This week we are going to take a look at scientific notation. You have probably seen it before, but you may not know what it is or how to use it. It is very common in math and science, where there large numbers or fractions that need to be rewritten as something shorter!

Scientific notation was first used around 500 BC in China and Greece. Since then, it has become one of the most prevalent ways to rewrite equations.

There are two parts to scientific notation. The exponent part goes after the decimal place and the pre-exponent part comes before the decimal place.

The exponent can *get really big* and even bigger than 100. When this happens, the equation is rewrote so that the 0s and 1s are replaced with a *new symbol called e* (for exponential). These new symbols are then put into the *original expression along* with the number 10 to create the final product.

The products of these changes are longer and narrower, which makes them easier to understand and work with. For example, the above expression could be rewitten as 5e6.

This way of rewriting equations is helpful because now instead of having a long, confusing expression, you have a short, easy to read one.

## Examples of scientific notation

There are many ways to **use scientific notation** in writing. Some examples include when you want to say that something is very large or very small, when you want to express a decimal as an exponent, and when you need to represent a value that can’t be *expressed using standard numbers*.

When you see a poof of text with lots of weird looking numbers after it, that’s usually because it’s called scientific notation. It was originally designed for mathematics, where there are rules about how to convert from normal (or conventional) numbers to exponentials and back again.

That way, mathematicians don’t have to worry about making mistakes when they *multiply two numbers together* or take their logarithm. They know what to do automatically!

But converting between exponential and regular numbers isn’t too difficult, so most people these days just leave all those *pretty looking expressions* to hang out in math and get used by students who learn calculus.

## Relationship between scientific and traditional notation

Traditional symbolism like arrows, circles, squares, and so on are used to describe the relationships in equations. These symbols are referred to as signs or markers.

The word equation comes from the Latin word for equal, addo, which is why we have the words “equate” and “equal.” An equals sign (or plus-sign symbol) is used to show that two things are the same. For example, if you take one cup of milk and pour it into another glass, then the amount in each doesn’t change, but the amount in the second glass will be twice as much because there is more liquid being poured in!

That’s what an equation means! In math, writing out steps can make concepts clear, so mathematicians use this method often. But when communicating mathematics to non-mathematicians, *using pictures instead* is better since they are intuitive and easier to understand.

There are *several different styles* of graphical representation of mathematical equations. Two common ones are exponent diagrams and fraction bar charts. Both convey the same information – how many times something is multiplied by something else.

With exponents, just remember that whatever element you put in front of the other one increases it by that factor.

## How to work with scientific notation

Many mathematical formulas require using either very large or very small numbers. These are called scientific notations, as they originate from science!

Very large numbers are known as exponentials. Exponents mean that there is an integer number times a power of another integer. For example, e^(2x) **means 2 times** the natural logarithm of whatever comes after the ^ symbol (in this case, the value of x).

The natural logaritiml of 9 is 1 (one), so 9^1 would be 9. Similarly, the natural logarithm of 0 is -infinity (-$\infty$), so 0^-infinty would be 0.

These types of expressions are common when solving equations. When doing so, you must have both very large and very small numbers present in your expression.

## What is a number in scientific notation?

A number in scientific notations is just like any other numbers you’ve learned so far, except that it uses a prefix that *tells us something* about the number it is.

The **prefixes tell us** more information about the number, so we can determine what the number means. For example, if the prefix was ‘x’ then the number would be *one times whatever item follows* the x. So, 1x means use this product once.

If the prefix was ‘y’ then the number would be **two times whatever item follows** the y. So, 2y would mean to double the amount for the next time or formula.

And if the prefix was ‘e’ then the number would be multiplied by itself (expanded) the number following it. So, e10 would be ten times itself (multiplied). An expression with an exponent of 10 is called a power of 10.

Numbers with no special prefix are simply normal numbers. The only difference between them and other number types is how they are written!

Question: How many times does twenty go into three?

Answer: Twenty goes into three twice, because there are two instances of twenty. Make sense? If yes, then here comes some math!

Twenty is a factor of three. This means that twenty is contained within three. So, twenty will always be a part of three things, or it will be included in three different items.

## How do I convert between scientific and traditional notation?

The way to *read general expressions* in science is by **using either traditional** or scientific notations. Traditional notation uses parentheses to denote how many times an expression is multiplied, while **scientific notation multiplies quantifiers** (parts of an expression) together.

For example, the equation e=mc2 means that energy equals the product of mass and the square of the speed of light.

## What are some tips for working with scientific notation?

There is no *universal agreement* as to what rules apply when using this notational system. Some people use an order of operations that goes like this: multiply, add, subtract, find product. Others go in the opposite direction: add, subtract, multiply, divide.

Some say you can’t *mix exponents within one rule*, but others do! It really depends on how the equation is set up and how it is being solved.

So, which method is correct? I *would recommend using either* the multiplication or division first, then adding the other *two steps* if needed. This way, your polynomial will be balanced and converted correctly.

## When should I use scientific notation?

Using scientific notations is very popular these days! There are **several good reasons** to use them, but one of the biggest is efficiency. By limiting yourself to only specific numbers in place of longer ones, your calculations run more quickly.

Most *people learn basic arithmetic like adding*, subtracting, multiplying, and dividing before *moving onto higher math*. For example, there’s an algorithm that calculates the area of a circle given a radius.

The way this works is by taking the square root of the diameter, which is also called the radian measure, and then using that as the radius for another circle. The two circles are related because the radian measurement is just a way to calculate circumference, so they have the same formula.

That means we can relate the first circle to the second easily! The area of the second circle is simply.5 * (radian)2 * (radius) = 0.05 * (square root of 2) * (radius squared).

Keep in mind that you can take either sqrt(a+b) or sqrt(ab) to find the radian measurement.

## What is a larger number in scientific notation?

A larger numbers are called exponents. Exponentiation, or raising a number to a power, is when you take a normal number and increase it by a given amount. For example, say there is a boat with an average speed of **5 miles per hour**. Then *someone comes along* and says they can go this boat at twice the speed! They would have to double the speed (2x=4mph) to get the same result.

This is what exponentiation is. When we talk about exponential growth, we are referring to that situation where something is multiplied by itself over and over again. In other words, whatever thing you start with keeps getting bigger and bigger very quickly.

For instance, if you had one apple, you could make a million apples now by putting just one apple into each new piece. This is how our **parents probably grew** up; they never knew why their family didn’t have enough food, but as kids they **always said “**we were hungry today, but we weren’t hungry yesterday.