Scientific Notation and Engineering Notation
Scientific notation and Engineering notation are used to express very large or small numbers.
Scientific Notation
A number is written in scientific notation if it is written in the form:
a × 10 r
where a is a nonzero number with one nonzero digit between -10 and 10 to the left of the
decimal point, and r is a power of 10.
In other words, a number in scientific notation is written as the product of a number and a power
of 10. The number has one digit to the left of the decimal point. The power of ten indicates how
many places the decimal point was moved.
The decimal number 650,000 written in scientific notation would be 6.5 × 10 5 because the
decimal point was moved 5 places to the left to form the number 6.5. It is equivalent to
6.5(10)(10)(10)(10)(10).
Engineering Notation
A number is written in engineering notation if it is written in the form:
a × 10 r
where a is a nonzero number with one, two or three digits between -999 and 999 to the left of the
decimal point, and r is a power of 10 that is a multiple of three.
The decimal number 650,000 written in engineering notation would be 650 × 10 3 because the
decimal point was moved 3 places to the left to form the number 650, and the power of ten is a
multiple of 3. It is equivalent to 650(10)(10)(10).
Entering numbers in a calculator using scientific or engineering notation.
Press the MODE key and set the calculator to Sci, or Eng. Scientific calculators have the
capacity to accept and perform calculations using either notation. Once the mode is set, enter a
number using the "EE" key. Do not type in "x10", or “^”, the "EE" key replaces that part of the
number.
Example: Enter 235000 on a scientific calculator.
Scientific Notation: Set the mode to Sci, enter 2.35 "EE" 5.
Engineering notation: Set mode to Eng, enter 235 “EE” 3.
Perform calculations using scientific or engineering notation.
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Example: Multiply using scientific notation: 2.35 × 10 5 5.8 × 10 7
)
enter 2.35 "EE" 5, press the multiply “X” key, enter 5.8 ”EE” 6, press “ENTER”
The answer, in scientific notation is: 1.363 × 1013
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Example: Multiply using engineering notation: 235 × 10 3 58 × 10 6
)
enter 235 "EE" 3, press the multiply “X” key, enter 58 ”EE” 6, press “ENTER”
The answer, in engineering notation is: 13.63 × 1012 . Notice the power on the ten is 12, which
is divisible by 3.
Practice:
1. Write each number in scientific notation, then in engineering notation:
a) 245,000,000,000
b) 15,000
c) 71,890,000
d) .00034
2. Write each number in decimal notation:
a) 6.98 × 10 4
b) 32 × 10 5
c) 47 × 10 −4
c) −6.98 × 10 4
Perform each of the calculations using scientific notation on a calculator. Leave your answer in
scientific notation.
4.2 × 1019
5
4
15
5
4.62 × 10 ÷ 2.4 × 10
3. 4.53 × 10 3.98 × 10
4.
5.
2.1× 10 3 5.2 × 10 4
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Perform each of the calculations using engineering notation on a calculator. Leave your answer
in engineering notation.
195 × 1015
3
3
12
6
6. 35 × 10 4.8 × 10
7. 85 × 10 ÷ 2.25 × 10
8.
( 4.25 × 10 3 )( 230 × 106 )
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)
Answers:
1. a) 2.45 × 1011 , 245 × 10 9
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b) 1.5 × 10 4 , 15 × 10 3
)
c) 7.189 × 10 7 , 71.89 × 10 6
d) 3.4 × 10 −4 , 34 × 10 −3
2. a) 69,800
b) 3,200,000
c) -69,800
d) .0047
3. 1.8 × 1010
4. 1.93 × 1010
5. 3.85 × 1011
6. 168 × 10 6
7. 37.8 × 10 6
8. 199 × 10 3