Scientific Notation (Standard Form)

Definition of Scientific Notation (Standard Form)

Scientific notation (standard form) is commonly used to simplify writing very large numbers or very small numbers. Large numbers or small numbers are converted into multiplication of numbers to exponents. In addition to simplifying writing, another benefit of scientific notation (standard form) is to simplify the operation of the number.

General Form of Scientific Notation (Standard Form)

Scientific notation (standard form) of a positive numbers are written in the following form:

\[\boxed{a \times {10^n}}\]

where $1 \le a < 10$ and $n$ are integers.

Converting Common Numbers to Scientific Notation (Form Standard)

State the following numbers in scientific notation (standard form)!

1. $146,000,000,000$
2. $35,000,000,000 .000$
3. $4,476,000,000$
4. $0.00000000000000234$
5. $0.0000000363535$

Answer

1. Problem a
$\begin{array}{l} 146.000.000.000 &= 1.46 \times 1000000.000.000\ \ &= 1.46 \times {10^{11}} \end{array}$
2. Problem b
$\begin{array}{l} 35,000,000,000,000,000 &= 3.5 \times 10,000,000,000,000\\ &= 3.5 \times {10^{13}} \end{array}$
3. Problem c
$\begin{array}{l} 4,476,000,000 & = 4,476 \times 1.000.000.000\\ &= 4,476 \times {10^9} \end{array}$
4. Problem d
$\begin{array}{l} 0.00000000000000234 &= \frac {{2,34}}{{1,000,000,000,000,000}}\\ &= \frac{{2,34}}{{{{10}^{15}}}}\\ &= 2.34 \times {10^{ - 15}} \end{array}$
5. Problem e
$\begin{array}{l} 0.0000000363535 &= \frac{{3,63535}}{{1,000,000,000}}\\ &= \frac{{3,63535}}{{{ {10}^9}}}\\ &= 3,63535 \times {10^{ - 9}} \end{array}$

Converting Scientific Notation (Standard Form) to Common Numbers

Express scientific notation (standard form) into ordinary numbers!

1. $2.5 \times {10^6}$
2. $1.5 \times {10^4}$
3. $7.57 \times {10^{11}}$
4. $4.59 \times {10^{ - 7}}$
5. $3.51 \times { 10^{ - 5}}$

Answer

1. Problem a
$ \begin{array}{l} 2.5 \times {10^6} &= 2.5 \times 1,000,000\\ &= 2,500,000 \end{array}$
2. Problem b
$\begin{array}{l} 1.5 \times {10^4} &= 1.5 \times 10,000\\ &= 15,000 \end{array}$
3. Problem c
$\begin {array}{l} 7.57 \times {10^{11}} &= 7.57 \times 1000000.000.000\\ &= 757.0000.000.000 \end{array}$
4. Problem d
$\begin{array}{l} 4.59 \times {10^{ - 7}} &= \frac{{4.59}}{{{{10}^7}}}\\ &= \frac {{4.59}}{{10,000,000}}\\ &= 0.000000459 \end{array}$
5. Problem e
$\begin{array}{l} 3.51 \times {10^{ - 5}} &= \frac{{3.51}}{{{{10}^5}}}\\ &= \frac{{3.51}}{{100,000}}\\ &= 0.00000351 \end{array}$

Thus the discussion about scientific notation (standard form) and some examples. If you have any questions, please doodle in the comments column. Hope it's useful