Standard Notation

Julia asked her kids, "Do you know how old the Earth is?"

They thought and said that the Earth was older than her, them, and their grandparents. Julia told them that the Earth was believed to have been formed over 4.5 billion years ago. She asked them if they could try counting this number on their fingers. When they replied that they couldn't, she told them that a number as huge as this can be expressed mathematically in its standard notation only.

In this mini-lesson, we will explore the world of scientific and standard notations of the numbers by finding the answers to the questions like what is a standard notation and how to convert scientific notation to standard notation.

Let us explore the chapter Standard Notation in detail.

Lesson Plan

A standard notation is a form of writing a given number, an equation, or an expression in a form that follows certain rules.

For example, 4.5 billion years is written as 4,500,000,000 years

As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few zeros less or more while writing in the number form.

So, to represent very large or very small numbers concisely, we use the standard notation.

In Britain, standard notation is also known as scientific notation where a large number is written in the form of power of 10. 

Depending upon which mathematical concept we are dealing with, the standard notation will vary.

The standard form has different meanings depending on which country you are in.

In the United States and countries using US conventions, the standard form is the usual way of writing numbers in decimal notation.

Standard form = \(3890\)

Expanded form = \(3000 + 800 + 90\)

Written form = Three thousand eight hundred and ninety

Let us now look at how to convert standard notations to scientific notation and vice-versa.

---

How To Convert Scientific Notation To Standard Notation?

Why in the above section 800 is written as \(8\times10^2\) in scientific notation?

\(800\) = \(8\times100\)

\(100=10\times10\ or 10^2\)

Hence, 800 can be written in both the ways.

Power to the 10 shows the exact place of decimal to be moved.

To find the place of decimal you should follow the below points:

a) If the power on 10 is greater than or equal to 10. The decimal point will move to the left side. This case implies only if the power of 10 is positive. For example, \(10^{2}\)
b) If the power on 10 is lesser than 1. The decimal point will move to the right side. This case implies only if the power of 10 is negative. For example, \(10^{-2}\) 

Let us look at the sample problem,

\(6.5\times10^{-3}\) can be written as \(6.5\times0.001\) or \(0.0065\)

Here one decimal before \(5\) represents one zero. If exponent in standard notation or scientific notations are in minus \(10^{-3}\) it is represented as \(\dfrac{1}{1000}\).

Example 1

 

 

The mass of Neptune is \(83,000,000,000,000,000,000,000\) \(lb\). Help Jamie to write this in scientific notation.

Solution

Here, standard notation \(83,000,000,000,000,000,000,000\) is written as \(83\) followed by 21 zeroes.

To write the number in scientific notation, Zoe needs to change \(83\) to \(8.3\).

Therefore, Jamie has to think of how many decimal places to be moved to the left side, i.e., 22 places.

So, the scientific notation for the number is \(8.3 \times 10^{22}\) \(lb\).

\(\therefore\) \(8.3 \times 10^{22}\) \(lb\).

Example 2

 

 

Can you help Zoe to write \(7.56\times10^{11}\) in the standard notation?

Solution

Here, \(7.56\) is \(756\). Now, Zoe has to move the decimal point 11 places to the right and add zeroes accordingly.

The standard notation for \(7.56\times10^{11}\) is \(756,000,000,000\). 

\(\therefore\) \(756,000,000,000\)

Example 3

 

 

The speed of a sound in the air is \(1.53645\times10^{5}\) miles per hour. Help Tim to write this in standard notation.

Solution

The power of \(10\) is \(5\), so move the decimal point \(5\) places to the right:

\(1.53645\) \(→\) \(15.3645\) \(→\) \(153.645\) \(→\) \(1536.45\) \(→\) \(15364.5\) \(→\) \(153645\)

So the speed of sound in the air is \(153645\ mph\).

\(\therefore\) \(153645\) miles per hour.