
- Kofi’s biology notes give the length of a bacterium as 0.0000032 m. Write this length in scientific notation.
- A census report from a made-up country gives its population as 1.42 \times 10^{7}. Write the population as a plain number.
Reasoning
- Explain why 10 is not allowed as a leading number, using 10 \times 10^{4} as the example. Which number in scientific notation does it equal?
- Uzma writes 0.00045 = 4.5 \times 10^{4}, “because the point moved four places”. Explain what is wrong, and give the correct scientific notation.
- Explain why every number between 0 and 1 has a negative exponent in scientific notation, and every number of 10 or more has a positive one.
Answers
- 9.3 \times 10^{4}. The point moves from the end to just after the 9, four places; five digits, exponent 4.
- 6.2 \times 10^{6}. Seven digits, so the exponent is 6.
- 4.05 \times 10^{8}. Nine digits, exponent 8; the zero between 4 and 5 stays.
- 7 \times 10^{1}. Two digits, exponent 1, and 7 \times 10 = 70.
- 1.2345 \times 10^{4}. No digit is a trailing zero, so all five are kept; five digits, exponent 4.
- 280{,}000. The point moves five places right: one move uses the 8, and four zeros fill the rest.
- 90{,}100{,}000. Seven places right: two moves give 901, and five zeros follow.
- 1{,}000{,}000{,}000. A 1 followed by nine zeros, one billion.
- 673.4. Two places right, with one digit left over after the point.
- 3.6 \times 10^{5}. 36 = 3.6 \times 10, so 36 \times 10^{4} = 3.6 \times 10 \times 10^{4} = 3.6 \times 10^{5}.
- 7 \times 10^{5}. 0.7 = 7 \div 10: the leading number is made ten times bigger, so the exponent goes down by one.
- 2 \times 10^{13} = 20{,}000{,}000{,}000{,}000. 20 = 2 \times 10 raises the exponent by one to 13; written out it is a 2 followed by thirteen zeros.
- 8.1 \times 10^{-3}. The 8 is in the third decimal place.
- 5.02 \times 10^{-7}. Six zeros follow the point, so the 5 is in the seventh decimal place; the zero between 5 and 2 stays.
- 2 \times 10^{-1}. The 2 is in the first decimal place, and \displaystyle 2 \times \frac{1}{10} = 0.2.
- 9.5 \times 10^{0}. It is already between 1 and 10, and 10^{0} = 1.
- 0.00039. The point moves four places left, so the 3 lands in the fourth decimal place.
- 0.00000707. Six places left, so the first 7 lands in the sixth decimal place.
- 6.5 \times 10^{-8} = 0.000000065. The 6 lands in the eighth decimal place, after seven zeros.
- 2.47 \times 10^{11} = 247{,}000{,}000{,}000. Eleven places right: two moves use the 4 and 7, nine zeros follow, 12 digits in all.
- 3.2 \times 10^{-6} m. Five zeros follow the point, so the 3 is in the sixth decimal place.
- 14{,}200{,}000. Seven places right: two moves give 142, and five zeros follow.
- 10 \times 10^{4} = 10^{5} = 1 \times 10^{5}. If 10 were allowed, 100{,}000 would have two forms, 10 \times 10^{4} and 1 \times 10^{5}. Leaving 10 out keeps exactly one form for each number, which is what lets the exponent tell the size at a glance.
- 4.5 \times 10^{-4}. The count of four places is right but the sign is not: the point moved right, which multiplied by 10^{4}, so 10^{-4} is needed to undo it. A size check shows the slip, since 4.5 \times 10^{4} = 45{,}000, far bigger than 1.
- Write the number as x = a \times 10^{n} with 1 \le a < 10, so 10^{n} = x \div a. If 0 < x < 1, then dividing by a \ge 1 cannot make it bigger, so 10^{n} < 1; but 10^{n} \ge 1 whenever n \ge 0, so n is negative. If x \ge 10, then dividing by a < 10 leaves more than 1, so 10^{n} > 1, which needs n > 0.