Converting a repeating decimal into a standard form (also known as p/q) can sometimes be challenging for some individuals who are not familiar with the correct steps. Nevertheless, with consistent practice, one will definitely find it quite an easy task to perform. To commence, we shall acknowledge what a repeating decimal is prior to understanding the steps involved in converting it into the standard form.
A repeating decimal is a decimal that contains a sequence of numbers that repeats itself infinitely. For example, 0.333… (where the 3s repeat endlessly) is a repeating decimal. It should be noted that, not all decimals are repeating decimals. Some decimals, like 0.123, terminate meaning the decimal has a finite number of digits, while others do not. To convert a repeating decimal into a standard form, there are a few steps that one must follow. The steps are quite simple and easy to follow, as illustrated below.
First, one will need to determine the repeating pattern, then subtract the terminating part (if there is any) from the original decimal and multiply it by 10 to the power of the number of repeating digits. The next step is subtracting the result from the original number again, and finally, solve for the variable (x), which is the decimal part of the standard from. For instance, to convert 0.333… to a standard form, we first determine the repeating pattern, which is 3. We then subtract the terminating part (none) from the original decimal, getting 0.333… We then multiply this by 10 to the power of the number of repeating digits (1), giving us 3.333… We then subtract this from the original number again, getting 3.000… Finally, we solve for x, getting 0.333… = x/9. Therefore, 0.333… in standard form is 1/3.
Dividing Both Sides by the Coefficient
Once we have moved all the variables to one side of the equation and the constants to the other side, we can divide both sides of the equation by the coefficient of the variable. The coefficient is the number that is being multiplied by the variable. For example, in the equation 2x + 5 = 11, the coefficient of x is 2.
When we divide both sides of an equation by a number, we are essentially dividing everything in the equation by that number. This means that we are dividing the variable, the constants, and the equals sign.
Dividing both sides of an equation by the coefficient of the variable will give us the value of the variable. For example, if we divide both sides of the equation 2x + 5 = 11 by 2, we get x + 5 = 5.5. Then, if we subtract 5 from both sides, we get x = 0.5.
Here is a table that shows how to divide both sides of an equation by the coefficient of the variable:
| Original Equation | Divide Both Sides by the Coefficient | Simplified Equation |
|---|---|---|
| 2x + 5 = 11 | Divide both sides by 2 | x + 5 = 5.5 |
| 3y – 7 = 12 | Divide both sides by 3 | y – 7/3 = 4 |
| 4z + 10 = 26 | Divide both sides by 4 | z + 2.5 = 6.5 |
Simplifying the Result
Simplifying the result of converting to standard form involves transforming the expression into its simplest possible form. This process is crucial to obtain the most concise and meaningful representation of the expression.
There are several steps involved in simplifying the result:
- Combine like terms: Group terms with the same variable and exponent and add their coefficients.
- Remove unnecessary parentheses: Eliminate redundant parentheses that do not affect the value of the expression.
- Simplify coefficients: Express coefficients as fractions in their simplest form, such as reducing a fraction to its lowest terms or converting a mixed number to an improper fraction.
- Rearrange the terms: Order the terms in the expression according to the descending power of the variable. For example, in a polynomial, the terms should be arranged from the highest power to the lowest power.
By following these steps, you can simplify the result of converting to standard form and obtain the most straightforward representation of the expression. The table below provides examples to illustrate the simplification process:
| Original Expression | Simplified Expression | ||
|---|---|---|---|
| (3x + 4) + (2x – 1) | 5x + 3 | ||
| 5 – (2x + 3) – (x – 4) | 5 – 2x – 3 – x + 4 | 5 – 3x + 1 | 4 – 3x |
| 2(x – 3) + 3(x + 2) | 2x – 6 + 3x + 6 | 5x |
Writing the Equation in the Form Ax + B = 0
To write an equation in the form Ax + B = 0, we need to get all the terms on one side of the equation and 0 on the other side. Here are the steps:
- Start by isolating the variable term (the term with the variable) on one side of the equation. To do this, add or subtract the same number from both sides of the equation until the variable term is alone on one side.
- Once the variable term is isolated, combine any constant terms (terms without the variable) on the other side of the equation. To do this, add or subtract the constants until there is only one constant term left.
- If the coefficient of the variable term is not 1, divide both sides of the equation by the coefficient to make the coefficient 1.
- The equation is now in the form Ax + B = 0, where A is the coefficient of the variable term and B is the constant term.
| Example | Steps |
|---|---|
| Solve for x: 3x – 5 = 2x + 7 |
|
Identifying the Value of A
To convert a complex number from polar form to standard form, we need to identify the values of A and θ first. The value of A represents the magnitude of the complex number, which is the distance from the origin to the point representing the complex number on the complex plane.
Steps to Find the Value of A:
- Convert θ to Radians: If θ is given in degrees, convert it to radians by multiplying it by π/180.
- Draw a Right Triangle: Draw a right triangle in the complex plane with the hypotenuse connecting the origin to the point representing the complex number.
- Identify the Adjacent Side: The adjacent side of the triangle is the horizontal component, which represents the real part of the complex number. It is denoted by x.
- Identify the Opposite Side: The opposite side of the triangle is the vertical component, which represents the imaginary part of the complex number. It is denoted by y.
- Apply the Pythagorean Theorem: Use the Pythagorean theorem to find the hypotenuse, which is equal to the magnitude A:
Pythagorean Theorem Expression for A A² = x² + y² A = √(x² + y²)
Substituting the Value of A
To substitute the value of a variable, we simply replace the variable with its numerical value. For example, if we have the expression 2x + 3 and we want to substitute x = 5, we would replace x with 5 to get 2(5) + 3.
In this case, we have the expression 2x + 3y + 5 and we want to substitute x = 2 and y = 3. We would replace x with 2 and y with 3 to get 2(2) + 3(3) + 5.
Simplifying this expression, we get 4 + 9 + 5 = 18. Therefore, the value of the expression 2x + 3y + 5 when x = 2 and y = 3 is 18.
Here is a table summarizing the steps for substituting the value of a variable:
| Step | Description |
|---|---|
| 1 | Identify the variable that you want to substitute. |
| 2 | Find the numerical value of the variable. |
| 3 | Replace the variable with its numerical value in the expression. |
| 4 | Simplify the expression. |
Simplifying the Expression
The expression 4 + (5i) + (7i – 3) can be simplified by combining like terms. Like terms are those that have the same variable, in this case, i. The expression can be simplified as follows:
4 + (5i) + (7i – 3) = 4 + 5i + 7i – 3
= 4 – 3 + 5i + 7i
= 1 + 12i
Therefore, the simplified expression is 1 + 12i.
| Step | Expression |
|---|---|
| 1 | 4 + (5i) + (7i – 3) |
| 2 | 4 + 5i + 7i – 3 |
| 3 | 4 – 3 + 5i + 7i |
| 4 | 1 + 12i |
Writing the Final Standard Form
The final standard form of a complex number is a+bi, where a and b are real numbers and i is the imaginary unit. To write a complex number in standard form, follow these steps:
- Separate the real and imaginary parts of the complex number. The real part is the part that does not contain i, and the imaginary part is the part that contains i.
- If the imaginary part is negative, then write it as -bi instead of i.
- Combine the real and imaginary parts using the + or – sign. The sign will be the same as the sign of the imaginary part.
For example, to write the complex number 3-4i in standard form, we would first separate the real and imaginary parts:
| Real Part | Imaginary Part |
|---|---|
| 3 | -4i |
Since the imaginary part is negative, we would write it as -4i. We would then combine the real and imaginary parts using the – sign, since the imaginary part is negative:
“`
3-4i = 3 – (-4i) = 3 + 4i
“`
Therefore, the standard form of the complex number 3-4i is 3+4i.
Checking for Accuracy
Once you have converted your equation to standard form, it’s important to check for accuracy. Here are a few tips:
- Check the signs: Make sure that the signs of the terms are correct. The term with the largest absolute value should be positive, and the other terms should be negative.
- Check the coefficients: Make sure that the coefficients of each term are correct. The coefficient of the term with the largest absolute value should be 1, and the other coefficients should be fractions.
- Check the variable: Make sure that the variable is correct. The variable should be in the denominator of the term with the largest absolute value, and it should be in the numerator of the other terms.
Checking the Equation with 9
Here’s a more detailed explanation of how to check the equation with 9:
- Multiply the equation by 9: This will clear the fractions in the equation.
- Check the signs: Make sure that the signs of the terms are correct. The term with the largest absolute value should be positive, and the other terms should be negative.
- Check the coefficients: Make sure that the coefficients of each term are correct. The coefficient of the term with the largest absolute value should be 9, and the other coefficients should be integers.
- Check the variable: Make sure that the variable is correct. The variable should be in the denominator of the term with the largest absolute value, and it should be in the numerator of the other terms.
If all of these checks are correct, then you can be confident that your equation is in standard form.
Applying the Process to Additional Equations
The process of converting to standard form with i can be applied to a variety of equations. Here are some additional examples:
Example 1: Convert the equation 2x + 3i = 7 – 4i to standard form.
Solution:
| Step | Equation |
|---|---|
| 1 | 2x + 3i = 7 – 4i |
| 2 | 2x – 4i + 3i = 7 |
| 3 | 2x – i = 7 |
Example 2: Convert the equation x – 2i = 5 + 3i to standard form.
Solution:
| Step | Equation |
|---|---|
| 1 | x – 2i = 5 + 3i |
| 2 | x – 2i – 3i = 5 |
| 3 | x – 5i = 5 |
Example 3: Convert the equation 2(x + i) = 6 – 2i to standard form.
Solution:
| Step | Equation |
|---|---|
| 1 | 2(x + i) = 6 – 2i |
| 2 | 2x + 2i = 6 – 2i |
| 3 | 2x + 2i – 2i = 6 |
| 4 | 2x = 6 |
| 5 | x = 3 |
How To Convert To Standard Form With I
Standard form of a number is when the number is written using a decimal point and without any exponents. For example, 123,456 is in standard form, while 1.23456 * 10^5 is not.
To convert a number to standard form with I, you need to move the decimal point until the number is between 1 and 10. The exponent of the 10 will tell you how many places you moved the decimal point. If you moved the decimal point to the left, the exponent will be positive. If you moved the decimal point to the right, the exponent will be negative.
For example, to convert 123,456 to standard form with I, you would move the decimal point 5 places to the left. This would give you 1.23456 * 10^5.
People Also Ask About How To Convert To Standard Form With I
How do I convert a number to standard form with i?
To convert a number to standard form with i, you need to move the decimal point until the number is between 1 and 10. The exponent of the 10 will tell you how many places you moved the decimal point. If you moved the decimal point to the left, the exponent will be positive. If you moved the decimal point to the right, the exponent will be negative.
What is the standard form of a number?
The standard form of a number is when the number is written using a decimal point and without any exponents. For example, 123,456 is in standard form, while 1.23456 * 10^5 is not.
How do I move the decimal point?
To move the decimal point, you need to multiply or divide the number by 10. For example, to move the decimal point one place to the left, you would multiply the number by 10. To move the decimal point one place to the right, you would divide the number by 10.
