Question

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

Answer 1

Hint: We use the method of conversion of given numbers using base 10. Use the place of each digit to determine the power of$${10}^n$$.
* Base 10 number system is called the decimal system having ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We use place value and decimal point to separate whole numbers and decimal numbers. Example: $$345=3\times 100+4\times 10+5\times 1$$.
* Law of exponents’ states that when the base is same powers can be added. $$\underset{n}{\underbrace{10\times 10\times ....\times 10}}={10}^n$$

Complete step-by-step answer:
The place value of any digit in a number is the position ones, tens, hundreds, thousands, ten thousands, lakhs, and ten lakhs and so on starting from right to left.
We solve each part separately.
(i) 12345
We are given the number 12345
Place value of 5 is ones.
Place value of 4 is tens.
Place value of 3 is hundreds.
Place value of 2 is thousands.
Place value of 1 is ten thousands.
We can write the given number is
$$\Rightarrow 12345=1\times 10000+2\times 1000+3\times 100+4\times 10+5\times 1$$ … (1)
Use law of exponents we can write $$10000={10}^4;1000={10}^3;100={10}^2;10={10}^1;1={10}^0$$
Substitute the value in equation (1)
$$\Rightarrow 12345=1\times {10}^4+2\times {10}^3+3\times {10}^2+4\times {10}^1+5\times {10}^0$$
$$\therefore$$Conversion of 12345 to base 10 conversions is $$1\times {10}^4+2\times {10}^3+3\times {10}^2+4\times {10}^1+5\times {10}^0$$.
(ii) 1010.0101
Here we have digits after the decimal place as well. We simply write the place value of the digits after the decimal in the denominator of the fraction. Since we know when writing the place value of digits without decimal we move from right to left. When writing place value of digits after the decimal we move from left to right.
Before the decimal:
Place value of 0 is ones.
Place value of 1 is tens.
Place value of 0 is hundreds.
Place value of 1 is thousands.
After the decimal:
Place value of 0 is $$1/$$tens.
Place value of 1 is $$1/$$hundreds.
Place value of 0 is $$1/$$thousands.
Place value of 1 is $$1/$$ten thousands.
We can write the given number is
$$\Rightarrow 1010.0101=1\times 1000+0\times 100+1\times 10+0\times 1+0\times {\displaystyle \frac{1}{10}}+1\times {\displaystyle \frac{1}{100}}+0\times {\displaystyle \frac{1}{1000}}+1\times {\displaystyle \frac{1}{10000}}$$ … (2)
Use law of exponents we can write $$10000={10}^4;1000={10}^3;100={10}^2;10={10}^1;1={10}^0$$
Substitute the value in equation (2)
$$\Rightarrow 1010.0101=1\times {10}^3+0\times {10}^2+1\times {10}^1+0\times {10}^0+0\times {\displaystyle \frac{1}{{10}^1}}+1\times {\displaystyle \frac{1}{{10}^2}}+0\times {\displaystyle \frac{1}{{10}^3}}+1\times {\displaystyle \frac{1}{{10}^4}}$$
Remove all the multiplicative terms with zero.
$$\Rightarrow 1010.0101=1\times {10}^3+1\times {10}^1+1\times {\displaystyle \frac{1}{{10}^2}}+1\times {\displaystyle \frac{1}{{10}^4}}$$
$$\therefore$$Conversion of 1010.0101 to base 10 conversions is$$1\times {10}^3+1\times {10}^1+1\times {\displaystyle \frac{1}{{10}^2}}+1\times {\displaystyle \frac{1}{{10}^4}}$$.
(iii) 0.1020304
Here we have digits after the decimal place as well. We simply write the place value of the digits after the decimal in the denominator of the fraction. Since we know when writing the place value of digits without decimal we move from right to left. When writing the place value of digits after the decimal we move from left to right.
Before the decimal:
Place value of 0 is ones.
After the decimal:
Place value of 1 is $$1/$$tens.
Place value of 0 is $$1/$$hundreds.
Place value of 2 is $$1/$$thousands.
Place value of 0 is $$1/$$ten thousands.
Place value of 3 is $$1/$$lakhs.
Place value of 0 is $$1/$$ten lakhs.
Place value of 4 is $$1/$$Crore.
We can write the given number is
$$\Rightarrow 0.1020304=0\times 1+1\times {\displaystyle \frac{1}{10}}+0\times {\displaystyle \frac{1}{100}}+2\times {\displaystyle \frac{1}{1000}}+0\times {\displaystyle \frac{1}{10000}}+3\times {\displaystyle \frac{1}{100000}}+0\times {\displaystyle \frac{1}{1000000}}+4\times {\displaystyle \frac{1}{10000000}}$$ … (3)
Use law of exponents we can write $$10000000={10}^7;1000000={10}^6;100000={10}^5;10000={10}^4;1000={10}^3;100={10}^2;10={10}^1;1={10}^0$$
Substitute the value in equation (3)
$$\Rightarrow 0.1020304=0\times {10}^0+1\times {\displaystyle \frac{1}{{10}^1}}+0\times {\displaystyle \frac{1}{{10}^2}}+2\times {\displaystyle \frac{1}{{10}^3}}+0\times {\displaystyle \frac{1}{{10}^4}}+3\times {\displaystyle \frac{1}{{10}^5}}+0\times {\displaystyle \frac{1}{{10}^6}}+4\times {\displaystyle \frac{1}{{10}^7}}$$
Remove all the multiplicative terms with zero.
$$\Rightarrow 0.1020304=1\times {\displaystyle \frac{1}{{10}^1}}+2\times {\displaystyle \frac{1}{{10}^3}}+3\times {\displaystyle \frac{1}{{10}^5}}+4\times {\displaystyle \frac{1}{{10}^7}}$$
$$\therefore$$Conversion of 0.10202304 to base 10 conversions is $$1\times {\displaystyle \frac{1}{{10}^1}}+2\times {\displaystyle \frac{1}{{10}^3}}+3\times {\displaystyle \frac{1}{{10}^5}}+4\times {\displaystyle \frac{1}{{10}^7}}$$.

Note: Students many times make mistakes in converting the values of digits after the decimal place as they think that like we move from right to left in writing the place value, we will start from the end of the number after the decimal and move in right to left manner which is wrong. Students can opt for understanding the concept of writing the place value for digits after the decimal using the concept that when converting a decimal to fraction, we write denominator as$${10}^n$$, where n is the number of digits after the decimal.