1.3 Standard Form
SL 1.1
- how to write very large and very small numbers in standard form \(a \times 10^k\)
- how to change standard form back to an ordinary number
- how to multiply, divide, add and subtract numbers in standard form without a calculator
- how to read the “E” notation on your GDC
What is standard form?
Scientists often work with very large numbers (the distance to the Sun) and very small numbers (the size of a cell). Writing all the zeros is slow, and it is easy to miscount them. Standard form (also called scientific notation) solves this problem.
Standard form Not in the formula booklet
A number is in standard form when it is written as
\[ a \times 10^{k}, \qquad \text{where } 1 \le a < 10 \text{ and } k \in \mathbb{Z}. \]
- \(a\): the number in front (\(1 \le a < 10\))
- \(k\): the power of 10 (an integer)
Here \(\mathbb{Z}\) is the set of integers \(\{\ldots, -2, -1, 0, 1, 2, \ldots\}\). The number \(a\) has exactly one non-zero digit before the decimal point.
- \(3.86 \times 10^{5}\) is in standard form.
- \(38.6 \times 10^{4}\) is not, because \(38.6 \ge 10\).
- \(0.386 \times 10^{6}\) is not, because \(0.386 < 1\).
Changing to standard form
Move the decimal point until exactly one non-zero digit is in front of it. Count how many places it moved.
- A large number (10 or more): the point moves left, and \(k\) is positive.
- A small number (less than 1): the point moves right, and \(k\) is negative.
\[ 386\,000 = 3.86 \times 10^{5} \qquad\qquad 0.000\,712 = 7.12 \times 10^{-4} \]
A quick check: \(10^{-4} = \frac{1}{10\,000}\), so \(7.12 \times 10^{-4}\) is a small number. ✓
Example 1 Write each number in standard form.
(a) \(259\,000\)
(b) \(0.0503\)
(c) \(0.000\,000\,8\)
(d) \(45.2 \times 10^{3}\)
(a) The point moves 5 places left: \(259\,000 = 2.59 \times 10^{5}\).
(b) The point moves 2 places right: \(0.0503 = 5.03 \times 10^{-2}\).
(c) The point moves 7 places right: \(0.000\,000\,8 = 8 \times 10^{-7}\).
(d) \(45.2\) is too big for \(a\). Write \(45.2 = 4.52 \times 10^{1}\), then use the laws of exponents:
\[ 45.2 \times 10^{3} = 4.52 \times 10^{1} \times 10^{3} = 4.52 \times 10^{4}. \]
Example 2 Write each number as an ordinary number.
(a) \(6.02 \times 10^{4}\) \(\qquad\) (b) \(3.7 \times 10^{-3}\)
(a) Move the point 4 places right: \(6.02 \times 10^{4} = 60\,200\).
(b) Move the point 3 places left: \(3.7 \times 10^{-3} = 0.0037\).
Calculating with standard form
For multiplication and division, deal with the numbers and the powers of 10 separately. At the end, check that the answer is still in standard form.
For addition and subtraction, first write both numbers with the same power of 10.
Example 3 Without using a calculator, work out the following. Give each answer in standard form.
(a) \(\left(4 \times 10^{5}\right) \times \left(6 \times 10^{-2}\right)\) \(\qquad\) (b) \(\dfrac{9 \times 10^{7}}{1.5 \times 10^{3}}\) \(\qquad\) (c) \(3.2 \times 10^{6} + 4.5 \times 10^{5}\)
(a) \(4 \times 6 = 24\) and \(10^{5} \times 10^{-2} = 10^{3}\), so the product is \(24 \times 10^{3}\). But \(24\) is not between 1 and 10:
\[ 24 \times 10^{3} = 2.4 \times 10^{1} \times 10^{3} = 2.4 \times 10^{4}. \]
(b) \(\dfrac{9}{1.5} = 6\) and \(\dfrac{10^{7}}{10^{3}} = 10^{4}\), so the answer is \(6 \times 10^{4}\).
(c) Write the second number with \(10^{6}\): \(4.5 \times 10^{5} = 0.45 \times 10^{6}\).
\[ 3.2 \times 10^{6} + 0.45 \times 10^{6} = 3.65 \times 10^{6} \]
Standard form on the GDC
On the TI-Nspire CX II, a very large or very small answer is shown with the letter E:
| GDC screen | Meaning |
|---|---|
3.65E-7 |
\(3.65 \times 10^{-7}\) |
2.4E12 |
\(2.4 \times 10^{12}\) |
“E” is only how the calculator displays the number. Never write E in your answer — always write \(\times 10^{k}\).
To type \(2.4 \times 10^{12}\), press [EE] (the key marked EE), so the screen shows 2.4E12.
Example 4 A satellite moves in a circular orbit \(35\,800\) km above the Earth’s surface. Take the radius of the Earth to be \(6370\) km.
(a) Write down the radius of the satellite’s orbit.
(b) Find the distance the satellite travels in one orbit, correct to the nearest km.
(c) Write your answer to (b) in the form \(a \times 10^{k}\), where \(1 \le a < 10\) and \(k \in \mathbb{Z}\), giving \(a\) to 3 significant figures.
(d) One orbit takes 24 hours. Find the speed of the satellite in km per hour, in standard form to 3 significant figures.
(a) The radius is measured from the centre of the Earth: \(6370 + 35\,800 = 42\,170\) km.
(b) The distance is the circumference: \(2\pi \times 42\,170 = 264\,961.92\ldots\), so 264 962 km.
(c) \(264\,962 = 2.649\,62 \times 10^{5} \approx 2.65 \times 10^{5}\) km.
(d) Speed \(= \dfrac{264\,961.92\ldots}{24} = 11\,040.08\ldots \approx 1.10 \times 10^{4}\) km/h.
Use the unrounded value from (b) in (d), so that the rounding error does not grow.
Common errors
| ✗ Wrong | ✓ Right | Why |
|---|---|---|
| \(0.000\,4 = 4 \times 10^{4}\) | \(0.000\,4 = 4 \times 10^{-4}\) | A small number needs a negative power of 10. |
| \(24 \times 10^{3}\) left as the answer | \(2.4 \times 10^{4}\) | In standard form, \(a\) must satisfy \(1 \le a < 10\). |
| \(3.2 \times 10^{6} + 4.5 \times 10^{5} = 7.7 \times 10^{11}\) | \(3.65 \times 10^{6}\) | Adding does not add the exponents; first use the same power of 10. |
Writing 3.65E-7 as an answer |
\(3.65 \times 10^{-7}\) | E is calculator notation only. |
| Radius of orbit \(= 35\,800\) km | \(42\,170\) km | The radius goes from the centre, not from the surface. |
Exercises
1 Write each number in standard form.
(a) \(52\,000\)
(b) \(0.000\,93\)
(c) \(7\)
(d) \(0.1\)
(a) \(5.2 \times 10^{4}\) \(\qquad\) (b) \(9.3 \times 10^{-4}\) \(\qquad\) (c) \(7 \times 10^{0}\) \(\qquad\) (d) \(1 \times 10^{-1}\)
2 Write each number as an ordinary number.
(a) \(8.05 \times 10^{6}\) \(\qquad\) (b) \(2.4 \times 10^{-5}\)
(a) \(8\,050\,000\) \(\qquad\) (b) \(0.000\,024\)
3 Write each number in standard form.
(a) \(27.5 \times 10^{-3}\) \(\qquad\) (b) \(0.06 \times 10^{8}\)
(a) \(27.5 = 2.75 \times 10^{1}\), so \(27.5 \times 10^{-3} = 2.75 \times 10^{-2}\).
(b) \(0.06 = 6 \times 10^{-2}\), so \(0.06 \times 10^{8} = 6 \times 10^{6}\).
4 Without a calculator, find \(\left(3 \times 10^{4}\right) \times \left(5 \times 10^{7}\right)\) in standard form.
\(15 \times 10^{11} = 1.5 \times 10^{1} \times 10^{11} = 1.5 \times 10^{12}\)
5 Without a calculator, find \(\dfrac{8 \times 10^{-3}}{4 \times 10^{5}}\) in standard form.
\(\dfrac{8}{4} = 2\) and \(10^{-3-5} = 10^{-8}\), so the answer is \(2 \times 10^{-8}\).
6 Without a calculator, find \(\dfrac{7.2 \times 10^{5}}{9 \times 10^{-2}}\) in standard form.
\(\dfrac{7.2}{9} = 0.8\) and \(10^{5-(-2)} = 10^{7}\), so we get \(0.8 \times 10^{7}\). Since \(0.8 < 1\): \(0.8 \times 10^{7} = 8 \times 10^{-1} \times 10^{7} = 8 \times 10^{6}\).
7 Without a calculator, find \(6.4 \times 10^{5} - 3.9 \times 10^{4}\) in standard form.
\(3.9 \times 10^{4} = 0.39 \times 10^{5}\), so \(6.4 \times 10^{5} - 0.39 \times 10^{5} = 6.01 \times 10^{5}\).
8 Without a calculator, find \(\left(2 \times 10^{3}\right)^{3}\) in standard form.
\(2^{3} \times \left(10^{3}\right)^{3} = 8 \times 10^{9}\)
9 Light travels at \(3.00 \times 10^{5}\) km per second. The distance from the Sun to the Earth is \(1.50 \times 10^{8}\) km. Find the time light takes to travel from the Sun to the Earth,
(a) in seconds, in standard form, \(\qquad\) (b) in minutes and seconds.
(a) time \(= \dfrac{\text{distance}}{\text{speed}} = \dfrac{1.50 \times 10^{8}}{3.00 \times 10^{5}} = 0.5 \times 10^{3} = 5.00 \times 10^{2}\) seconds.
(b) \(500\) seconds \(= 8\) minutes \(20\) seconds.
10 A red blood cell has a diameter of \(7.5 \times 10^{-6}\) metres. How many red blood cells, placed side by side, would make a line 3 cm long? Give your answer in standard form.
First use the same unit: \(3\) cm \(= 0.03\) m \(= 3 \times 10^{-2}\) m.
\[ \frac{3 \times 10^{-2}}{7.5 \times 10^{-6}} = 0.4 \times 10^{4} = 4 \times 10^{3} \]
So about \(4 \times 10^{3}\) (4000) cells.