Writing numbers in scientific notation
The population of Ireland is about 5,100,000. The Sun is about 150,000,000,000 m from Earth. Counting all those zeros is slow, and it is easy to lose one. Scientists, engineers and your calculator use a shorter way to write big numbers. By the end of this page you will be able to write the population of Ireland in just a few symbols. This page is an extension: scientific notation is usually met later in Junior Cycle.
Powers of 10 and writing big numbers
Multiplying by 10 moves every digit one place to the left. So powers of 10 are very easy to read, because the index tells you how many zeros there are.
| Power | Ordinary number |
|---|---|
| $10^1$ | $10$ |
| $10^2$ | $100$ |
| $10^3$ | $1{,}000$ |
| $10^4$ | $10{,}000$ |
| $10^6$ | $1{,}000{,}000$ |
Now look at $45{,}000$. We can write it as $4.5 \times 10{,}000$, which is $4.5 \times 10^4$. We have swapped four zeros for a power of 10.
The rule $1 \leq a < 10$ means there is exactly one non-zero digit before the decimal point. This makes sure every number has only one correct scientific-notation form.
Write in scientific notation, $a \times 10^n$ with $1 \leq a < 10$: a) $4,500$ b) $72,000$ c) $300,000,000$
Write in scientific notation, $a \times 10^n$ with $1 \leq a < 10$:
- a) $4,500$
- b) $72,000$
- c) $300,000,000$
Which of these is written correctly in scientific notation? Rewrite the others correctly. a) $25 \times 10^3$ b) $4.8 \times 10^6$ c) $0.7 \times 10^2$
Which of these is written correctly in scientific notation? Rewrite the others correctly.
- a) $25 \times 10^3$
- b) $4.8 \times 10^6$
- c) $0.7 \times 10^2$
Back to an ordinary number
To go the other way, move the decimal point to the right by the number of places in the index. Add zeros if you run out of digits.
The same idea works for units. To change $2.6 \times 10^5$ m into kilometres, divide by $1{,}000$, which is the same as lowering the index by 3: $2.6 \times 10^5 \div 10^3 = 2.6 \times 10^2$ km $= 260$ km.
Write as an ordinary number: a) $3 \times 10^4$ b) $6.2 \times 10^3$ c) $9.05 \times 10^5$
Write as an ordinary number:
- a) $3 \times 10^4$
- b) $6.2 \times 10^3$
- c) $9.05 \times 10^5$
The distance from Dublin to Cork by road is about $2.6 \times 10^5$ m. a) Write this as an ordinary number of metres. b) Convert it to kilometres and write in scientific…
The distance from Dublin to Cork by road is about $2.6 \times 10^5$ m.
- a) Write this as an ordinary number of metres.
- b) Convert it to kilometres and write in scientific notation.
Comparing and ordering
Scientific notation makes big numbers easy to compare. Compare in this order:
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The index first: a bigger index means a bigger number (when $a$ is between 1 and 10).
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If the indices are the same, compare the values of $a$.
So $8.9 \times 10^4$ (about 89,000) is smaller than $3.2 \times 10^5$ (about 320,000), even though $8.9 > 3.2$.
When you compare two numbers, you can also ask "how many times bigger?" If the indices differ by 1, one number is about 10 times the other. If they differ by 2, it is about 100 times, and so on.
The population of Ireland is about $5,100,000$ and the population of Cork city is about $224,000$. a) Write both in scientific notation. b) Which is bigger, and by what…
The population of Ireland is about $5,100,000$ and the population of Cork city is about $224,000$.
- a) Write both in scientific notation.
- b) Which is bigger, and by what power of ten approximately?
Put these in order from smallest to largest: $3.2 \times 10^5$, $8.9 \times 10^4$, $1.1 \times 10^6$, $450,000$
Put these in order from smallest to largest:
$3.2 \times 10^5$, $8.9 \times 10^4$, $1.1 \times 10^6$, $450,000$
Multiplying and dividing
Scientific notation also makes calculations with huge numbers short. Multiply or divide the front numbers, then deal with the powers of 10 using the index laws:
$$10^m \times 10^n = 10^{m+n} \qquad 10^m \div 10^n = 10^{m-n}$$Afterwards, check that $a$ is still between 1 and 10. If it is not, adjust it.
Dividing
Very small numbers
A grain of sand has a mass of about $0.00002$ kg, which is $2$ hundred-thousandths of a kilogram. To compare it with a tonne, divide: $1{,}000 \div 0.00002$. Multiply top and bottom by $100{,}000$ to get $100{,}000{,}000 \div 2 = 50{,}000{,}000$ grains. Powers of 10 with natural-number indices are all you need.
Light travels at about $3 \times 10^8$ m/s. The Sun is about $1.5 \times 10^{11}$ m from Earth. a) Find the time in seconds for light to reach Earth. b) Write the answer…
Light travels at about $3 \times 10^8$ m/s. The Sun is about $1.5 \times 10^{11}$ m from Earth.
- a) Find the time in seconds for light to reach Earth.
- b) Write the answer in minutes, correct to one decimal place.
A stadium holds $5.1 \times 10^4$ people. A ticket costs €$4.5 \times 10^1$. a) Find the total income from a full house, in scientific notation. b) Explain why…
A stadium holds $5.1 \times 10^4$ people. A ticket costs €$4.5 \times 10^1$.
- a) Find the total income from a full house, in scientific notation.
- b) Explain why $a \times 10^n$ with $a = 12$ is not in standard form, using this answer if it had been written as $51 \times 45 \times 10^3$.
Back to the start: the population of Ireland, $5{,}100{,}000$, is just $5.1 \times 10^6$, with one digit in front of the point and the point moved 6 places.
Summary
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Scientific notation: $a \times 10^n$ with $1 \leq a < 10$ and $n$ a whole number. Example: $72{,}000 = 7.2 \times 10^4$.
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Large to scientific: move the decimal point until one non-zero digit is in front. The number of places moved is the index.
-
Scientific to ordinary: move the point right by the index, adding zeros if needed. Negative indices are for later years.
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Comparing: compare the index first, then the front number.
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Multiplying and dividing: multiply or divide the front numbers, add or subtract the indices, then tidy $a$ so it is between 1 and 10.