Scientific Notation Vs Standard Form
Scientific Notation
Hi, and welcome to this video about scientific notation! In this video, nosotros will explore what scientific notation is and how to write and compare numbers in scientific annotation.
Offset, what is scientific notation? The idea of scientific notation was adult past Archimedes in the tertiary century BC, where he outlined a system for computing the number of grains of sand in the universe, which he institute to exist 1 followed by 63 zeroes. His piece of work was based on place value, a novel concept at the fourth dimension.
What is Scientific Note?
Scientific notation is just a mode of writing numbers. It is especially useful in expressing very large or very small numbers because it is shorter and more efficient and information technology shows magnitude very hands.
Every real number can exist written as a product of two parts: a decimal part times an integer power of ten.
\(grand \times x^n\), where \(1\) ≤ \(m\) < \(ten\) and \(northward\) is an integer
Why 10? Our number system is based on 10, and each identify value is x times the previous place value. One ten equals ten ones; one hundred equals 10 tens, etc.
Permit's look at writing large numbers using this notation system:
Nosotros can write the number 1 every bit \(1 \times 10^0\). Retrieve, \(10^0=i\), so \(i \times ane=ane\).
We can also write the number thirteen every bit \(1.3 \times 10^one\) considering \(1.iii \times 10=13\).
We tin write the number 134 as \(1.34 \times 10^2\) because \(ane.34 \times 10 \times x=134\).
Let's look again at Archimedes' findings. He expressed the number of grains of sand in the universe every bit "1 followed by 63 zeroes." We could write that out, but that would take way too long and exist highly inefficient.
In scientific notation, this would be \(one \times 10^{63}\), a much more meaty and efficient way of expressing this number. The number of zeros in the gigantic number is represented by the exponent. In the fully written number, it's important to realize each time we multiply by 10, we movement to a new place value. And then adding a zero means multiplying by ten.
A light-yr (the altitude light travels in a year) is 5 trillion 878 billion 600 million miles. Let'due south express this in scientific notation. Often, this is described as "moving the decimal point," which doesn't really happen. We simply need to count the number of times we multiply by 10.
The decimal role is created from the first block that begins and ends with a non-zero number (in other words, the block tin contain a 0, simply we don't employ the zeros at the end).
\(5,878,600,000,000\)
Our decimal must be greater than or equal to 1 and less than ten. So we always first from the ones place. Hither, nosotros accept:
\(five.8786×10^?\)
This is where we count the number of times our decimal is multiplied by 10. We need to count from the decimal we created to the end of the number:
\(5.876 \times ten^ane=58.786\)
\(5.876 \times x^2=587.86\)
\(5.876 \times x^3=5,878.6\)
and and so on then on until we accomplish
\(5.876 \times 10^{12}\)\(=5,876,000,000,000\)
And so, in scientific notation, a low-cal-year tin can exist expressed as \(5.876 \times ten^{12}\) miles.
But what if you lot wanted to take a number written using scientific notation and change information technology into standard form?
A calorie-free-year can also be expressed equally \(ix.4607 \times ten^{fifteen}\) meters. We tin can easily change this number to standard form.
\(9,460,700,000,000,000\)
Start with the decimal office (9.4607) and multiply by ten a total of xv times. This tells us that a light-year is 9 quadrillion 460 trillion 700 billion meters.
Allow'southward expect at a couple more examples before we move on:
If I google "lite year," I'yard given about 7 billion 380 million search results. Allow'southward express this number in scientific annotation.
\(7,380,000,000\)
\(seven.380000000\)
There are nine digits later on the decimal, and then our exponent will be 9.
\(7.38 \times 10^9\)
As you can see, I was given \(seven.38 \times 10^9\) search results.
On average, there are \(3.72 \times 10^{13}\) cells in a homo trunk. Express this number in standard form.
The exponent thirteen tells united states that we have thirteen numbers after the decimal, which gives the states three.72 followed by 11 zeros. If we multiply this by ten xiii times, nosotros see that there are 37.2 trillion cells in the human being body.
So now we know how to write big numbers using scientific note, but what about pocket-size numbers?
First, let's recall how negative exponents work. For case:
\(10^{-1}\text{ }\)\(= \frac{ane}{10^ane}= \frac{one}{10}\)
\(10^{-2} \text{ }\)\(=\frac{1}{10^2}=\frac{ane}{100}\)
Where positive exponents stand for multiplication, negative exponents represent division.
The number ane tin can exist written as \(1 \times x^0=1 \times 1=1\).
The number 0.one can be written every bit \(1 \times 10^{-1}=i \times \frac{one}{ten}=0.i\).
The number 0.01 tin be written equally \(1 \times 10^{-ii}=ane \times \frac{i}{100}=0.01\).
The wavelength of green light is 0.00000055 meters. Permit's come across this in scientific notation.
Nosotros begin the same way as with big numbers—creating the decimal from the chunk bookended by not-null numbers. Our issue will resemble \(5.5 \times 10^{-?}\).
At present, nosotros count the number of times our decimal is divided by 10.
\(v.five \times 10^{-1}=.55\)
\(v.5 \times 10^{-2}=.055\)
\(5.five \times 10^{-three}=.0055\)
and so on and so on until nosotros accomplish
\(5.5 \times 10^{-7}=.00000055\)
So, in scientific annotation, the wavelength of green light tin can be expressed as \(five.five \times ten^{-7}\) meters.
The radius of a hydrogen atom is \(2.v \times 10^{-11}\) meters. We tin express this in standard form by starting with the decimal office 2.5 and dividing by 10 a total of 11 times.
This tells u.s. that the radius of a hydrogen atom is 0.000000000025 meters.
I hope that this video helped you understand how to piece of work with numbers in scientific note!
Cheers for watching, and happy studying!
Ofttimes Asked Questions
Q
What are the five rules of scientific notation?
A
- Scientific notation will e'er consist of a coefficient multiplied by a power of ten.
- The coefficient needs to exist greater than or equal to i, just less than 10
- The exponent is a non-zero integer, positive or negative.
- A positive exponent means you will movement the decimal that many places to the right.
- A negative exponent means y'all will move the decimal that many places to the left.
Q
What are the iii parts of scientific annotation?
A
Numbers in scientific notation are written with three parts. A coefficient, a base, and a power. For example, \(3.5\times10^four\) has a coefficient of \(3.5\), a base of \(10\), and a power of \(iv\). The power of \(4\) indicates that \(3.5\) should be multiplied by \(10^iv\), which is \(35{,}000\).
Q
Where is scientific annotation used?
A
Scientific notation is often used in the fields of science and math. Instead of writing a number in standard form with many zeros, mathematicians and scientists oftentimes prefer to condense numbers into scientific annotation because it is much more compact. For case, the number \(4{,}000{,}000{,}000{,}000\) can be written as \(4\times10^{12}\). This form makes extremely large and extremely modest numbers easier to piece of work with.
Q
What is the correct way to write in scientific notation?
A
A number in scientific note requires a coefficient to exist multiplied past a ability of 10. When converting from standard form to scientific annotation, start by placing the decimal point later on the first pregnant digit. For example, in the number \(54{,}000{,}000{,}000\) the decimal would be placed between the \(five\) and the \(4\). At present multiply this \((5.4)\) by a power of ten. The power is adamant past the number of decimal movements needed in order to get back to the original number. \(5.4\) requires \(10\) decimal movements to the right in order to go back to \(54{,}000{,}000{,}000\). In scientific annotation this would be \(5.4\times10^{10}\). The same process works for negative exponents, except the decimal movements are to the left.
Q
What types of answers are best written in scientific notation?
A
Scientific notation is a convenient manner to handle extremely large or extremely pocket-size numbers. For example, instead of writing \(0.0000000043\), nosotros can use scientific annotation and simply write \(four.three\times10^{-9}\). Scientific notation can be used to express numbers like the diameter of a ruby-red claret jail cell \(viii.0\times10^{-6}\) chiliad, or the distance between the Earth and the Moon \(iii.84\times10^5\) km. Writing numbers in scientific notation conveniently eliminates large numbers of zeros.
Q
What is the benefit of writing numbers in scientific annotation?
A
The benefit of writing numbers in scientific note is that you eliminate the need for multiple zeros. Instead of writing an extremely large number like \(567{,}000{,}000{,}000{,}000{,}000{,}000\) you can express this same amount in scientific note as \(five.67\times10^{20}\). This is very user-friendly when making calculations with very small or very large numbers.
Q
When should you use scientific notation?
A
Scientific annotation should be used when doing calculations with numbers that are extremely large or extremely small. Chemists oft use scientific notation when dealing with extremely minor measurements. Astronomers often apply scientific annotation when dealing with extremely large measurements.
Q
What does a negative exponent mean in scientific notation?
A
A negative exponent indicates that the decimal bespeak volition be shifted that number of places to the left. For example, \(4.4\times10^{-four}\) indicates that the decimal betoken volition move \(4\) places to the left. \(four.four\times10^{−4}\) becomes \(0.00044\) in standard form.
Fact Sheet
Practice Questions
Question #1:
What is \(3.79×10^5\) in standard grade?
0.0000379
379,000
0.00379
3,790
Answer:
The correct respond is 379,000. When looking at a number written in scientific notation, the exponent on the x tells you lot how many places to motion the decimal either right or left. A positive 5 tells you to move the decimal point 5 places to the right, adding 0s when necessary. This results in 379,000.
Question #2:
What is 0.000413 in scientific notation?
\(4.13×10^{-2}\)
\(4.13×x^2\)
\(four.13×10^iv\)
\(4.13×10^{-four}\)
Answer:
The correct respond is \(iv.xiii×10^{-4}\). To write a number properly in scientific note, identify one non-zero digit in forepart of the decimal point and the remaining non-zero digits afterwards the decimal bespeak. Then, multiply the number by 10 raised to a ability that reflects how many places to move the decimal signal. If the decimal signal in the number 0.000413 is moved to exist right after the iv, then the decimal bespeak would need to be moved 4 places to the left to be in its original position, then multiply \(iv.13\) by \(x^{-4}\) to get your final answer.
Question #iii:
What is \(3.47×10^{-6}\) in standard grade?
0.00000347
0.00347
3,470,000
347,000
Answer:
The correct answer is 0.00000347. When looking at a number written in scientific notation, the exponent on the ten tells you how many places to motion the decimal either right or left. A negative half-dozen tells us to move the decimal indicate 6 places to the left, adding 0s when necessary. This results in 0.00000347.
Question #4:
What is 71,329,100,000 in scientific notation?
\(71,329.1×10^{-6}\)
\(7.13291×10^{-10}\)
\(71,329.1×x^half dozen\)
\(7.13291×10^{x}\)
Answer:
The correct answer is \(seven.13291×ten^{10}\). To write a number properly in scientific annotation, place i not-cypher digit in front end of the decimal bespeak and the remaining non-nada digits afterwards the decimal point. Then, multiply the number past 10 raised to a ability that reflects how many places to move the decimal point. If the decimal point in the number 71,329,100,000 is moved to exist right subsequently the vii, then the decimal point would demand to be moved 10 places to the right to be in its original position. This results in a terminal reply of \(7.13291×10^{10}\).
Question #5:
Which of the following is written in proper scientific note?
\(413×10^{-iii}\)
\(0.413×10^{-3}\)
\(41.3×x^{-3}\)
\(4.thirteen×10^{-3}\)
Respond:
The right answer is \(4.13×10^{-3}\). For a number to exist properly written in scientific notation, there must be only one digit in front of the decimal bespeak.
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Scientific Notation Vs Standard Form,
Source: https://www.mometrix.com/academy/scientific-notation/
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