1.1 Scientific notation

Introduction

The distance from the Sun to the Earth is about 150,000,000,000 metres, and a single water molecule is about 0.0000000003 metres across. Numbers like these are hard to read, easy to copy wrongly and slow to calculate with, because most of their digits are zeros that only show the position of the decimal point. Standard form replaces those zeros with a power of ten, so that $1.50 \times 10^{11}$ and $3 \times 10^{-10}$ show at a glance both the digits that matter and the size of the number.

Scientists, engineers, economists and computer designers all work with quantities that span many powers of ten: national budgets in trillions of dollars, cell sizes in micrometres, data storage in terabytes. Standard form is the common language for all of them, and it is also how a graphic display calculator handles numbers that are too long for its screen.

The number sets

Every number used in this course belongs to one or more standard sets. The sets are named with special capital letters, and each larger set contains the smaller ones inside it. Knowing which set a number belongs to matters in practice: a count of people must be a natural number, a temperature can be a negative integer, and a length such as the diagonal of a unit square cannot be written as a fraction at all.

Subtracting one natural number from another can give a result that is not a natural number, such as $3 - 5 = -2$. Including the negatives of the natural numbers gives the integers, which are enough for any addition, subtraction or multiplication of whole numbers.

Dividing one integer by another usually leaves the integers: $7 \div 2 = 3.5$. Numbers that can be written as one integer divided by another form the rational numbers. Every terminating decimal is rational ($0.75 = \frac{3}{4}$), and so is every recurring decimal ($0.333\ldots = \frac{1}{3}$).

Some numbers cannot be written as a fraction of two integers. The diagonal of a square with side 1 has length $\sqrt{2} = 1.414\,213\,56\ldots$, and the ratio of a circle's circumference to its diameter is $\pi = 3.141\,592\,65\ldots$. Their decimal expansions never end and never settle into a repeating pattern.

Together, the rational and irrational numbers fill the whole number line with no gaps. Every length, mass, time or amount of money that can be measured is a real number.

The nested diagram of the number sets shows each set sitting inside the next: every natural number is an integer, every integer is rational, and every rational number is real. The irrational numbers are the real numbers left over outside $\mathbb{Q}$.

Standard form

Powers of ten and place value

In the decimal system each place is worth ten times the place to its right. Multiplying by 10 moves every digit one place to the left (or, equivalently, moves the decimal point one place to the right), and dividing by 10 moves the decimal point one place to the left. Repeated multiplication by 10 is written with an exponent: $10^{3} = 10 \times 10 \times 10 = 1000$.

Negative powers continue the pattern downwards. Each step down the table divides by 10, so $10^{0} = 1$, $10^{-1} = \frac{1}{10}$ and $10^{-3} = \frac{1}{1000}$. A negative exponent therefore means a small positive number, not a negative number: $10^{-3} = 0.001$. The reasons are set out with the laws of exponents in subtopic 1.5.

Writing a number in standard form

Any positive number can be written as a number between 1 and 10 multiplied by a power of ten. The number between 1 and 10 holds the significant digits, and the power of ten fixes the size of the number.

The two parts of the form have names. The power of ten is the exponent $k$, and the number in front is the mantissa.

The condition $1 \le a < 10$ makes the form unique. The number 28,800 could be written $28.8 \times 10^{3}$, $2.88 \times 10^{4}$ or $0.288 \times 10^{5}$, and all three are equal, but only $2.88 \times 10^{4}$ is in standard form. Because every number has exactly one standard form, two numbers can be compared by looking first at the exponents, then at the mantissas.

The method for large numbers

1. Place the decimal point after the first non-zero digit to form the mantissa $a$.

2. Count how many places the decimal point has moved to the left to get there. That count is $k$.

3. Write $a \times 10^{k}$, dropping zeros at the end of $a$ that only held places.

For 4,070,000 the decimal point sits after the last zero. Moving it six places to the left gives 4.070000, so $4\,070\,000 = 4.07 \times 10^{6}$. The justification is that $4.07 \times 10^{6}$ means 4.07 multiplied by 10 six times, which moves the decimal point six places back to the right.

The method for small numbers

For a number between 0 and 1, the decimal point has to move to the right to reach the first non-zero digit, and the exponent is negative.

The decimal point moves five places to the right, from 0.0000523 to 5.23, so $k = -5$. As a check, $5.23 \times 10^{-5} = 5.23 \div 100\,000 = 0.000\,052\,3$.

Numbers given in words or with prefixes

News reports and data tables often write large numbers with words: "149.6 million kilometres", "US$2.3 trillion", "8.1 billion people". Each word is a power of ten (million $= 10^{6}$, billion $= 10^{9}$, trillion $= 10^{12}$), so the number can be written as a product and then adjusted to standard form.

The mantissa 149.6 is too large, so it is written as $1.496 \times 10^{2}$, and the two powers of ten combine by adding exponents.

Converting back to an ordinary number

Converting from standard form reverses the process. A positive exponent $k$ moves the decimal point $k$ places to the right, filling empty places with zeros; a negative exponent moves it $|k|$ places to the left.

For $6.1 \times 10^{-4}$, the decimal point moves four places to the left: 6.1 → 0.61 → 0.061 → 0.0061 → 0.00061. Counting the zeros after the decimal point gives a quick check: there are three zeros before the 6, one fewer than the 4 in the exponent, because the first move only takes the point to just before the 6.

Significant figures and the mantissa

The mantissa holds all the digits that carry information about a number's precision. Zeros that only mark the position of the decimal point disappear into the power of ten, which is one reason standard form is used for measurements.

In standard form the number of significant figures is the number of digits in the mantissa. A measurement written as $2.50 \times 10^{3}$ m has three significant figures, which says it was measured to the nearest 10 m; $2.5 \times 10^{3}$ m has two, which says only to the nearest 100 m. Standard form removes the ambiguity of a number such as 2500, where the zeros may or may not be significant. The full rules for counting significant figures, for rounding and for choosing an accuracy are taught in subtopic 1.6, Approximation and error.

Unless a context says otherwise, a final answer in standard form has its mantissa given to 3 significant figures, as in $1.50 \times 10^{11}$. Rounding the mantissa never changes the exponent, except when it rounds up to 10: $9.997 \times 10^{4}$ to 3 significant figures is $10.0 \times 10^{4}$, which must then be rewritten as $1.00 \times 10^{5}$.

Calculating in standard form

Multiplying and dividing

Multiplication and division in standard form use two facts about powers of the same base. Multiplying $10^{m}$ by $10^{n}$ is multiplying 10 by itself $m$ times and then $n$ more times, so the exponents add. Dividing cancels $n$ of the tens, so the exponents subtract.

Because multiplication can be done in any order, the mantissas and the powers of ten can be dealt with separately and then combined. The method is:

1. Multiply (or divide) the mantissas.

2. Add (or subtract) the exponents.

3. Adjust the result so that the mantissa lies between 1 and 10.

The product of the mantissas, 24, is too large, so it becomes $2.4 \times 10^{1}$ and the exponent rises by 1. Division can give a mantissa that is too small instead:

Subtracting a negative exponent adds: $4 - (-2) = 6$. The mantissa 0.5 becomes $5 \times 10^{-1}$, and the exponent falls by 1.

Source: the astronomical unit was fixed at exactly 149,597,870,700 m by the International Astronomical Union in 2012; the speed of light is exact by the SI definition of the metre (BIPM).

Adding and subtracting

Addition and subtraction do not have a shortcut like adding exponents, because $10^{5} + 10^{4}$ is not a single power of ten. Two numbers can be added directly only when they are multiplied by the same power of ten, in the same way that 3 hundreds plus 4 hundreds is 7 hundreds. The method is:

1. Rewrite one number so that both have the same power of ten (usually the larger power).

2. Add or subtract the numbers in front.

3. Adjust the result to standard form.

An alternative that always works is to convert both numbers to ordinary numbers, add, and convert back: $320\,000 + 46\,000 = 366\,000 = 3.66 \times 10^{5}$. When the exponents are far apart, the smaller number may make almost no difference. Adding $4.6 \times 10^{4}$ to $3.2 \times 10^{12}$ gives $3.200\,000\,046 \times 10^{12}$, which is still $3.20 \times 10^{12}$ to 3 significant figures.

Calculating with a GDC

For anything beyond simple mantissas, the GDC does the arithmetic. Numbers in standard form are entered with the calculator's power-of-ten key: on a TI-84 Plus CE it is 2nd , (printed EE), and on a Casio fx-CG50 it is ×10^x. Entering 2.5E-3 this way is safer than typing 2.5×10^-3 by hand, because the calculator treats the whole entry as one number and there is no risk of the order of operations splitting it, for example in a division.

A GDC switches to this display automatically when a number is too large or too small for the screen, and it can be set to show every answer in standard form (on most models, a Sci setting in the mode menu). Whatever the display, the written answer must be rewritten in mathematical notation:

The full GDC value should be kept in the calculator's memory (with Ans or a stored variable) and only the final answer rounded. Rounding the mantissa of an intermediate result to 3 significant figures and then using the rounded value can change the third figure of the answer, an example of premature rounding.

Units and standard form

SI units and prefixes

Measurements in science and in this course use one agreed set of units, so that a length measured in Tokyo means the same as one measured in Toronto.

Prefixes such as kilo, milli and micro are powers of ten, so converting between them is multiplying or dividing by a power of ten, which standard form handles neatly. A distance of 3.4 km is $3.4 \times 10^{3}$ m; a cell 12 μm across is $12 \times 10^{-6} = 1.2 \times 10^{-5}$ m.

Converting units of area and volume

Units of area and volume need special care. A square of side 1 m is 100 cm by 100 cm, so it contains $100 \times 100 = 10^{4}$ square centimetres, not 100. A cube of side 1 m contains $100^{3} = 10^{6}$ cubic centimetres.

The conversion factor for length is squared for area and cubed for volume, because an area has two lengths multiplied together and a volume has three. The same idea gives $1\text{ km}^{2} = (10^{3})^{2}\text{ m}^{2} = 10^{6}\text{ m}^{2}$ and $1\text{ litre} = 1000\text{ cm}^{3} = 10^{-3}\text{ m}^{3}$.

Very small quantities: the Avogadro constant

Chemistry counts atoms and molecules, which are so small that any visible amount of a substance contains an enormous number of them. Chemists count them in moles, and the number of particles in one mole is a fixed constant.

Since the 2019 redefinition of the SI units, the Avogadro constant has had an exact defined value. It links the very large (the number of molecules in a glass of water) to the very small (the size of one molecule), and standard form makes both calculations short.

Orders of magnitude

When two quantities are very different in size, they are usually compared by the number of powers of ten that separate them. The exponent in standard form gives this immediately.

Two quantities with the same exponent are of the same order of magnitude even if one is several times the other ($2 \times 10^{6}$ and $9 \times 10^{6}$). Some scientists round to the nearest power of ten instead, treating $9 \times 10^{6}$ as being of order $10^{7}$, so the comparison is a rough one by design. It still shows at once whether two quantities are similar in size, a few powers of ten apart or very far apart.

The scale of lengths places four real objects on a line where each step is one order of magnitude, so equal distances along the line represent equal ratios. On an ordinary linear scale reaching $1.5 \times 10^{11}$ m, the molecule, the person and even the Earth would all be squeezed into the first ten-thousandth of the line.

The order-of-magnitude difference between two quantities is found by dividing them and reading the exponent of the result. The Earth–Sun distance divided by the Earth's diameter is

so the Earth–Sun distance is about four orders of magnitude larger: more than eleven thousand Earths would fit side by side between the Earth and the Sun. Comparing quantities that differ by many orders of magnitude on a graph needs a logarithmic scale, introduced in subtopic 1.5.

Naming very large numbers

Some very large numbers have their own names. The best-known are two invented for fun, which are still used to give a feeling for size.

A googol is far larger than any count of physical objects in daily life, but in standard form it is simply $1 \times 10^{100}$. The googolplex is far larger again, too large to be written out in full.

A googolplex in ordinary form would need a googol of zeros, more digits than could ever be written down, yet its definition in powers of ten fits on one line. Writing numbers with powers of ten keeps the size of a number in its exponent, which stays short even when the number itself is enormous.

Case study: world GDP per person

Gross domestic product (GDP) measures the total value of goods and services an economy produces in a year. The World Bank publishes world GDP and world population every year, and both are numbers far too large to handle comfortably in ordinary form.

Source: World Bank, World Development Indicators (NY.GDP.MKTP.CD and SP.POP.TOTL), retrieved 2026.

GDP per person is GDP divided by population. By hand, the mantissas and the powers of ten can be divided separately, which gives a quick estimate:

The GDC with the full values gives a more accurate answer:

World GDP per person in 2024 was about US$13,700. The estimate from the rounded values, $13,760, differs in the third significant figure, because each input had been rounded to 3 significant figures before dividing. Keeping the full values until the last step avoids this loss of accuracy.

The figure needs careful interpretation. It is an average over the whole world, so it hides enormous differences between countries and within them: a country with GDP per person ten times the world average is one order of magnitude above it. It is also measured in US dollars at market exchange rates, which do not account for prices being much lower in some countries than in others, so it describes living standards only roughly.

Exam focus

Standard form usually appears as one part of a longer question, very often the last part of a calculation ("give your answer in the form $a \times 10^{k}$, where $1 \le a < 10$ and $k \in \mathbb{Z}$"), for 1–3 marks on Paper 1 or Paper 2. It is combined with areas and volumes (subtopic 3.1), unit conversions, pH and other logarithmic formulas (subtopic 1.5), and time, speed and distance. The command terms are write down (no working needed: a direct conversion), calculate and find (show the calculation entered). Nothing on standard form is in the formula booklet: the form $a \times 10^{k}$ with $1 \le a < 10$, the rules $10^{m} \times 10^{n} = 10^{m+n}$ and $10^{m} \div 10^{n} = 10^{m-n}$, and the area and volume conversion factors must all be known.

• Write the unrounded answer before converting it to standard form.

• Give the mantissa to 3 significant figures unless told otherwise.

• Convert all lengths to the same unit before multiplying, and convert areas and volumes with the squared or cubed factor.

• Never leave calculator notation in an answer.

Most marks lost on these parts come from unit conversions: multiplying by 100 instead of $10^{6}$ to change $\text{m}^3$ to $\text{cm}^3$, or forgetting to change cm² to m² before using a price per square metre.

Practice questions

1. Write 0.000 073 6 in the form $a \times 10^{k}$, where $1 \le a < 10$ and $k \in \mathbb{Z}$. [1]

2. Write down $7.05 \times 10^{6}$ as an ordinary number. [1]

3. Explain why $24.5 \times 10^{6}$ is not in standard form, and write it correctly. [2]

4. Calculate $(5.2 \times 10^{8}) \times (3.0 \times 10^{-3})$, giving your answer in standard form. [2]

5. Calculate $\frac{4.8 \times 10^{5}}{1.6 \times 10^{9}}$, giving your answer in standard form. [2]

6. Calculate $6.4 \times 10^{7} + 3.9 \times 10^{6}$, giving your answer in standard form. [2]

7. A GDC displays the answer 3.0471E-6. Write down this number in standard form correct to 3 significant figures. [1]

8. A storage drive holds $2 \times 10^{12}$ bytes. Each photograph uses $3.5 \times 10^{6}$ bytes. Find the greatest number of photographs the drive can hold. [3]

9. A year of 365.25 days is used to define the light-year. Taking the speed of light as $3.00 \times 10^{8}\text{ m s}^{-1}$, calculate the distance light travels in this year, in metres, in standard form. [3]

10. A rectangular solar panel measures 1.7 m by 1.1 m. Find its area in $\text{cm}^2$, in the form $a \times 10^{k}$, where $1 \le a < 10$

Summary

• The number sets nest: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$; irrational numbers are the real numbers that are not rational.

• Standard form is $a \times 10^{k}$ with $1 \le a < 10$ and $k \in \mathbb{Z}$; numbers below 1 have a negative exponent.

• To multiply or divide, combine the mantissas and add or subtract the exponents, using $10^{m} \times 10^{n} = 10^{m+n}$ and $10^{m} \div 10^{n} = 10^{m-n}$

• To add or subtract, first rewrite the numbers with the same power of ten (or convert to ordinary numbers).

• In standard form, the significant figures are the digits of the mantissa; give 3 significant figures unless told otherwise.

• Calculator notation (2.88E4) is never a written answer; write $2.88 \times 10^{4}$.

• Area conversions use the length factor squared ($1\text{ m}^2 = 10^{4}\text{ cm}^2$) and volume conversions the length factor cubed ($1\text{ m}^3 = 10^{6}\text{ cm}^3$

• The order of magnitude is the power of ten; two quantities differ by $n$ orders of magnitude when their ratio is about $10^{n}$.

Self-check

• To do:

  I can say which of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ a number belongs to, and recognise an irrational number.

• To do:

  I can write a large number such as 52,600,000 in standard form.

• To do:

  I can write a small number such as 0.000 408 in standard form with the correct negative exponent.

• To do:

  I can rewrite numbers such as $36.5 \times 10^{-4}$ or $0.82 \times 10^{9}$ so the mantissa is between 1 and 10.

• To do:

  I can convert numbers written with "million", "billion" or SI prefixes into standard form.

• To do:

  I can convert a number in standard form back to an ordinary number.

• To do:

  I can multiply and divide numbers in standard form by hand.

• To do:

  I can add and subtract numbers in standard form by matching the powers of ten.

• To do:

  I can enter standard form on my GDC with the power-of-ten key.

• To do:

  I can rewrite calculator notation as $a \times 10^{k}$ with a 3 significant figure mantissa.

• To do:

  I can convert units of length, area and volume using powers of ten.

• To do:

  I can find how many orders of magnitude apart two quantities are.

• To do:

  I can write the unrounded answer before converting it to standard form.

[2]2 marks

(c) $\sqrt{9} = 3$. A square root sign does not make a number irrational: 3 belongs to $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.

(d) 10 is not a perfect square, so $\sqrt{10} = 3.162\,277\ldots$ never terminates or repeats. It is irrational: it belongs to $\mathbb{R}$ only.

(e) $\frac{22}{7}$ is a fraction of two integers, so it is rational ($\mathbb{Q}$ and $\mathbb{R}$). It is close to $\pi$ but not equal to it: $\frac{22}{7} = 3.142\,857\ldots$, while $\pi = 3.141\,592\ldots$

(c) One billion is $10^{9}$, so 7.3 billion $= 7.3 \times 10^{9}$. The mantissa is already between 1 and 10.

(d) The mantissa 36.5 is too large. Writing $36.5 = 3.65 \times 10^{1}$:

(e) The mantissa 0.82 is too small. Writing $0.82 = 8.2 \times 10^{-1}$:

In (d) and (e), making the mantissa smaller by a factor of 10 is balanced by making the exponent larger by 1, and the reverse.

Check: $0.0071 - 0.0009 = 0.0062$.

(b) Multiply the mantissas and add the exponents:

(c) Squaring means multiplying the number by itself, so the mantissa is squared and the exponent is doubled:

Check by converting the lengths first: $24\,000\text{ cm} \times 8500\text{ cm} = 204\,000\,000\text{ cm}^2 = 2.04 \times 10^{8}\text{ cm}^2$.

So one molecule occupies about $2.99 \times 10^{-23}\text{ mL}$.

(b) First convert to cubic metres. Since $1\text{ cm} = 10^{-2}\text{ m}$, $1\text{ cm}^3 = (10^{-2})^{3} = 10^{-6}\text{ m}^3$:

Similarly $1\text{ nm}^3 = (10^{-9})^{3} = 10^{-27}\text{ m}^3$, so divide by $10^{-27}$:

A water molecule takes up about $2.99 \times 10^{-2}\text{ nm}^3$, the volume of a cube roughly 0.3 nm along each side.

(b) $25.4469\ldots \times 10^{6}$ (M1)

$= 2.54 \times 10^{7}\text{ cm}^3$ A1

Note: award (M1) for multiplying their (a) by $10^{6}$ (or $100^{3}$). Accept $2.54 \times 10^{7}$ from use of 25.4. Award A0 for 2.54E7.

(c) $\frac{2.544\,69\ldots \times 10^{7}}{0.05}$ (M1)

$= 5.09 \times 10^{8}$ drops $(508\,938\,009.9\ldots)$ A1

(d) $\frac{2.544\,69\ldots \times 10^{7}}{12 \times 1000} = 2120.57\ldots$ minutes (M1)

$\frac{2120.57\ldots}{60} = 35.3$ hours $(35.3429\ldots)$ A1

Total [8]

[2 marks]

[3]3 marks

[2]2 marks