Standard Form in Maths: Rules, Worked Examples, Mistakes

Standard form in mathematics: rules and examples at a glance

Standard form in maths is a way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. It is written a x 10^n, where a is at least 1 and less than 10, and n is a whole number. For example, 340,000 is 3.4 x 10^5 and 0.00072 is 7.2 x 10^-4. England's national curriculum calls this standard form. NASA's glossary calls it scientific notation. Both names describe the same thing.

Key details
Item	Detail
Form	a x 10^n
Rule for a	At least 1 and less than 10
Rule for n	A whole number: positive, negative or zero
Other name	Scientific notation
Used for	Very large and very small numbers
England national curriculum	Key stage 3: interpret and compare. Key stage 4: calculate
Notation in this article	x means multiply, ^ means "to the power"

What standard form means

Every number in standard form has a coefficient a, a multiplication by 10, and an exponent n. The coefficient holds the digits. The power of 10 holds the size. The exponent n counts how many places the decimal point moved. It is positive for numbers of 10 or more, negative for numbers between 0 and 1, and zero for numbers from 1 up to 10.

The wording differs between sources. The England key stage 3 programme of study of September 2013 asks pupils to interpret and compare numbers in standard form A x 10^n, with A at least 1 and less than 10 and n a positive or negative integer or zero. The key stage 4 programme of July 2014 asks pupils to "calculate with numbers in standard form". NASA's Imagine the Universe dictionary defines scientific notation as a decimal fraction, usually between 1 and 10, times a power of ten.

Which name your textbook uses depends on its board and edition, so check the chapter heading. The method is identical. One caution: "standard form" also names other things. The same England key stage 3 document uses it for the equation y = mx + c, so read the context.

Writing large numbers in standard form

Follow three steps. Put the decimal point after the first non-zero digit. Count how many places it moved to the left. Write the result as a x 10^n with that count as n.

Take 340,000. The point sits after the last zero. Moving it five places left gives 3.4, so 340,000 = 3.4 x 10^5.

Indian digit grouping works the same way. The number 5,60,00,000 is 56,000,000, or five crore sixty lakh. The point moves seven places, so it is 5.6 x 10^7.

NASA's dictionary defines the speed of light in a vacuum as 299,792,458 metres per second. The point moves eight places, so the speed is 2.99792458 x 10^8 m/s. Rounded to one significant figure, that is 3 x 10^8 m/s.

Writing small numbers in standard form

For a number below 1, move the decimal point to the right until one non-zero digit sits before it. The exponent is the negative of the number of places moved.

Take 0.00072. The point moves four places right, giving 7.2, so 0.00072 = 7.2 x 10^-4. Likewise 0.0000053 = 5.3 x 10^-6, because the point moves six places.

A quick check helps. If the number is big, n is positive. If the number is a small decimal, n is negative.

Key details
Ordinary number	Standard form	Decimal point moves
340,000	3.4 x 10^5	5 places left
5,60,00,000	5.6 x 10^7	7 places left
299,792,458	2.99792458 x 10^8	8 places left
6.2	6.2 x 10^0	0 places
0.0409	4.09 x 10^-2	2 places right
0.00072	7.2 x 10^-4	4 places right
0.0000053	5.3 x 10^-6	6 places right
0.000000075	7.5 x 10^-8	8 places right

Converting back to an ordinary number

Move the decimal point the number of places the exponent shows. Move right for a positive exponent and left for a negative one, adding zeros where needed.

For 4.07 x 10^6, move the point six places right to get 4,070,000. For 2.5 x 10^4, four places right gives 25,000. A negative exponent sends the point the other way: 6.1 x 10^-3 becomes 0.0061 after three places left, and 9.2 x 10^-5 becomes 0.000092 after five.

Multiplying in standard form

Multiply the coefficients, add the exponents, then check whether the new coefficient is still between 1 and 10.

(3 x 10^4) x (2.5 x 10^3): 3 x 2.5 = 7.5 and 4 + 3 = 7, so the answer is 7.5 x 10^7.
(4 x 10^5) x (6 x 10^3): 4 x 6 = 24 and 5 + 3 = 8, giving 24 x 10^8. The coefficient is too big, so rewrite it as 2.4 x 10^9.
(3.2 x 10^5) x (4 x 10^-3): 3.2 x 4 = 12.8 and 5 + (-3) = 2, giving 12.8 x 10^2 = 1.28 x 10^3, which is 1,280.

To raise a number to a power, apply the power to both parts. (2 x 10^3)^2 = 4 x 10^6.

Dividing in standard form

Divide the coefficients, subtract the exponents, then adjust.

(9.6 x 10^8) / (3 x 10^3): 9.6 / 3 = 3.2 and 8 - 3 = 5, so the answer is 3.2 x 10^5.
(2.4 x 10^6) / (8 x 10^2): 2.4 / 8 = 0.3 and 6 - 2 = 4, giving 0.3 x 10^4. The coefficient is too small, so rewrite it as 3 x 10^3.

A longer example comes from the exponents chapter of the NIOS secondary maths course, which asks how long light takes to reach Earth from the star Sirius. Take the exercise's figures: a distance of 8.1 x 10^13 km and a light speed of 3.0 x 10^5 km per second. Time = distance / speed = (8.1 x 10^13) / (3.0 x 10^5) = 2.7 x 10^8 seconds. With 31,536,000 seconds in a 365-day year, that is about 8.6 years.

Adding and subtracting in standard form

Addition and subtraction need matching exponents. Rewrite one number so both use the same power of 10, combine the coefficients, then tidy up.

(3.6 x 10^5) + (4.8 x 10^4): rewrite 4.8 x 10^4 as 0.48 x 10^5. Then 3.6 + 0.48 = 4.08, so the answer is 4.08 x 10^5. Check: 360,000 + 48,000 = 408,000.
(7.5 x 10^6) + (6.2 x 10^6): the exponents already match, and 7.5 + 6.2 = 13.7, giving 13.7 x 10^6 = 1.37 x 10^7.
(5.2 x 10^3) - (3.9 x 10^2): rewrite 3.9 x 10^2 as 0.39 x 10^3. Then 5.2 - 0.39 = 4.81, so the answer is 4.81 x 10^3. Check: 5,200 - 390 = 4,810.

Key details
Operation	Rule	Example
Multiply	Multiply coefficients, add exponents	(3 x 10^4) x (2.5 x 10^3) = 7.5 x 10^7
Divide	Divide coefficients, subtract exponents	(9.6 x 10^8) / (3 x 10^3) = 3.2 x 10^5
Add or subtract	Match exponents, then combine coefficients	(3.6 x 10^5) + (4.8 x 10^4) = 4.08 x 10^5
Power	Apply the power to both parts	(2 x 10^3)^2 = 4 x 10^6

Common mistakes

Leaving the coefficient outside the range. Answers such as 24 x 10^8 or 0.5 x 10^4 are not in standard form, because a must be at least 1 and less than 10.
Using the wrong sign for small numbers. 0.0006 is 6 x 10^-4. It is not 6 x 10^4 or 6 x 10^-3.
Counting places from the wrong starting point. Mark the original decimal point, then count each jump.
Adding exponents in an addition. 3 x 10^4 + 2 x 10^4 is 5 x 10^4, not 5 x 10^8.
Forgetting to adjust after multiplying. A coefficient of 12.8 needs one more step.
Losing the minus sign on negative numbers. -4,500 is -4.5 x 10^3, and the sign stays with the coefficient.
Copying a calculator display. Calculators and spreadsheets often show 3.4E5 or 3.4E+05, which means 3.4 x 10^5. Write the full form in your answer.
Dropping zeros that count as significant figures. Follow the rounding instruction in the question.

Work through each mistake on a few numbers of your own, then check the answer by converting back to an ordinary number. If the converted number does not match the original, the exponent is the first thing to recheck.

Next steps

Official sources

Frequently asked questions

What is standard form in maths?

Standard form is a way of writing numbers as a x 10^n, where a is at least 1 and less than 10 and n is a whole number. It makes very large and very small numbers short to write and easy to compare. For example, 5,000,000 is 5 x 10^6.

Is standard form the same as scientific notation?

For this topic, yes. England's national curriculum uses the name standard form, and NASA's glossary uses scientific notation, but both describe a x 10^n. Check which name your own textbook uses.

How do you write 0.00072 in standard form?

Move the decimal point four places to the right to get 7.2, then write the exponent as -4. The answer is 7.2 x 10^-4.

Why must the first number be between 1 and 10?

The rule gives every number exactly one standard form. It also makes comparison quick, because you look at the exponents first. A number with exponent 8 is larger than one with exponent 5 when both coefficients are in range.

How do you multiply numbers in standard form?

Multiply the coefficients and add the exponents. If the new coefficient is 10 or more, divide it by 10 and add 1 to the exponent. For example, (4 x 10^5) x (6 x 10^3) = 24 x 10^8 = 2.4 x 10^9.

Can a number in standard form be negative?

Yes. The negative sign goes with the coefficient. For example, -4,500 is -4.5 x 10^3. The exponent shows size, not sign.

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