Standard Form

What standard form is, and the three rules that go with it

Standard form writes a number as $A\times {10}^n$. The number in front, $A$, must be at least 1 and less than 10, and the index $n$ must be an integer, positive or negative. Both halves of the definition are examined: a question can ask you to write a number in standard form, or to say whether a given number already is in standard form.

The index counts how many places the decimal point moves to leave exactly one non-zero digit in front of it. A large number moves the point to the left, so the index is positive: $47000=4.7\times {10}^4$. A number smaller than 1 moves the point to the right, so the index is negative: $0.00032=3.2\times {10}^{-4}$.

The value never changes when you convert, only the way it is written. If $A$ comes out as 47 or as 0.6, the number is not yet in standard form, however equal it is to the original. That single check catches most lost marks in this subtopic.

Converting large numbers into standard form

Take the number, find its decimal point, and move it left until exactly one digit is left in front of it. The number of places moved is the index, and for a large number the index is positive. $5800000$ needs 6 moves, so it is $5.8\times {10}^6$, and $64000$ needs 4 moves, so it is $6.4\times {10}^4$.

Every digit is kept. $902000000$ is $9.02\times {10}^8$: the zero between the 9 and the 2 is part of the number and must not be dropped. Count the places rather than the digits, and check the index by multiplying the power of ten out again.

Going the other way, $3.4\times {10}^5$ is written in full by moving the point 5 places to the right, filling the gaps with zeros: 340000. The index tells you the size of the number, so ${10}^5$ means the answer is in the hundred thousands.

• $5800000=5.8\times {10}^6$
• $64000=6.4\times {10}^4$
• $902000000=9.02\times {10}^8$
• $3.4\times {10}^5=340000$

Small numbers and the negative index

For a number smaller than 1 the decimal point moves to the right, so the index is negative. $0.00047$ becomes $4.7\times {10}^{-4}$, because the point moves 4 places right to leave the 4 in front. $0.0083=8.3\times {10}^{-3}$ and $0.0000006=6\times {10}^{-7}$.

A negative index is not a negative number. ${10}^{-4}$ means $\frac{1}{{10}^4}=0.0001$, so $2.5\times {10}^{-3}=2.5\times 0.001=0.0025$. The answer is small and positive; only the index carries the minus sign.

Count the places the point moves, not the zeros. $0.00047$ has three zeros after the point but the index is $-4$, because the point passes the 4 as well as the zeros. Counting zeros instead of places is the commonest error in this subtopic.

Multiplying and dividing in standard form

The $A$ values are multiplied together and the indices are added: $(A_1\times {10}^{n_1})\times (A_2\times {10}^{n_2})=(A_1{A}_2)\times {10}^{n_1+n_2}$. So $(3\times {10}^4)\times (2\times {10}^5)=6\times {10}^9$, because $3\times 2=6$ and ${10}^4\times {10}^5={10}^9$.

The product of the $A$ values often falls outside the allowed range, and it must be adjusted before you write the answer down. $(4\times {10}^6)\times (5\times {10}^{-2})=20\times {10}^4$, which is $2\times {10}^5$ once the 20 is written as $2\times 10$.

For division the $A$ values are divided and the indices are subtracted: $(8\times {10}^9)÷(2\times {10}^3)=4\times {10}^6$. Subtracting a negative index adds it, so $(6\times {10}^{-3})÷(3\times {10}^{-5})=2\times {10}^2=200$.

Adding and subtracting in standard form

Addition and subtraction are different: the powers of ten must match before the $A$ values can be added or subtracted. $3\times {10}^4+2\times {10}^4=5\times {10}^4$, because both terms are counts of ten thousands.

When the indices differ, rewrite the smaller power as a larger one. $4.5\times {10}^5+3\times {10}^4$ becomes $4.5\times {10}^5+0.3\times {10}^5=4.8\times {10}^5$. The value of the term has not changed: $0.3\times {10}^5$ is still 30000.

The same works for negative indices: $7\times {10}^{-3}+5\times {10}^{-4}$ becomes $7\times {10}^{-3}+0.5\times {10}^{-3}=7.5\times {10}^{-3}$. If you prefer, write both numbers out in full, add them, and convert the answer back at the end.

• $3\times {10}^4+2\times {10}^4=5\times {10}^4$
• $4.5\times {10}^5+3\times {10}^4=4.5\times {10}^5+0.3\times {10}^5=4.8\times {10}^5$
• $6\times {10}^{-2}-2\times {10}^{-2}=4\times {10}^{-2}$
• $7\times {10}^{-3}+5\times {10}^{-4}=7.5\times {10}^{-3}$

Standard form in a word problem, and the calculator

Standard form turns up wherever the numbers are very large or very small, and the calculation is usually a division or a multiplication in disguise. A satellite that travels $2.4\times {10}^4$ km in $3\times {10}^2$ seconds has an average speed of $2.4\times {10}^4÷3\times {10}^2=0.8\times {10}^2=8\times {10}^1$ km per second. Write the speed as a distance divided by a time, then do the two parts separately.

If the question asks for the answer to a given accuracy, round the $A$ value and leave the power of ten alone: $3.457\times {10}^5$ to 2 significant figures is $3.5\times {10}^5$. Rounding the index would change the size of the answer by a factor of ten.

On the calculator, the power of ten is entered with the key marked EXP or $\times {10}^x$, not by typing 10 then the index. The syllabus note for this subtopic says that Core candidates are expected to calculate with values in standard form only on Paper 3, the calculator paper, so a standard form question on a non-calculator paper is about converting rather than calculating.

• Average speed $=2.4\times {10}^4÷3\times {10}^2=8\times {10}^1$ km/s
• $3.457\times {10}^5$ to 2 significant figures is $3.5\times {10}^5$
• Leave the index alone when you round; round only the $A$ value
• Core: calculation with standard form is a Paper 3 skill