Scientists and engineers often work with very large and very small numbers. The ordinary practice of using commas and leading zeroes proves to be very cumbersome in this situation. Scientific notation is more compact and less error prone method of representation. The number is split into two portions: a precision part (the mantissa) and a magnitude part (the exponent, being a power of ten). For example, the value 23,000 could be written as 23 times 10 to the 3rd power (that is, times one thousand). The exponent may be thought of in terms of how places the decimal point is moved to the left. Spelling this out is awkward, so a shorthand method is used where “times 10 to the X power” is replaced by the letter E (which stands for exponent). Thus, 23,000 could be written as 23E3. The value 45,000,000,000 would be written as 45E9. Note that it would also be possible to write this number as 4.5E10 or even 0.45E11. The only difference between scientific notation and engineering notation is that for engineering notation the exponent is always a multiple of three. Thus, 45E9 is proper engineering notation but 4.5E10 isn’t. On most scientific calculators E is represented by either an “EE” or “EXP” button. The process of entering the value 45E9 would be depressing the keys 4 5 EE 9.

For fractional values, the exponent is negative and may be thought of in terms how many places the decimal point must be moved to the right. Thus, 0.00067 may be written as 0.67E−3 or 6.7E−4 or even 670E−6. Note that only the first and last of these three are acceptable as engineering notation.

Engineering notation goes one step further by using a set of prefixes to replace the multiples of three for the exponent. The prefixes are:

E12 = Tera (T) |
E9 = Giga (G) |
E6 = Mega (M) |
E3 = kilo (k) |

E−3 = milli (m) |
E−6 = micro (\(\mu\)) |
E−9 = nano (n) |
E−12 = pico (p) |

Table \(\PageIndex{1}\)

Thus, 23,000 volts could be written as 23E3 volts or simply 23 kilovolts.

Besides being more compact, this notation is much simpler than the ordinary form when manipulating wide ranging values. When multiplying, simply multiply the precision portions and add the exponents. Similarly, when dividing, divide the precision portions and subtract the exponents. For example, 23,000 times 0.000003 may appear to be a complicated task. In engineering notation this is 23E3 times 3E−6. The result is 69E−3 (that is, 0.069). Given enough practice it will become second nature that kilo (E3) times micro (E−6) yields milli (E−3). This will facilitate lab estimates a great deal. Continuing, 42,000,000 divided by 0.002 is 42E6 divided by 2E−3, or 21E9 (the exponent is 6 minus a negative 3, or 9).

When adding or subtracting, first make sure that the exponents are the same (scaling if required) and then add or subtract the precision portions. For example, 2E3 plus 5E3 is 7E3. By comparison, 2E3 plus 5E6 is the same as 2E3 plus 5000E3, or 5002E3 (or 5.002E6).

Perform the following operations. Convert the following into scientific and engineering notation.

1. 1,500

2. 63,200,000

3. 0.0234

4. 0.000059

5. 170

Convert the following into normal longhand notation:

6. 1.23E3

7. 54.7E6

8. 2E−3

9. 27E−9

10. 4.39E7

Use the appropriate prefix for the following:

11. 4E6 volts

12. 5.1E3 feet

13. 3.3E−6 grams

Perform the following operations:

14. 5.2E6 + 1.7E6

15. 12E3 − 900

16. 1.7E3 \(\cdot\) 2E6

17. 48E3 / 4E6

18. 20 / 4E3

19. 10 M \(\cdot\) 2 k

20. 8 n / 2 m