Express large numbers in the standard form

Introduction

The simplest form in which a number can be written is called standard form. The goal is to simplify number calculations as well as reading and writing. Every mathematical concept, including integers, fractions, equations, and expressions, has a specified standard form. The standard form can be thought of as the most basic approach to represent a mathematical element. Any value between \(1.0\) and \(10.0\) that can be expressed as a decimal number and multiplied by a power of 10 is referred to as being in standard form.

Expressing large numbers in the standard form

The easiest form of decimal numbers to read and write is in standard form. For instance, \(9 \times {10^{ – 3}}\) is the standard form of the decimal number \(0.009\). Very large or very small numbers might be challenging to read or write at times. We therefore use standard form while writing them. Any number can be written in standard form, not just decimals. Some fractions result in decimal numbers with additional digits at the thousandth, hundredth, or tenth places.

In general, we can say that it is the standard form representation of rational numbers. The definition of a rational number is any number that can be written as p/q, where p and q are both integers. As an example, \(1/13,{\rm{ }}4/15,{\rm{ }}8/9\) , etc.

Standard form

In mathematics, the most usual way to represent a specific element is called standard form. Every mathematical concept, including big numbers, small numbers, equations, and lines, has a standard form. Explore this exciting idea of standard form as it relates to many math concepts, including fractions, equations, algebra, slope, and learning the standard from a formula.

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What are standard form

A standard form is a way to express a particular mathematical idea, such as an equation, number, or expression, in prose that follows to a set of rules. The standard form is used to express very big or very small numbers clearly. For instance, \(4,500,000,000\) years is how \(4.5\) billion years is written. As you can see, it is difficult and time-consuming to write a large number like \(4.5\) billion in its number form. There is also a chance that we may write a few of more or less than necessary. It is quite beneficial in this situation to write the number in standard form. For instance, \(4,500,000,000\) in standard form equals  \(4.5 \times {10^9}\) . Additionally to integers, other mathematical constructs like fractions, equations, expressions, and polynomials can.

Standard form of a number

Writing a very large or extremely small number using powers of \(10\)  multiplied by values between  \(1\)  and \(10\) is known as scientific notation. 3890, for instance, can be written as \(3.89 \times {10^3}\) . Use positive powers of \(10\)  to express these values, which are bigger than \(1\) . The negative power of ten is used for numbers lower than one. For instance, \(0.0451\) may be expressed as \(4.51 \times {10^{ – 2}}\).

Express the following numbers in standard form

(1.) \(0.0000000000069\)

\( = \frac{{69}}{{10000000000000}}\)

\( = \frac{{69}}{{{{10}^{13}}}} = 69 \times {10^{ – 13}}\)

\( = 6.9 \times 10 \times {10^{ – 13}}\)

\( = 6.9 \times {10^{ – 12}}\)

(2.) \(90000000\)

\(\; = 9 \times {10^7}\)

(3.)  \(2650000000\)

\(\; = 2.65 \times {10^9}\)

How to write numbers in standard form 

The stages to writing a number in its standard form are as follows:

Step 1: Write the first digit of the supplied number in step one.

Step 2: After the first number, add the decimal point.

Step 3: Next, count how many digits there are in the supplied number after the first one and express that number as a power of \(10\) .

For instance, the number is \(52300000000\) . Thus, the following is how a number is represented in standard form:

The initial number is \(5\) .

Step 2: Input the decimal point to make \(5\) into “\(5\)”.

Step 3: There are \(10\) digits after the number \(5\) .

Consequently, \(5.23 \times {10^{10}}\) is the conventional form of \(52300000000\) .

Express the following numbers in usual form

Shift the decimal to the left by the number of places equal to the power of  \(10\) to convert a smaller number (negative powers of \(10\) ) to its standard form. The decimal point must be moved to the right by the number of places equal to the power of 10 in order to change a large number (positive powers of \(10\) ) to its basic format.

4. \(4.5{\rm{ }} \times {10^5} = {\rm{ }}450000\)
5. \(7.8 \times {10^6} = 7800000\)
6. \(9.7 \times 109 = 9700000000\)

Standard form definition 

A standard form is a way to express mathematical ideas like an equation, an expression, or some numbers. Example 2,500,000,000 years is the same as \(2.5\) billion years. As you can see, it is challenging and time-consuming to read or write a large number like \(2.5\) billion. Therefore, we utilize the standard form to precisely write large or tiny integers.

Solved examples 

Example 1: Use an exponential function to represent the separation between the Earth and the Sun.

Answer: The Earth’s distance from the Sun is \(1496000000km\) .

Therefore,

\(1496000000Km{\rm{ }} = 1.496 \times 109Km\)

Example 2: Use the conventional form to describe the size of blood cells.

Answer: Human blood cells typically measure \(0.000015\) m in size.

Therefore,

\(0.000015 = 1.5 \times {10^{ – 5}}m\)

Example 3.Write \(3253\) in standard form.

Answer: \(3.253 \times 1000\) can be used to represent the number 3253. The standard form of 3253 is \(3.253 \times {10^3}\) .

Conclusion

Technically speaking, large numbers are defined as those that are greater than those observed frequently. Large numbers in the number system are those that are typically bigger or greater than the other numbers. For instance, large figures like 1 lakh, 1 million, 1 billion, etc. are not ones we use frequently. In essence, standard form is used to indicate these large integers.

Frequently Asked Questions

1. What is the canonical form of a number?

Ans. The canonical form of a number is a way of writing numbers in a form

That follows certain rules. Numbers that can be written as decimal numbers

Between 1.0 and 10.0 multiplied by powers of 10 are called canonical forms.

2. What is the standard form of a decimal number?

Ans. The standard form of a decimal number is the representation of a given Decimal number raised to the tenth power in order to simplify to the original Value.

3. Are the standard form and the standard notation of decimal numbers the Same?

Ans. The standard form is also called the standard notation. So, both are the

Same.