Wize High School Algebra I Textbook (Common Core) > Foundations of Algebra

Scientific Notation

Popular Courses

Algebra I

US High School

Grade 9 Math (De-streamed)

Ontario High School

MATH 122

McGill University

Find My Course

0:00 / 0:00

Scientific Notation

  

By using the number 10 and properties of exponents we can express really large numbers, or really small numbers.  This is the idea behind scientific notation

Scientific Notation
We say that a number is written in scientific notation if it has the form

a×10n\colorOne{a} \times10^{\colorThree{n}}a×10n

We must have that
a\colorOne{a}a is greater than or equal to 1 and less than 10
n\colorThree{n}n is an integer (positive or negative)
Example 1
Which of the following numbers are in proper scientific notation?

1.  2.11×2362.11 \times 23^62.11×236

ANSWER:  This is not in scientific notation.  The number 10 must be raised to a power.

2.  34.56×10434.56\times10^434.56×104

ANSWER:  This is not in scientific notation.  The value for aaa is larger than 10.

3.  4.51×100.234.51 \times 10^{0.23}4.51×100.23

ANSWER:  This is not in scientific notation.  The power must be an integer.

4.  2.98×10−32.98 \times 10^{-3}2.98×10−3

ANSWER:  This is in scientific notation.  a=2.98a = 2.98a=2.98 and n=−3n = -3n=−3.
This is the same number as 0.002980.002980.00298

Converting to scientific notation
We can put a number into scientific notation by doing the following
Determine where the decimal point should be so that the number is between 1 and 10. this will be a\colorOne{a}a.
Count how many places the decimal must be moved.  This will be n\colorThree{n}n.
If we have to move the decimal to the left (as for large numbers) n\colorThree{n}n is positive.  If we have to move the decimal to the right (as for small numbers) n\colorThree{n}n is negative.
Example 2
Express the following numbers in scientific notation

1.  45,000,00045,000,00045,000,000

ANSWER:  4.5×1074.5 \times 10^74.5×107

2.  0.000000009780.000000009780.00000000978

ANSWER:  9.78×10−99.78 \times 10^{-9}9.78×10−9

0:00 / 0:00

Working with Scientific Notation

In 2008 scientist created the worlds smallest transistor at approximately 0.000000001 meters at its thickest spot.  

1.  Express the thickness of the transistor in scientific notation.

ANSWER:  1×10−91 \times 10^{-9}1×10−9

2.  If the average sheet of paper is 1×10−41 \times 10^{-4}1×10−4 meters thick, how many of these transistors could we fit across the edge?

ANSWER:  
We can solve this be dividing the thickness of the paper, but the width o the transistor.

1×10−41×10−9=1×10−4−(−9)=1×105=100,000\begin{aligned}
\displaystyle\frac{1\times10^{-4}}{1 \times10^{-9}} &= 1 \times 10^{-4 - (-9)} \\
&= 1 \times 10^{5} \\
&= 100,000
\end{aligned}1×10−91×10−4​​=1×10−4−(−9)=1×105=100,000​
This shows that on average you could fit 100,000 transistors.

Scientific Notation and the Internet

On an average day in 2021, there are approximately 533 billion searches performed.

1.  Express this number in scientific notation.

2.  There are approximately 8.64×1058.64 \times 10^58.64×105 seconds in a day.  About how many searches are performed every second?

1.  Express this number in scientific notation.

A)533×109533 \times 10^9533×109

B) 533×10−9533 \times 10^{-9}533×10−9

C)5.33×10−115.33\times10^{-11}5.33×10−11

D)5.33×10115.33\times10^{11}5.33×1011

I don't know

Scientific Notation and Hamburgers

On average Americans consume about 3.2 hamburgers a week.  If we assume that there are about 52 weeks in a year, and that there are about 300 million Americans, how many hamburgers are consumed in a year?  (Express your answer in scientific notation, with aaa rounded to two decimal places)

Answer

I don't know