Standard Form: Numbers In Standard Form Or Scientific Notation
Standard Form: Numbers In Standard Form Or Scientific Notation
Standard form means the regular way to write out a number, like 4,300, as opposed to scientific notation, which is 4.3 x 10^3, or in expanded form, like 4,000 + 300. You can convert numbers large and small to standard form through some steps. Standard form is a way of writing down very large or very small numbers readily. 103 = 1000, so 4 × 103 = 4000. Therefore, 4000 can be written as 4 × 103. This idea can be made use of in writing even larger numbers down readily in standard form.
Small numbers can as well be written in standard form. Nevertheless, instead of the index being positive (in the above example, the index was 3), it will be negative. The rules when writing a number in standard form is that first you write down a number between 1 and 10, then, you write × 10 (to the power of a number).
Now standard form is also popularly referred to as scientific notation. It's a way of making big numbers to look smaller and easier to handle. Therefore, lets take a big number such as 8,143,260. Now if we wish to convert this into standard form we want it to be just 8 before a decimal point, all these other numbers after a decimal and then we're going to multiply it by 10 to an exponent. Therefore, in this regard at the moment the decimal is over here. And we want it to be over here. Therefore we have to move it over 1,2,3,4,5,6 places. So we have 8.143,26 and you don't have to say the zero, times 10 to the 6th power. Now it's the 6th due to the fact that it is moved over 6 places.
We can make use of a simpler number merely to illustrate its variation. If we take the number 8,143, for example. To convert the number to standard form we would require to move it over 1,2,3 places. Therefore, it's 8.143 times 10 to the 3rd. Therefore, bear in mind that the exponent is the number of places you have to move.
Standard Form With big Numbers
To convert a number from scientific notation to standard form, consider 1.5625 x 10^4. First, focus on the exponential number, which is 10^4. Multiply the base number, which is 10, by itself the number of times shown by the exponent, which is four. You would carry out the operation as shown below:
To convert a number from scientific notation to standard form, consider 1.5625 x 10^4. First, focus on the exponential number, which is 10^4. Multiply the base number, which is 10, by itself the number of times shown by the exponent, which is four. You would carry out the operation as shown below:
10 x 10 x 10 x 10 = 10, 000
Fix the product -- which is 10,000 -- into the original equation to obtain the answer below:
1.5625 x 10,000 = 15, 625
Thus, the standard form of this number is 15,625.
For a number in expanded form, convert the number, with the use of basic addition. For instance, consider the expanded number 10,000 + 2000 + 100 + 10 + 9. Add up these numbers to obtain the standard form. Your equation would appear as shown below:
10,000 + 2,000 + 100 + 10 + 9 = 12,119
Thus, the standard form of the number is 12,119.
1. Standard Form With Small Numbers
Small numbers, like those less than one, can as well be converted to standard form. For instance, consider the scientific notation 5 x 10^-2. Begin with 10^-2, and observe that the exponent is negative, which in this case, is negative two. This shows that the number will be a decimal, with the decimal point shifting twice to the left of the base number, which in this instance is 10. This would offer you 0.01. Your equation will appear like this:
10^2-1 = 0.01
Plug in this value, which, in this regard is 0.01, into your original equation to obtain the result below:
5 x 0.01 = 0.05
Therefore, the standard form of this number is 0.05.
Convert small numbers obtained in expanded form, like 0.0003 + 0.001 + 0.02 by adding like this:
0.0003 + 0.001 + 0.02 = 0.0213
Thus, 0.0213 is the standard form of this number.
Standard Form is not the "correct form", it is merely a handy agreed-upon style. You may discover a few other form to be more essential.
Standard Form of a Decimal Number is another name for Scientific Notation, where you write down a number in the manner shown in the examples below:
In this example, 5326.6 is written as 5.3266 × 103,because 5326.6 = 5.3266 × 1000 = 5.3266 × 103
In some countries standard form means "not in expanded form"
561=Standard Form
500 + 60 + 1=Expanded form
Standard Form of an Equation
The "Standard Form" of an equation is:
(some expression) = 0
Put in another way, "= 0" is on the right, and everything else is on the left.
Example: Put x2 = 7 into Standard Form
Answer: x2 - 7 = 0
Standard Form of a Polynomial
The "Standard Form" for writing down a polynomial is to put the terms with the highest degree first (such as the "2" in x2 if there is one variable).
Example: Put this in Standard Form:
3x2 - 7 + 4x3 + x6
The highest degree is 6, therefore that goes first, then 3, 2 and then the constant last:
x6 + 4x3 + 3x2 -7
Standard Form of a Linear Equation
The "Standard Form" for writing down a Linear Equation is
Ax + By = C
A is not supposed to be negative, A and Boughtn't to be both zero, and A, B and Cought to be integers.
Example: Put this in Standard Form:
y = 3x + 2
Move 3x to the left hand side:
-3x + y = 2
Multiply all by -1:
3x - y = -2
Observe: A=3, B=-1, C=-2
This form:
Ax + By + C = 0
is sometimes known as "Standard Form", but is more correctly known as the "General Form".
Standard Form of a Quadratic Equation
The "Standard Form" for writing down a Quadratic Equation is
ax2 + bx + c = 0
(a not equal to zero)
Example: Put this in Standard Form:
x(x-1) = 3
Expand "x(x-1)":
x2 - x = 3
Move 3 to left:
x2 - x - 3 = 0
Observe that: a =1, b = -1, c = -3
Example
Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 1013
It’s 1013 due to the fact that the decimal point has been moved 13 places to the left to get the number to be 8.19
Example
Write 0.000 001 2 in standard form:
0.000 001 2 = 1.2 × 10-6
It’s 10-6 due to the fact that the decimal point has been moved 6 places to the right to obtain the number to be 1.2
On a calculator, you would normally enter a number in standard form as shown below:
Type in the first number (the one between 1 and 10), Press EXP and type in the power to which the 10 is expanded.
Manipulation in Standard Form
This is best explained with an example:
Example
The number p written in standard form is 8 × 105
The number q written in standard form is 5 × 10-2
Calculate p × q. Provide your answer in standard form.
Solution
Multiply the two first parts of the numbers together and the two second parts together:
8 × 5 × 105 × 10-2
= 40 × 103 (Remember 105 × 10-2 = 103)
The question asks you to provide the answer in standard form, but this is not standard form because the first part (the 40) ought to be a number between 1 and 10.
= 4 × 104
Calculate p ÷ q.
Provide your answer in standard form.
This time, divide the two first parts of the standard forms. Divide the two second parts. (8 ÷ 5) × (105 ÷ 10-2) = 1.6 × 107
More examples :
375300000 = 3.753 x 10
8
0.00000035 = 3.5 x 10
-7
Use of tables of squares, square roots and reciprocals
Simple examples of Negative and fractional indices
Introduction
Indices are a simple way of expressing large numbers more simply. They as well offer us with a lot of essential properties for manipulating numbers with the use of indices law. For example in the following expression 25 is defined as illustrated below:
We call "2" the base and "5" the index.
Law of Indices
To manipulate expressions, we can consider the use of the Law of Indices. These laws only apply to expressions with the same base, for example, 34 and 32 can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 35 and 57 as their base differs (their bases are 3 and 5, respectively).
Six rules of the Law of Indices
Rule 1:
Any number, except 0, whose index is 0 is always equal to 1, irrespective of the value of the base.
An Example:
Simplify 20:
Rule 2:
An Example:
Simplify 2-2:
Rule 3:
To multiply expressions with the same base, copy the base and add the indices.
An Example:
Simplify : (note: 5 = 51)
Rule 4:
To divide expressions with the same base, copy the base and subtract the indices.
An Example:
Simplify:
=5y4
Rule 5:
(am)n=amn
An Example:
simpify (y2)6:
Rule 6
Example:
Simplify 1252/3:
This can be summarized as shown below:
Indices and Multiplication
Bearing in mind that:
Examples
>
Indices and Division
Bearing in mind that:
Examples:
Indices and Powers
Bearing in mind that:
Examples:
Indices and Roots and Reciprocals
Bearing in mind that:
and
Examples:
The Imaginary Unit is defined as
i = .
Why it is called "imaginary" numbers is that when these numbers were first proposed many hundred years ago, people could not "imagine" such a number.
It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, very clearly there is no such number in existence. Today, we discover that the imaginary unit being used in mathematics and science. Electrical engineers make use of the imaginary unit (which they represent as j ) in the study of electricity.
Imaginary numbers come into existence when a quadratic equation has no roots in the set of real numbers.
An imaginary number is a number whose square is a negative number.
* i = or - i = -
A pure imaginary number can be written in bi form where
b is a real number and i is .
A complex number is any number that can be written in the standard form a + bi, where a and b are real numbers and i is the imaginary unit. A complex number is a real number a, or a pure imaginary number bi, or the addition of both. Observe these examples of complex numbers written in standard a + bi form: 2 + 3i, -5 + 0i
| Complex Number:standard a + biform | A | Bi |
| 7 + 2i | 7 | 2i |
| 1 - 5i | 1 | - 5i |
| 8i | 0 | 8i |
| -2+3i/5=-2/5+3i/5 | -2/5 | 3i |
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