While writing numbers that have many zeroes, there is a chance to occur an error by omitting zeros or writing more than the actual number of zeros. To overcome this problem, Archimedes developed a concise and convenient method of writing small or large numbers in the simplest form by using base 10. Standard form is used when we want to concisely write a very large or very small number. Weare going to discuss the standard form of a number with examples. We will learn how to convert ordinary numbers into standard form. To understand it more, we will solve some examples.
Standard form of a number in mathematics
“Scientific notation” is another name for the standard form. In Mathematics, a number is written as m ×10nand 1 ≤ m < 10, and n is any integer is called standard form or scientific notation. For example,
The mass of a proton ≈0.000000000000000000000001672 gram
In standard form, the mass of a proton ≈ 1.6726 × 10-24 g
How to convert ordinary numbers into standard form?
Let’s learn how to write any ordinary number that should be greater or equal to 1.
- Shift the decimal point after the first non-zero digit of the given number.
- Count digits between the first non-zero digit and the decimal point, say n. Then write it as 10n.
- If the decimal point moves to the left, then we will write it as 10n.If the decimal point moves to the right, we will write a negative sign on a power of 10i.e. 10-n. We get 100 when the decimal point does not need to move.
- Neglect all zeroes, when the decimal point moves to the right side.
- To change the standard form into an ordinary number, we will take the inverse of the above steps.
The standard form of a Fraction or rational number
A rational number is defined as “Any number that can be written in the form of p / q. where p and q are any integers but q ≠ 0”. If the denominator and the numerator have no common divisor except 1 then it is said to standard form of a rational number. In other words, the GCD (greatest common divisor) of p and q should be 1. For example, a standard form of 6 / 12 is 1 / 2 because the greatest common divisor of 1 and 2 is 1.
How to convert the expanded form of a number into standard form?
In expanded form, each digit of the number shows its place value. For example, the expanded form of 20356 is 20000 + 0000+ 300 + 50 + 6, if we want to change it to standard form then add them.
We can obtain a standard form by writing the first value of each term of the expanded form. i.e.
4000 + 600 + 70+ 0= 4670
| Expanded form | Standard form |
| 5000 + 400 + 10 + 9 | 5419 |
| 60000 + 9000 + 300 + 60 + 2 | 69362 |
| 900000 + 70000 + 5000 + 300 + 20 + 1 | 975321 |
| 800000 + 00000 + 0000 + 200 + 30 + 5 | 800235 |
Some solved Examples
Example1.
Write -33 / 39 in standard or simplest form.
Solution
-33 and 39 both are divisible by 3, so by dividing 3 we get
-11 / 13
So the common divisor of 11 and 13 is 1, hence -11 / 13 is the simplest form of -33 / 39
Example2.
Write 0.000000059 in standard form.
Solution
First, move the decimal point after first non-zero digit, here 5 is first non-zero digit so place the point after 5 and neglect all zeros. The decimal point moves to the right, we will write a negative sign on a power of 10.
0.000000059 = 5.9 × 10-8
The above problem of converting smaller numbers in standard form can also be evaluated with the help of a standard form calculator by Meracalculator.
Example3.
Change 6.678 × 109 in ordinary notation.
Solution
As, 109 = 1000000000
To remove the decimal point from 6.678, divide it by 1000,
After a cancellation, we get, 6678000000
Some other solved examples are in the following table.
| Ordinary number | Standard form (scientific notation) |
| 79043 | 79043 × 104 |
| 7.890534 | 7.890534 × 100 |
| 0.000000008 | 0.8 × 10-9 |
| 760096754.40 | 76009675440 × 104 |
| 0.000000006666789 | 6.666789 × 10-9 |
Unsolved question
Solve the following question by yourself
- Write 78.324 in standard form.
- Write 0.0000000657 in standard form.
- Write 0.897 × 10-11 in the ordinary form.
- Write 6.7894 × 109 inthe ordinary form.
Some important questions related to a standard form of a number
First time when and who used the standard form?
In the 3rd century, Archimedes used it first time to write the number of grains of sand in our universe.
When do we need the standard form of a number?
Sometimes, we face very small or very large values, with many numbers of zeroes. There is a chance to occur an error by omitting zeros or writing more than the actual number of zeros. Therefore, we use the standard form of a number.
Is any difference between standard form and scientific notation?
There is no difference between standard form and scientific notation. Standard form, scientific notation, and standard index form are the same.
Can you write 1 in a standard form?
Yes, 1 can be written in standard form, as we know that any integer that has power 0 is always equal to 1.
In standard form 1 can be written as 1 × 100
How I can write 25000000000000 in standard form?
The standard form of 25000000000000 is 2.5× 1013
Conclusion
In this article, we’ve discussed the standard form of a number with its examples. We learned how we could convert ordinary numbers into standard notation. We have discussed the standard form of any rational numbers. Then we learned how to convert the expanded form into the standard notation.
In the example section, we solved many examples related to scientific notation for you. Further, we answered some questions that can arise in your mind while reading this article. After reading this article, you can solve any question related to standard form.