Standard form

What standard form is

The mass of the Earth is about $5970000000000000000000000$ kg. A virus may be $0.0000001$ m across. Numbers like these are hard to read and easy to copy wrongly, so scientists and engineers write them in standard form:
$$A\times {10}^n,\text{where}1⩽A<10\text{and}n\text{is an integer.}$$

So $5970000000000000000000000=5.97\times {10}^{24}$ and $0.0000001=1\times {10}^{-7}$.

Both conditions matter:

• $A$ is at least 1 and less than 10. It has exactly one non-zero digit before the decimal point. $3.8\times {10}^5$ is in standard form; $38\times {10}^4$ and $0.38\times {10}^6$ are equal to it but are not.
• $n$ is a whole number, positive for large numbers and negative for numbers between 0 and 1.

Why it works. Each power of 10 shifts the digits one place. ${10}^3=1000$, so multiplying by ${10}^3$ moves every digit three places to the left. From the indices lesson, ${10}^{-2}=\frac{1}{100}$, so multiplying by ${10}^{-2}$ moves every digit two places to the right. The power $n$ just counts how many places the decimal point has moved.

A useful by-product: the power tells you the size at a glance. $4\times {10}^9$ is bigger than $9\times {10}^8$ however the first parts compare, because ${10}^9$ is ten times ${10}^8$.

Converting large numbers into standard form

1. Place the decimal point after the first non-zero digit. This gives $A$.
2. Count how many places the point has moved to get there. That number is $n$.
3. A large number (10 or more) gives a positive $n$.

For $47000000$: the first digit is 4, so $A=4.7$. The point has to move 7 places from the end of $47000000$ to sit between the 4 and the 7. So $47000000=4.7\times {10}^7$.

Check by counting digits: a number with 8 digits before its decimal point is between ${10}^7$ and ${10}^8$, so its power is 7. The power is always one less than the number of digits in front of the point.

Converting small numbers into standard form

For a number between 0 and 1 the method is the same, but the point moves to the right, so $n$ is negative.

For $0.00036$: the first non-zero digit is 3, so $A=3.6$. The point moves 4 places to the right to get from $0.00036$ to $3.6$. So $0.00036=3.6\times {10}^{-4}$.

A quick check: ${10}^{-4}=0.0001$, and $3.6\times 0.0001=0.00036$. Another way to count: the number of zeros before the first non-zero digit, including the one before the point, is the size of the negative power. $0.00036$ has four zeros in front of the 3, so the power is $-4$.

Converting out of standard form reverses this. For a positive power move the point right; for a negative power move it left, filling with zeros.

• $2.05\times {10}^6$: move the point 6 places right to get $2050000$.
• $7.1\times {10}^{-3}$: move the point 3 places left to get $0.0071$.

Numbers that are nearly in standard form

A calculation often produces something like $34\times {10}^5$ or $0.6\times {10}^{-3}$. These are correct values but not standard form, and the final answer mark needs standard form.

Adjust $A$ by a factor of 10 and the power by 1 the opposite way, so the value does not change:

• $34\times {10}^5=3.4\times 10\times {10}^5=3.4\times {10}^6$. Making $A$ ten times smaller makes the power one bigger.
• $0.6\times {10}^{-3}=6\times {10}^{-1}\times {10}^{-3}=6\times {10}^{-4}$. Making $A$ ten times bigger makes the power one smaller.

If you are unsure which way the power goes, write the number out in full and convert again from scratch.

Comparing and ordering. To put numbers in standard form in order, compare the powers first; only when the powers are equal do you compare the values of $A$. With negative powers, remember that ${10}^{-5}$ is smaller than ${10}^{-4}$, so $9\times {10}^{-5}<1.2\times {10}^{-4}$.

Calculating with a calculator

On Paper 3 (Core) and Paper 4 (Extended) you calculate with standard form on a calculator. Use the $\times {10}^x$ key (labelled EXP or EE on some calculators) to enter the power. To enter $6.2\times {10}^{-5}$, press 6.2, then the $\times {10}^x$ key, then $-5$.

Two things go wrong:

• Typing $\times 10$ and then the power key. This is fine only if you put brackets around the whole number, $(6.2\times {10}^{-5})$. The $\times {10}^x$ key avoids the problem, because it keeps the number together.
• Misreading the display. A calculator may show a long answer as 7.462686567E6 or with a small raised 06. That means $7.462686567\times {10}^6$. Never write the answer as ${7.46}^6$; that is a different number.

The question will often say "Give your answer in standard form" and "correct to 3 significant figures". Round $A$, and keep the power: $7.462686\dots \times {10}^6$ becomes $7.46\times {10}^6$.