Question

Write the following numbers using base $10$ and exponents:
(i) $12345$
(ii) $1010.0101$
(iii) $0.1020304$

Answer 1

(i) ${12345}_{\left(10\right)}$

$=1\times {10}^4+2\times {10}^3+3\times {10}^2+4\times {10}^1+5\times {10}^0$

$={10}^4+2\times {10}^3+3\times {10}^2+4\times {10}^1+5$

(ii) ${1010.0101}_{\left(10\right)}=1\times {10}^3+0\times {10}^2+1\times {10}^1+0\times {10}^0+0\times \frac{1}{10}+1\times \frac{1}{{10}^2}+0\times \frac{1}{{10}^3}+1\times \frac{1}{{10}^4}$

$={10}^3+{10}^1+\frac{1}{{10}^2}+\frac{1}{{10}^4}$

(iii) ${0.102030}_{\left(10\right)}^4=1\times \frac{1}{10}+0\times \frac{1}{{10}^2}+2\times \frac{1}{{10}^3}+0\times \frac{1}{{10}^4}+3\times \frac{1}{{10}^5}+0\times \frac{1}{{10}^6}+4\times \frac{1}{{10}^7}$

$=\frac{1}{10}+2\times \frac{1}{{10}^3}+3\times \frac{1}{{10}^5}+4\times \frac{1}{{10}^7}$