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Standard Form of Numbers and Equations

The Notion of Standard Form

The method of reduction to standard form depends on what you are working with (in mathematics or natural sciences). It is possible to bring to the standard form can individual numbers, equations, decimals and fractions, polynomial, linear equation, and quadratic equation.

Standard Form of Numbers

If you need to bring the number written in words to the standard form, write the number in the form of numbers. For example, you need to bring the number «seven thousand four hundred thirty-eight» to the standard form. In this example, the number is given in words (i.e. written in words). Copy this number into a digital form.

To do this, in the number, which is written in words, determine the number of units, tens, hundreds, thousands, and so on, and then write the number in the form of their sum. Please note that at this stage you are going to write the number down in the expanded form. Once you learn the process, you can skip this step and move on to the next one. For example, seven thousand, four hundred and thirty-eight: seven thousand = 7000; four hundred= 400; thirty = 30; eight = 8. The expanded form is 7000 + 400 + 30 + 8.

To write a number in a standard form, add the members of its expanded form. For example, 7000 + 400 + 30 + 8 = 7438. Write down the final answer, because you have got the number in the standard form. For example, the standard appearance of the long number above is 7438.

Standard Form of Decimals and Fractions (Scientific Notation)

In case with decimals and fractions, the standard form is used for compact representation of very large or very small numbers. This is known as the exponential representation of numbers.

As a rule, you will be given a very large number or a very small number, but in any case it can be represented in the form of exponential notation. For example, 429000000000 or 0,0000000078.

Move the decimal point and put it directly after the first digit of the number. Keep in mind the original position of the decimal point. For example: 429000000000 => 4.29. Please note that this number doesn’t have a decimal point, but it means that it is at the end of the number (after the last digit). For example, 0,0000000078 => 7,8.

Count the number of positions, on which you moved the decimal point. This amount will be equal to the exponent. If you have moved the decimal point to the left, the exponent is positive, and if you have transferred the decimal point to the right side, then the exponent is negative. For example, you have moved the decimal point for 11 positions to the left, so the exponent is 11. If you have moved the decimal point to 9 positions to the right, so the exponent is equal to -9.

Write down the final answer. To write down this number in the exponential form, write a new number multiplied by 10. For example: the standard view (exponential form) of 429000000000 is 4.29 x 1011. The standard form of 0.0000000078 is 7.8 x 10-9.

Standard Form of Equation

If you are given an equation with one variable, you need to rewrite it so that on the right side of the equation there is only 0. For example, bring the following equations to the standard form: x5 = -9; y4 = 24. In order to do that, transfer all the terms of the equation on the left side. To do this, you need to add or subtract them from both sides of the equation.

The type of the mathematical operation will depend on the type of the equation you have.

  • If a member on the right side is negative, add it to both sides of the equation.
  • If a member on the right side is positive, subtract it from both sides of the equation.

In our example: x5 + 9 = -9 + 9 the term on the right was negative (-9), so you need to add 9 to both sides of the equation.

In the other example: y4 – 24 = 24 – 24 the term on the right was positive (24), so you need to subtract 24 from both sides of the equation.

Write down the final answer, if on the right side of the equation you have 0: x5 + 9 = 0; y4 – 24 = 0.

Standard Form of a Polynomial

If you are given a polynomial (or equation) with multiple variables, its standard form is a record of a polynomial in descending order of exponents of its members. For example, 8x + 2×3 – 4×4 + 7×2 + x5 = 10. If it is necessary, move all the members to the left side so the zero will remain on the right side (this step is optional and depends on the conditions of the problem).

In order to write down such a polynomial in the standard form, write down its members in the descending order, starting with the variable with the greatest exponent. The free member should be written down the last. When recording the members, signs that stand in front of them in the original polynomial, do not change. For example, 8x + 2×3 – 4×4 + 7×2 + x5 – 10: x5 – 4×4 + 2×3 + 7×2 + 8x – 10 = 0. Write down the final answer when you write the polynomial in descending order of exponents of its members: x5 – 4×4 + 2×3 + 7×2 + 8x – 10 = 0.

Standard Form of a Linear Equation

The standard form of a linear equation is Ax + By = C. The coefficient A can’t be negative, while A and B can’t be equal to 0 and «A», «B» and «C» are integers (not fractions). This application can also be named as a linear equation or a «general form».

A linear equation has three members. The first member has x variable, the second member has y variable, and the third member is free. For example: write the following equation in the standard form: y/2 = 7x – 4.

Get rid of fractions, since all the coefficients of the linear equation must be integers. To do this, multiply both sides of the fraction by the denominator. For example, 2 x (3y/2) = (7x – 4) x 2; 3y = 14x – 8.

Isolate the free member «C» on the right side from the equal sign. If on the right side of the equation there are other members other than a free member, transfer them to the left side of the equation. For example: 3y = 14x – 8. Here, the number -8 is a constant member. As the member 14x is located on the right side of the equation, subtract it from the equation’s sides: 3y – 14x = 14x – 8 – 14x; 3y – 14x = -8.

Rewrite the equation so that its members comply with the formula Ax + By = C. Make sure the signs of the members do not change. For example: 3y – 14x = -8; – 14x + 3y = -8.

The coefficient of «x» must be positive. If it is not positive, multiply both sides by -1.For instance, -1 x [-14x + 3y = -8]; 14x – 3y = 8. Write down the final answer, since you have brought the linear equation to the standard form: 14x – 3y = 8.

Standard Form of a Quadratic Equation

Standard view of a quadratic equation (i.e. equation containing a member with x2) is Ax2 + Bx + C = 0. Note that the coefficient of A can’t be zero.

If there was a member with x2in the elementary equation, then you can bring the equation to the standard form, as described above.

Sometimes the member with x2 is not obvious at first sight. But if as a result of mathematical operations you get a member with x2, you can bring the equation to the standard form, as described above. For example, bring the equation x * (2x + 5) = -11 to standard form. Open the brackets, to identify the term with x2. If it is not required to open the brackets, you can skip this step. For example: x * (2x + 5) = -11. In order to open the brackets, multiply the member that is in front of the brackets on each member standing in parentheses: 2×2 + 5x = -11.

Move all members on the left side of the equation so that only 0 remains on the right side: 2×2 + 5x + 11 = -11 + 11; 2×2 + 5x + 11 = 0.

Write down the final answer if its form corresponds to the type of Ax2 + Bx + C = 0. For example, the standard form of the equation 2×2 + 5x + 11 = 0.

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