Standard form in maths is the method of representing a particular element in the most common manner. From large numbers to small numbers to equations to lines, every element in maths is denoted in a standard form. Let us explore this interesting concept of standard form in various elements of maths such as fractions, equations, algebra, slope along with learning the standard from formula. Solving examples and understanding the basic thumb rule will help in understanding the concept better.
What is Standard Form?
A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules. Representing very large or very small numbers concisely, the standard form is used. For example, 4.5 billion years is written as 4,500,000,000 years. As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few 0’s less or more while writing in the number form. In this case, writing the number in standard form is very helpful. For example, the standard form of 4,500,000,000 = 4.5 × 109. Not only numbers but the fractions, equations, expressions, polynomials, etc also can be written in the standard form.
Let us study the standard form of each of these in detail.
Standard Form of Number
The standard form of numbers (which is also known as "scientific notation" of numbers) has different meanings depending on which country you are in. In the United Kingdom and countries using UK conventions, the standard form is another name for scientific notation. Scientific notation is the process of writing a very large or very small number using numbers between 1 to 10 multiplied by the power of 10. For example, 3890 is written as 3.89 × 103. These are numbers that are greater than 1 use positive powers of 10. Numbers less than 1 use the negative power of 10. For example, 0.0451 is written as 4.51 × 10-2.
In the United States and countries using US conventions, the standard form is the usual way of writing numbers in decimal notation. Using the same example,
Standard form = 3890, Expanded form = 3000 + 800 + 90, and Written form = Three thousand eight hundred and ninety.
Standard Form of Fraction
In the case of fractions, we need to ensure that in the standard form of fractions, the numerator and denominator must be co-prime numbers. That means they have no common factor other than 1, hence the standard form is also called the simplest form of a fraction. For example, 14/22 and 13/6. The simplest form of 14/22 = 7/22 and 13/6 is already in its simplest form as 13 and 6 are co-prime.
Standard Form of Equations
The standard form of an equation is where zero goes on the right and everything else goes on the left. i.e., it is of the form
- Expression = 0.
This helps in solving the equation in a simple manner. Equations used for polynomials, linear and quadratic have a standard form, let us look at what they are.
Standard Form of Polynomial
The standard form polynomial is written with the exponents in decreasing order to make calculations easier. A polynomial is said to be in its standard form, if it is expressed in such a way that the term with the highest degree is placed first, followed by the term which has the next highest degree, and so on. There are two very simple rules of writing a polynomial in a standard form, they are:
- Write the terms in the descending order of their powers (also called exponents).
- Ensure the polynomial contains no like terms.
Hence, the standard form is anxn + an-1xn-1 + an-2xn-2 + ... + a1x1 + a0. For example, the standard form of equation y2 + 7y6 - 8y - y9 is written as -y9 + 7y6 + y2 - 8y.
Note: The thumb rule for writing a polynomial in its standard form is D-U. D stands for Descending and U stands for unlike terms.
Standard Form of a Linear Equation
The standard form of linear equations also known as the general form is a method of writing linear equations. A linear equation can be written in different forms like the standard form, the slope-intercept form, and the point-slope form. The standard form of a linear equation is expressed in two ways, with one variable and with two variables. The standard form of linear equation with one variable is expressed as ax + b = 0 where a and b are constants and the letters x is the variable. The standard form of a linear equation with two variables is expressed as ax + by = c. where a, b, and c are real numbers and the letters x and y are the variables. Look at the image below.
Let’s see how to convert the two lines into the standard form of a linear equation ax + by = c.
Line 1: x + y = 7 i.e. 1x +1y = 7. Here, a = 1, b =1 , c = 7
Line 2: y = 3x i.e. 3x -1y = 0. Here, a = 3, b = -1 , c = 0
Therefore, what we see here is the standard form of equation which is linear i.e. ax + by = c.
Standard Form of Slope of a Line
The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. Representing a line geometrically, we use the standard form of a linear equation (mentioned above). To determine the slope of a line that is expressed graphically, the equation must be converted to a slope-intercept form. To do this, we must solve the equation for y, and the standard form of a slope is expressed as y = mx + c, where m is the slope of the line. This formula is used when the line is straight.
When there are two points in a plane, the slope can be defined as the ratio of change in the value of y to the change in the value of x. The standard form of slope of a line is expressed as m = (y2 – y1)/(x2 – x1). The image below represents both the coordinates on a graph.
Standard Form of Quadratic Equations
Apart from the standard form of a quadratic equation, a quadratic equation can be written in several other forms.
- Vertex Form: a (x - h)2 + k = 0
- Intercept Form: a (x - p)(x - q) = 0
A parabola is a graph of a quadratic function that refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. The following graphs are two typical parabolas; their x and y-intercepts are marked by green dots, and the vertex of each parabola is marked by a blue dot.
The graph will have one of the two shapes as shown above.
- When a > 0, it represents the first parabola (figure 1) which opens upward (U-shaped).
- When a < 0, we obtain a parabola that opens downward (inverted U-shaped) as shown in figure 2.
When we put y = 0 in the above equation, we get the x-intercept which is also called the roots of the equation. Finding the roots provides us with the turning point of the parabola and the vertical line which is drawn from the turning point is called the axis of symmetry. Look at the image below for reference.
Standard Form Formula
The standard form formula represents the standard form of an equation which is the commonly accepted form of an equation. The formula to represent the standard form formula is based on the degree of the equations.
The standard form of a linear equation is the basic form of an equation. The standard form of a linear equation with two variables and more than two variables is presented below. Here x, y, or x1, x2, x3,.... represent the variables and a, b, a1, a2, a3, ........ an are referred to as the coefficients. The numerics placed to the right of the equals to sign are called constants.
ax + by = c
a1x1 +a2 x2+ a3 x3 + ........ + an xn = D
The standard form of a quadratic equation is a second-degree equation and has a variable, coefficients, and constant term. Here it is a single variable x of degree 2. The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0.
Further, we have standard form formulas for equations of higher degrees. Also in coordinate geometry, we have a standard form for different geometric representations such as a straight line, circle, ellipse, hyperbola, and parabola.
- Straight-line: ax + by = c, where a is a positive integer, and b, and c are integers.
- Circle: (x - h)2 + (y - k)2 = (r)2, where ( h, k) is the center and r is the radius.
- Ellipse: x2/a2 + y2/b2 = 1
- Hyperbola: (x-x0)2/a2 - (y-y0)2/b2 = 1, where x0, x0 are the center points, a = semi-major axis and b = semi-minor axis.
- Parabola: (x - h)2 = 4p(y - k)
Listed below are a few topics that are related to a standard form.
Standard Form Examples
Example 1: Liza is trying to find out which of the following equations represent the given graph. As there are no values of the coordinates given, she is not able to decide. How can we use the standard form concept to solve her problem?
1. x4 + 7x2 - 5x = 4
2. 4x + 5y = 0
3. y2 + 7y6 - 9y = y3
4. 3z4 + 7z5 = -12z
The graph given here represents a line. Now, a line represents a linear equation in two variables which has a degree of 1. Out of the four equations given above, only option (b) is linear.
Answer: Therefore, the graph represents the equation, 4x + 5y = 0.
Example 2: Anna showed her class notes on polynomials to her teacher. The teacher returned her notes with a remark "Write the polynomial x2 -10x + 16- x2+ x5 - 3x4 + 3x2 in standard form." What is the correct format Anna should have used?
To write a polynomial in standard form, two rules must be taken care of.
1. Write the terms in descending order of their powers.
2. All terms must be unlike.
Let us first arrange in descending order.
x2 - 10x + 16- x2+ x5 - 3x4 + 3x2 = x5 - 3x4 + x2 - x2 + 3x2 - 10x +16
After adding like terms, we get
x5 - 3x4 +3x2 - 10x + 16.
Answer: Therefore, the standard form is x5 - 3x4 +3x2 - 10x + 16.
Example 3: Convert the following quadratic equation into standard form: (1/x) + x = 1.
(1/x) + x = 1
Multiplying with x on both sides,
1 + x2 = x
Shifting R.H.S. terms to L.H.S.,
x2 - x + 1 = 0
Answer: Therefore, Standard form of the given quadratic equation is x2 - x + 1 = 0.
FAQs on Standard Form
What is the Definition of Standard Form in Math?
Standard form in math is the method of representing a particular element (numbers, fractions, equations, etc) in the most common way. Very large numbers or very small numbers are expressed in the standard form. Mathematical elements such as equations are expressed in a standard form to better solve the problem. In other words, a standard form is a form of writing a given mathematical concept like an equation, number, or expression in a form that follows certain rules.
How Do We Write Standard Form in Maths?
The process of writing a given mathematical concept like an equation, number, or expression in certain rules is called the standard form. Depending upon which mathematical concept we are dealing with, the standard form formula will vary.
How Do You Write Standard Form of Number?
Standard form in math of numbers is written differently depending on the country. In the UK, numbers that are greater than 1 use positive powers of 10, and numbers less than 1 use the negative power of 10. For example, 3670000 is written as 3.67 × 106 and 0.0763 is written as 7.63 × 10-2. This is commonly known as scientific notation of numbers.
What is Standard Form Formula?
The standard form formula refers to the formula presenting the general representation for different types of notation. For example, the standard form of
- a linear equation is ax + by = c.
- a quadratic equation is ax2 + bx + c = 0
- a cubic equation is ax3 + bx2 + cx + d = 0
What is the Standard Form Formula for Parabola?
The standard form formula of the equation of the parabola is this: (y - k)2 = 4p(x - h), where p≠ 0 only in case a parabola has a horizontal axis.
What is the Standard Form for Slope Formula?
The standard form of the slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line whereas the standard form of a linear equation is Ax + By = C and the slope in this form is -A/B.
How to Use Standard Form Formula?
We can use the standard form formula depending on the equation, if it's linear, quadratic, etc. Just rewrite the given formulas in the standard form.
- Standard form of linear equation: ax + by = c
- Standard form of a quadratic equation is a second degree equation: ax2 + bx + c = 0
What is the Standard Form of Equation?
In the standard form of an equation, 0 is usually present on the right side, whereas the rest of the expressions are on the left. Also, the terms are arranged in the descending order of their exponents. To convert an equation into stadard form, just apply arithmetic operations on both sides to turn the right side to be 0.