STANDARD FORM OF A NUMBER

STANDARD FORM OF A NUMBER

The Standard Form of a number with examples: Standard form is an important topic in mathematics that all students MUST understand. In mathematics, you might be given a question and finally be requested to leave your answer in standard form. This article will explain ineptly everything you need to know on this topic.

STANDARD FORM DEFINITION

Standard form is used to express very large and very small numbers. Examples of such numbers are 237600000, 0.000009. The standard form is written as

\dpi{120} {\color{DarkOrange} a\times 10^{x}}

Where, \dpi{120} a is a number between 1 and 10, and, \dpi{120} x can be a positive (+Ve) or negative (-Ve) integer.

In the next two paragraphs, we will consider examples of large and small numbers expressed in standard form.

RECOMMENDED: Logarithm of Numbers and Permutation and Combination.

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STANDARD FORM OF LARGE NUMBERS

The number 237600000 can be expressed as 2.376\times 100000000 which will be equal to 2.376\times 10^{8}. These value is its standard form, that is in the format {\color{DarkOrange} a\times 10^{x}}.

Where a=2.376 which lies between 1 and 10 as defined above and n=8 which is a positive (+ve) integer.

Sometimes we can easily get the standard form of large value like the above, by simply counting the number of digits from the RIGHT to LEFT up to the point (at blue arrow) the decimal point will be placed. For example, the number 237600000 is

The digit count is 8 digit from the right to left. So we will have 2.376\times 10^{8}.

STANDARD FORM OF SMALL NUMBERS

Numbers that are very small are better expressed in standard form. Consider the number, 0.000038. This number is equal to \frac{3.8}{100000}. That is

{\color{DarkOrange} \frac{3.8}{100000}=3.8\times \frac{1}{100000}=3.8\times \frac{1}{10^{5}}=3.8\times 10^{-5}}

However, we can easily get the standard form of very small values by simply counting the number of digits from the decimal point in the left to the point between the first two real numbers.

Counting from the left to the right (negative counting), we get 3.8\times 10^{-5}.

Any number can be expressed in its standard form. But in most instances, it is usually more convenient to use the standard form for very large or very small numbers.

Astronomers use the standard form of number extensively because they use very large numbers and very small numbers in calculations. For example, the average distance between the Earth and the Sun is about 93 million miles (150 million kilometers).

Let us consider some examples on the standard form of large and small numbers as real examples and see the solutions to such questions.

STANDARD FORM EXAMPLES WITH REAL SOLUTIONS

Write in standard form the numbers: i.  178000 and ii.  0.0000082

  • Count the number 178000 from the right to the left (+ve counting), we will have 1.78\times 10^{5}. Note, this can also be 17.8\times 10^{4}, it all depends on where you place the decimal point.
  • Counting the number 0.0000082 from the decimal point to the right (the point between the first two real numbers), we get \frac{8.2}{1000000}=\frac{8.2}{10^{6}}=8.2\times 10^{-6}

Recall that \frac{1}{10^{6}}=10^{-6}

Also Read: Surds

In some instances, you might be given a question that will require you to add and subtract or multiply and divide two numbers in standard forms. In the next two paragraphs, we will learn how to solve such questions.

ADDITION AND SUBTRACTION IN STANDARD FORM

In order to add or subtract expressions in standard form, the powers of 10 of the two numbers must be made equal. See the examples below:

  • Add 6.02\times 10^{8}+ 2.68\times 10^{7}

In the above example, we have to make 2.68\times 10^{7} have 10^{8} by moving the decimal point. This will become 0.268\times 10^{8}.

∴ 6.02\times 10^{8} + 0.268\times 10^{8}=(6.02+0.268)\times 10^{8}

=6.308\times 10^{8}

  • Subtract.

Considering the two values, let’s make 2.45\times 10^{4} have same power as 10^{6}. We will have 0.0245\times 10^{6}.

∴ 9.22\times 10^{6}-0.0245\times 10^{6}=(9.22-0.0245)\times 10^{6}

= 9.1955\times 10^{6}

MULTIPLICATION AND DIVISION OF STANDARD FORM

The process of multiplication and division follows a similar rule as in the operation of indices. For example, if P=A\times 10^{x} and Q=M\times 10^{y}

∴ PQ=(A\times 10^{x})\times (M\times 10^{y}), which will further be PQ=AM\times 10^{x+y}

Also in the division,  \frac{P}{Q}=\frac{A\times 10^{x}}{M\times 10^{y}}

∴ \frac{P}{Q}=\frac{A}{M}\times 10^{x-y}

Furthermore, the practical examples below will explain how to carry out this operation.

  • Simplify (2.5\times 10^{9})\times (4.9\times 10^{2}) in standard form.

From the above, (2.5\times 4.9)\times 10^{9}\times 10^{-2}

=12.25\times 10^{9+(-2)}

=12.25\times 10^{9-2}=12.25\times 10^{7}

  • Evaluate 5.6\times 10^{16} ÷ 0.14\times 10^{11}

The above will give, \frac{5.6}{0.14}\times \frac{10^{16}}{10^{11}}, simplifying these will give,

40\times 10^{16-11}=40\times 10^{5}

For more quizzes on the standard form of numbers, download the worksheet. Click HERE to download.

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August 26, 2021
MATH ARTICLES

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