How to find the Standard Form of very large or small numbers?

Imagine how difficult it would be if you have to do math with an amount consisting of twenty digits. To perform any arithmetic operation on that kind of number, you would need a calculator that can accumulate that type of calculation.

Scientists deal with very big numbers, as well as extremely small numbers, by converting them to standard form, which is a decimal number followed by an exponent of 10. The decimal might be as precise as required, though it is commonly rounded to two. The exponent value denotes the magnitude of the number. The distance to the closest star in standard form is a considerably more reasonable 4.02 X 10 ¹³ kilometers.

What is Standard Form?

Standard form is also referred to as scientific notation. It is a technique of representing extremely big or extremely small integers. It’s used as shorthand in science and math, rather than writing down the whole number every time.

This form also makes it much easier to perform computations than working with a number that may have several place values. If you work with numbers with a lot of digits, either extremely large or very little, converting them to standard form will assist you in a big way.

The following notation is used to represent standard form,

B x 10 ^a

A is a number that’s known as the coefficient. The coefficient must be greater than or equal to 1 but less than 10.

• ‘B’ is the multiplication sign read as ‘times.’
• 10 is the base, and it must always be 10 in scientific notation.
• a is a number usually known as the exponent, also referred to as the power of 10.

For example,

0.00029876 = 2.9876 x 10 ^-4

Writing number in standard form could be much easier if you know how to convert a number into standard form. In the next section, we will explain the method to convert numbers in standard form.

How to convert numbers into Standard Form?

Before converting a number with an exponent, keep in mind another convention: use commas to separate number strings into groups of three or thousands. The number 2468561215, for example, is frequently expressed as 246,856,1215.

The first three digits of a number appear when the number is expressed in standard form. This is true even if the first group only has one or two numbers. The first three digits of the number 246,856,1215 for example, are 2, 4, and 6.

Small numbers, such as the radius of an atom, can be just as difficult to manage as huge ones. To convert those into standard form, you should employ the same technique. If the number is big, the decimal is placed after the first digit on the left, and the exponent is made positive. It is equal to the number of digits after the decimal. You can try out an online standard form calculator to see the steps of calculation if you are not sure about the answer.

In case the number is very small, the first three digits following the string of zeros are the same as the three used at the start of the number in normal form, and the exponent is negative. The exponent is the number of zeros multiplied by the first digit of the number series.

Here are several examples: Light travels at a speed of 299,792,458 meters per second. This is 3.00 X 10 ⁸ m/s in standard form. Be aware that you must round 299 to 300 because the fourth digit is more than 4. A hydrogen atom’s nucleus and electron are distant by 0.00000000005291772 meters. This is 5.29 X 10 ^-11 meters in standard form. You don’t need to round up because the digit after 9 in the original number is less than 5.

Performing arithmetic operations on Standard Form

·      Adding/Subtracting numbers in Standard Form

Adding and subtracting integers in standard form is simple as long as they have the same exponents. Simply add or subtract the digits. If the exponents of the numbers are not alike, convert one to the exponent of the other.

Add 1425.8 x 10 ⁵and 1.36 x 10 ⁸

As you can see, these numbers are in standard form. To add these numbers, we have to make the exponent of both numbers the same. To do so, let’s move the decimal in the number 1425.8 x 10 ⁵.

= 1425.8 x 10 ⁵ = 1.4258 x 10 ⁸

Now we can add both numbers because the exponents of both numbers are identical.

= 1.4258 x 10 ⁸ + 1.36 x 10 ⁸

Simply add the numbers on the left side of the multiplication sign and write the 10 raised to the power 8 with the result.

= 2.7858 x 10 ⁸

Subtraction is as same as addition. Just put the subtraction sign instead of the addition sign. Let’s use two standard form numbers for subtraction.

Subtract 3.93 x 10 ⁴ and 1.25 x 10 ²

To subtract these numbers, we have to make the exponent of both numbers the same as we did for adding numbers. To do so, let’s move the decimal point in the number 3.93 x 10 ⁴.

3.93 x 10 ⁴ = 393 x 10 ²

As you can see, both of the numbers now have the exponent of 2. Subtract the value before the multiplication sign and place the 10 raised to the power 2 as it is.

= 393 x 10 ² – 1.25 x 10 ²

Take 10 ² as common.

= (393 – 1.25) x 10 ²

= 391.75 x 10 ²

Move the decimal to convert the number into standard form for the final output.

= 3.9175 x 10 ^{2 + 2}

= 3.9175 x 10 ⁴

·      Multiplying/dividing numbers in Standard Form

When you multiply integers in standard form, you multiply the digits and add the exponents. When you divide one number by another, you perform the division on the number strings and subtract the exponents.

It is as simple as it gets. There is no need to complicate things when dealing with scientific form. In the case of multiplication and division, there is no need to make exponents equal. Add exponents in case of multiplication and subtract them in case of division.

Multiply 2.25 x 10 ⁴and 1.5 x 10 ⁵

Simply multiply numbers on the left side of the multiplication sign and add the exponents.

= 2.25 x 1.5 x 10 ⁴⁺⁵

= 3.375 x 10 ⁹

Divide 39.5 x 10 ⁶and 2.3 x 10 ³

Divide numbers on the left side of the multiplication sign and subtract the exponents of both of the 10s.

= (39.5 ÷ 2.3) x 10 ^{6 – 3}

= 1.7173913043 x 10 ⁴

Summary

If you’re reading this, you already understand why we need standard or scientific form when working with numbers. It is a very useful notion that is commonly utilized in many sectors of science. Aside from numbers, the standard form can also be applied to linear equations, quadratic equations, and polynomials. It has a wide range of applications that can be implemented based on the underlying needs.