Question

Write in standard form:-32000. \[\]

Answer 1

Hint: We recall the definition of decimal numbers, exponent, and then the standard form. We divide and multiply the given number 32000 by ${{10}^{4}}$so that we can express the given number 32000 with a decimal point after one digit from left.\[\]

Complete step-by-step solution
We know that the decimal representation of a number is a system of representing numbers, which has both integral and fractional parts. The integral and fractional part is separated by a dot called decimal separator or decimal point for example in 12.56 the integral part is 12 and the fractional part is .56.\[\]
We also know that the exponent of a number is the number of times the number multiplied with itself. If we multiply $b$ for $n$ times, we can write it in an exponential form
\[b\times b\times ....\left( n\text{ times} \right)\times b={{b}^{n}}\]
If the number $b=10$ then we have
\[10\times 10\times ....\left( n\text{ times} \right)\times 10={{10}^{n}}\]
 The standard form otherwise known as the scientific notation of numbers is used when we want to express large numbers multiplied with an exponential of 10 with the condition that the integral part is a single digit and not zero. If the number has $n$digits before the decimal point then we divide and multiply ${{10}^{n-1}}$ to express it in the standard form.
We are given the number 32000. We count the number of digits and find that $n=5$. So divide and multiply ${{10}^{n-1}}={{10}^{5-1}}={{10}^{5}}$ with the number. We have
\[\dfrac{32000}{{{10}^{4}}}\times {{10}^{4}}=\dfrac{32000}{10\times 10\times 10\times 10}\times {{10}^{4}}=\dfrac{32000}{10000}\times {{10}^{4}}=3.2\times {{10}^{4}}\]
The obtained form $3.2\times {{10}^{4}}$ is the standard form of number.

Note: We note that we cannot use this procedure for all decimal numbers for example in 0.0000123 where we multiply and divide ${{10}^{n}}$where $n$ number of digits after the decimal point has to be shifted to get make the integral part a single digit which is this case here is $n=5$ and then we have $0.0000123=\dfrac{0.0000123}{{{10}^{5}}}\times {{10}^{5}}=1.23\times {{10}^{-5}}$. The standard form is primarily used to represent very small or very large quantities for example the wavelength of microwave and the weight of the earth.