A way of writing a number as a×10k, where 1≤a<10 and k is an integer. It makes very large and very small numbers easy to read, compare and calculate with.
The digits of a number that carry meaning about its size, counted from the first non-zero digit. For example, 0.004070 has four significant figures: 4, 0, 7 and the final 0.
and
k∈Z
, and convert it back to an ordinary number.
Round a number to a given number of decimal places or significant figures.
Multiply, divide, add and subtract numbers in scientific notation without a calculator.
Read and enter scientific notation on a graphic display calculator (GDC), including displays such as `2.5E7`.
Compare quantities by their order of magnitude and solve problems involving very large and very small numbers.
Scientific notation is a compact way of writing numbers that are very large or very small. The distance from the Earth to the Sun is about 150000000000 metres, and the diameter of a typical virus is about 0.0000001 metres. Written as 1.5×1011 m and 1×10−7 m, both numbers are easier to read, to compare and to use in calculations. The same notation is used by scientists, engineers and calculators throughout the world.
Writing numbers in scientific notation
Every positive number can be written as a number between 1 and 10 multiplied by a power of 10. The power of 10 records how many places the decimal point has moved, and the number in front records the digits. This is the idea behind scientific notation, also called standard form.
The condition 1≤a<10 means that a has exactly one non-zero digit before the decimal point. So 3.2×105 is in scientific notation, but 32×104 and 0.32×106 are not, even though all three equal 320000. Negative numbers are written in the same way with a minus sign in front, for example −4.1×103.
Large numbers
For a number greater than or equal to 10, place the decimal point after the first non-zero digit and count how many places it has moved to the left. That count is the positive exponent k. For example, 320000=3.2×105 because moving the point five places to the left turns 320000 into 3.2, and multiplying by 105 moves it back.
Small numbers
For a number between 0 and 1, place the decimal point after the first non-zero digit and count how many places it has moved to the right. The exponent is then negative. For example, 0.00047=4.7×10−4, since 10−4=100001 and 4.7÷10000=0.00047. A number from 1 up to (but not including) 10 needs no movement at all, so k=0 and, for example, 6.8=6.8×100.
To convert back to an ordinary number, move the decimal point k places: to the right when k is positive and to the left when k is negative, filling any gaps with zeros. For example, 2.08×106=2080000 and 3.5×10−5=0.000035. A quick check is that a positive exponent gives a number of at least 10 and a negative exponent gives a number less than 1.
Adjusting a number into scientific notation
Calculations often produce answers such as 45.6×105 or 0.38×10−2, which have the right value but the wrong form. To fix the form, write the front number itself in scientific notation and then combine the powers of 10 with the rule 10m×10n=10m+n.
45.6×105=(4.56×101)×105=4.56×106
0.38×10−2=(3.8×10−1)×10−2=3.8×10−3
Making a ten times smaller requires making the power of 10 ten times larger, so the exponent goes up by 1, and the other way round.
Rounding: decimal places and significant figures
Most measurements and many calculated answers cannot be written exactly, so they are rounded. Two conventions are used. Rounding to a number of decimal places fixes how many digits appear after the decimal point. Rounding to a number of significant figures fixes how many meaningful digits appear, whatever the size of the number. Scientific notation and significant figures fit together naturally, because the digits of a are exactly the significant figures.
To round, look at the digit just after the last digit to be kept. If it is 5 or more, increase the last kept digit by 1; otherwise leave it as it is. Then remove the remaining digits after the decimal point, or replace the remaining digits before the decimal point with zeros so that the number keeps its size.
Number
Significant figures
Why
4.072
4
The zero lies between non-zero digits
0.00315
3
Leading zeros only fix the position of the point
2.50
3
A final zero in a decimal shows the value is known to that accuracy
36000
2, 3, 4 or 5
Trailing zeros of a whole number are ambiguous
3.60×104
3
Scientific notation removes the ambiguity
The last two rows show one advantage of scientific notation. The number 36000 could be exact, or it could be a value rounded to the nearest thousand. Writing 3.60×104 states that it is known to three significant figures, while 3.6×104 states two.
How accurate is a rounded value?
A rounded value stands for a whole interval of possible values. If a mass is given as 4.27 kg to 3 significant figures, the true mass m could be anything that rounds to 4.27, which is
4.265≤m<4.275
The smallest possible value is the lower bound and the value that the mass must stay below is the upper bound. Each bound is half a unit of the last kept digit away from the rounded value. The idea explains why answers should not be given to more figures than the data supports: a calculation from measurements known to 3 significant figures cannot produce an answer reliable to 8.
Rounding during a calculation
Rounding too early changes the final answer, because the small error in each rounded value is carried forward and can grow. A calculator keeps about 10 or more significant figures, so the safe method is to keep the unrounded value (in the calculator's memory or written to many figures) and round only at the end.
Multiplying and dividing in scientific notation
The two parts of a number in scientific notation can be handled separately. When multiplying, multiply the front numbers and multiply the powers of 10. The powers of 10 combine through the laws of exponents
10m×10n=10m+n10n10m=10m−n
so exponents are added when multiplying and subtracted when dividing. The result may then need adjusting into scientific notation.
(a×10m)(b×10n)=(a×b)×10m+nb×10na×10m=ba×10m−n
The rewriting in part (d) is a useful technique in general: choose the form of a number that makes the next step easy, then adjust back to scientific notation at the end.
Adding and subtracting in scientific notation
Adding and subtracting work differently. The digits can only be added when they have the same place value, so the two numbers must first be written with the same power of 10. It is usually easiest to rewrite the number with the smaller power so that it matches the larger one.
Scientific notation on a GDC
A graphic display calculator (GDC) shows very large and very small answers in scientific notation automatically, but it does not use the written form. A display such as `2.5E7` means 2.5×107, and `3.04E-5` means 3.04×10−5. The letter E stands for "times ten to the power". Most calculators can also be set to a scientific display mode with a chosen number of significant figures.
To enter a number in scientific notation, use the calculator's exponent key (often labelled EE or ×10x), which inserts the E. Entering `6E8` is treated as a single number, so `6E8 / 2E3` gives 300000=3×105 correctly. If the powers of 10 are typed out with × and ∧ instead, the whole denominator must be in brackets.
Very large and very small quantities
Scientific notation makes it possible to compare quantities that differ enormously in size. The exponent k gives a rough measure of size called the order of magnitude. Lengths in the natural world range from viruses to the distances between planets and beyond, and placing them on a scale in which each step is a factor of 10 shows these differences at a glance.
Equal steps on this scale are equal factors of 10. A red blood cell is about two orders of magnitude larger than a virus, and the Earth–Sun distance is about four orders of magnitude larger than the Earth's diameter. Typical values from standard references.
The scale is logarithmic: moving up one tick multiplies the length by 10, so a person and Mount Everest, which differ by a factor of about 5000, are only a few ticks apart. To compare two quantities, divide one by the other. Dividing the numbers is easy in scientific notation, and the difference in their exponents gives the number of orders of magnitude between them straight away.
Calculations with physical constants
Many scientific constants are so large or so small that they are only ever written in scientific notation. The speed of light in a vacuum is about 3.00×108 m s⁻¹, the mean distance from the Earth to the Sun is about 1.50×1011 m, and the Avogadro constant, the number of particles in one mole of a substance, is 6.022×1023 per mole. Calculations with these values follow exactly the rules for multiplying and dividing.
Comparing quantities in context
Questions about population, money and data often involve numbers in the millions, billions (109) and trillions (1012). A national debt quoted as "2.4 trillion dollars", for example, is 2.4×1012 dollars, and dividing it by the population gives the debt per person. Writing every value in scientific notation first keeps such calculations manageable.
The population figure comes from the United Nations, World Population Prospects 2024.
Exam-style questions on scientific notation
Questions on scientific notation appear on both papers. On Paper 1 (no calculator) they test the rules for the front number and the exponent; on Paper 2 (GDC allowed) they are usually set in a scientific or financial context, where the work is in reading the problem correctly and presenting the answer in the right form.
Self-check
To do:
I can write a large or a small number in scientific notation and convert it back.
To do:
I can adjust a result such as 45.6×105 into the form a×10k with 1≤a<10.
To do:
I can round to a given number of decimal places or significant figures, including final zeros such as 2.10.
To do:
I can state the interval of values that a rounded measurement could have come from.
To do:
I can multiply, divide, add and subtract numbers in scientific notation without a calculator.
To do:
I can read a GDC display such as `2.5E7` and enter numbers so that the calculator works them out correctly.
To do:
I can compare two quantities by dividing them and by their orders of magnitude.
To do:
I can solve problems in context and give the answer in scientific notation to 3 significant figures.
(M1)
r=3.50 (m) (3.50070…)A1
(b)
Find the capacity of the tank in litres, giving your answer in the form a×10k, with 1≤a<10 and k∈Z.
[3]3 marks
Markscheme (b)
V=34π(3.50070…)3=179.702… m3
Question 2 · AA-0086613 marks
The sound level, L decibels (dB), of a sound with intensity I W m−2 is given by
L(I)=10log10(10−12I),I≥10−12.
(a)
Find the sound level of a sound with intensity 3.2×10−5 W m−2.
[2]2 marks
(b)
At a concert hall, sounds have intensities in the interval 10−12≤I≤10
(b) Divide the front numbers and subtract the exponents.
96.3=0.7,103108=105
0.7×105=7×10−1×105=7×104
(c) Cube both parts. Cubing 103 multiplies the exponent by 3.
(2×103)3=23×109=8×109
(d) A square root halves the exponent, so the exponent should be even. Rewrite 1.6×109 as 16×108 first.
16×108=16×108=4×104
.
4.5×106+0.32×106=4.82×106
(b) The larger power is 10−3. Since 4×10−4=0.4×10−3,
7.1×10−3−0.4×10−3=6.7×10−3
A check with ordinary numbers: 0.0071−0.0004=0.0067.
The mass of one atom is 1.99×10−23 g to 3 significant figures.
The line would reach the Moon about 10.7 times. The line is about one order of magnitude longer than the Earth–Moon distance, which is why the answer is close to 10.
. [2 marks]
(b) Divide the front numbers and subtract the exponents.
ba=1.23.6×105−(−2)=3×107
(c) Write 2.4×104 with the power 105, then add.
3.6×105+0.24×105=3.84×105
In each part, one mark is for a correct method (adding or subtracting exponents, or matching the powers) and one for the answer in the correct form.
, where
1≤a<10
and
k∈Z
. [3 marks]
2.998×108×3.15576×107=9.4609…×1015 m
which is 9.46×1015 m to 3 significant figures. Because this is a "show that" question, both the unrounded value and the rounding to the given answer must be written.
(b) Divide by the length of one light-year, using the unrounded value from (a).
9.4609…×10154.02×1016=4.2490…≈4.25 light-years
(c) Find the time in seconds, then convert to years.
t=1.7×1044.02×1016=2.3647…×1012 s
3.15576×1072.3647…×1012=74933… years
In the form asked for, this is 7.49×104 years. Using the rounded 9.46×1015 in (b) still gives 4.25, but in longer chains the unrounded value avoids an accumulated error.
(A1)
1 m3=1000 litres, so 179702.8… litres (M1)
1.80×105 (litres) A1
Note: Award (A1)(M1)A0 for 180000 or 179703 (not in the required form).
. Find the range of
L
for these intensities.
[2]2 marks
(c)
Find an expression for L−1(x).
[3]3 marks
(d)
Hence find the intensity of a sound with a sound level of 85 dB, writing your answer in scientific notation.
[2]2 marks
(e)
Show that L−1(x+10)=10×L−1(x).
[2]2 marks
(f)
A sound is produced by a single loudspeaker. A second, identical loudspeaker is then switched on, which doubles the intensity of the sound. Find the increase in the sound level.
[2]2 marks
to show that
xdxdv=v2
.
[3]3 marks
(c)
Hence, by solving the differential equation, show that y=2−lnxx.
[5]5 marks
(d)
Find the percentage error in the estimate found in part (a).
[3]3 marks
(e)
Write down the equation of the vertical asymptote of the solution curve.
[1]1 mark
(f)
Find the exact coordinates of the point on the solution curve at which the gradient is 2.
