{"id":274,"date":"2021-12-06T09:14:06","date_gmt":"2021-12-06T09:14:06","guid":{"rendered":"https:\/\/visualfractions.com\/blog\/?p=274"},"modified":"2023-02-23T16:59:12","modified_gmt":"2023-02-23T16:59:12","slug":"standard-form","status":"publish","type":"post","link":"https:\/\/visualfractions.com\/blog\/standard-form\/","title":{"rendered":"Standard Form"},"content":{"rendered":"\n

The phrase standard form is one most common phrases you\u2019ll encounter in mathematics. We use these forms so that we have a uniform way of writing numbers, expressions, and equations down. This is why knowing the common standard forms referred to in math will come in handy not only in your math classes but in your sciences and advanced STEM classes too!<\/p>\n\n\n\n

In this article, we\u2019ll show you how to write common mathematical expressions in their standard forms and break down the steps for you. We\u2019ll show you an overview of how different standard forms are written and also give you enough examples to work on. By the end of our discussion, we want you to feel confident when asked to write numbers and equations in standard form! <\/p>\n\n\n\n

What Does Standard Form Mean in Math?<\/h2>\n\n\n\n

When we say standard form in math, we simply mean the standard rule we write a certain number, equation, or formula. From arithmetic to more complex math topics, you\u2019ll often encounter this phrase. This also means that the phrase \u201cstandard form\u201d will be relative to the mathematical concept that we are referring to. <\/p>\n\n\n\n

Mathematical Concept<\/strong><\/th>Standard Form Example<\/strong><\/th><\/tr><\/thead>
Number<\/td>[katex]12, 567[\/katex]<\/td><\/tr>
Rational Number<\/td>[katex]\\dfrac{12}{19}[\/katex]<\/td><\/tr>
Scientific Notation<\/td>[katex]45, 230 = 4.523 \\times 10^4[\/katex]<\/td><\/tr>
Equation of the Line<\/td>[katex] 3x + 4y = 12[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Here are some examples of what we mean when we say standard forms in math. Mathematicians have set up these rules so that the language of math remains constant. In textbooks and in fields that heavily rely on math, numbers and equations are normally written in standard form as well.  In the next section, we\u2019ll break down the processes of writing common expressions and equations in their standard form \u2013 starting with writing numbers in standard form.<\/p>\n\n\n\n

How To Write Numbers in Standard Form?<\/h2>\n\n\n\n

When learning about numbers, we\u2019ll be introduced to the numbers\u2019 standard and expanded form. The numbers we\u2019re working with are normally in standard form but we can also write numbers in expanded to standard form by simplifying them. <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

When given the expanded form of a number, follow the steps below to simplify them and write them in standard form:<\/p>\n\n\n\n

  1. Take the expanded form\u2019s sum vertically or horizontally. Here are examples of what we meant by simplifying the expanded form to its standard form:<\/li><\/ol>\n\n\n\n

    Vertical Method<\/h3>\n\n\n

    [katex display=block]\\underline{\\left.\\begin{matrix}
    \n\\phantom{x}40,000\\\\
    \n\\phantom{xx}3,000\\\\
    \n\\phantom{xxxx}100\\\\
    \n+\\phantom{xxxx}6\\\\
    \n\\end{matrix}\\right\\}} \\text{Expanded Form}\\\\\\underbrace{43, 106}_{\\displaystyle \\text{Standard Form}}\\phantom{xxxxxxxxxxx} [\/katex]<\/p>\n\n\n\n

    Horizontal Method<\/h3>\n\n\n

    [katex display=block] \\begin{aligned}\\overbrace{40,000 + 3, 000 + 100 + 6}^{\\displaystyle\\text{Expanded Form}}&= 43, 000+ 100 +6\\\\&= 43, 100 + 6\\\\&= \\underbrace{43, 106}_{\\displaystyle\\text{Standard Form}}\\end{aligned} [\/katex]<\/p>\n\n\n\n

    1. Simplify the sum and return it as the standard form of the number. For our example, the simplified form of the number is equal to [katex] 43,106[\/katex].<\/li><\/ol>\n\n\n\n

      Problem 1<\/h3>\n\n\n\n

      Why don\u2019t we go ahead and try writing down the standard form of the numbers given their expanded forms?<\/p>\n\n\n\n

      a. [katex]100, 000+ 40, 000 + 1, 000+ 400 + 20 + 1[\/katex]<\/p>\n\n\n\n

      As we have mentioned, we can use any approach but what matters is that we simplify the expanded form to its sum to find the number\u2019s standard form. We can add these numbers vertically for now, but we\u2019ll show you a faster way in the second item!<\/p>\n\n\n

      [katex display=block] \\underline{\\left.\\begin{matrix}
      \n100, 000\\\\ \\phantom{x}40, 000\\\\
      \n\\phantom{xx}1, 000\\\\
      \n\\phantom{xxxx}400\\\\\\phantom{xxxxx}20 \\\\
      \n+\\phantom{xxxx}1\\\\
      \n\\end{matrix}\\right\\}} \\text{Expanded Form}\\\\\\underbrace{141, 421}_{\\displaystyle \\text{Standard Form}}\\phantom{xxxxxxxxxxx}[\/katex]<\/p>\n\n\n\n

      This means that the number\u2019s standard form is equal to [katex]141, 421[\/katex].<\/p>\n\n\n\n

      b. [katex]500, 000+ 6, 000+ 300  + 5[\/katex]<\/p>\n\n\n\n

      Now, adding vertically or horizontally can sometimes be tedious and when asked to try different problems, it will take so much space and time. A simpler method would be taking note of the largest number from the expanded form. <\/p>\n\n\n\n

      [katex]\\begin{aligned}500, 000+ 6, 000+ 300  + 5 = 506, 305\\end{aligned}[\/katex]<\/p>\n\n\n\n

      Notice how some numbers are missing for some places? We simply add zero then move on. For our example, we add zero to the ten thousand and tens places. This means that [katex]500, 000+ 6, 000+ 300\u00a0 + 5[\/katex] is simply equal to [katex]506, 305[\/katex].<\/p>\n\n\n\n

      How To Write Numbers in Standard Form (Using Scientific Notation)?<\/h2>\n\n\n\n

      We can also write significantly large or small numbers in standard form using scientific notation. In the UK, this is what is meant when we say the standard form of numbers. It\u2019s still important that we know how to write numbers in scientific notation since this is how we report large and small numbers in science and engineering.\u00a0<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Similar to scientific notations, we can write numbers like 0.000045 in standard form using the general form shown below.<\/p>\n\n\n\n

      [katex] \\begin{aligned}\\text{Scientific Notation} &= b \\times 10^n\\\\b&: 1 \\leq b<10\\\\n&: \\text{Integer} \\end{aligned} [\/katex]<\/p>\n\n\n\n

      For a scientific notation and a number in standard form, the base must be between 1 and 10 and the power must also be an integer. Let\u2019s break down the steps we need to rewrite significantly large or small numbers in scientific notation:<\/p>\n\n\n\n

      1. Move the decimal point so that the resulting number is between 1 and 10. <\/li>
      2. Count the number of decimal places moved and the direction we moved. 
        1. If we moved [katex]n[\/katex] decimals places to the left, the exponent is [katex]n[\/katex].<\/li>
        2. If we moved [katex]n[\/katex] decimals places to the right, the exponent is [katex]-n[\/katex].<\/li>
        3. If the decimal or number is already between 1 and 10, [katex]n = 0[\/katex].<\/li><\/ol><\/li>
        4. Write the standard form (in scientific notation) using the form [katex]b \\times 10^n[\/katex].<\/li>
        5. If there are zeroes trailing after moving the decimal places in [katex]b[\/katex], remove them for a more simplified form.<\/li><\/ol>\n\n\n\n

          Problem 2<\/h3>\n\n\n\n

          Let\u2019s try writing down the numbers shown below in standard form using scientific notation. Write the base in two decimal places. <\/p>\n\n\n\n

          a. [katex]316, 001. 28[\/katex]<\/p>\n\n\n\n

          To write large numbers such as [katex]316, 001. 28[\/katex] in scientific notation, we move decimal places to the right so that we\u2019re left with a number or decimal between 1 and 10. We move five decimal places to the left of [katex]316, 001. 28[\/katex] and we\u2019ll have [katex]b = 3.1600128[\/katex]. <\/p>\n\n\n\n

          [katex]\\begin{aligned}316, 001. 28 &= 3.1600128 \\times 10^{5}\\\\&= 3.16 \\times 10^{5}\\end{aligned}[\/katex]<\/p>\n\n\n\n

          To account for the decimal places we\u2019ve moved, we multiply the base by [katex]10^{5}[\/katex]. This means that our decimal, in scientific notation, is equal to [katex] 3.16 \\times 10^{5}[\/katex].<\/p>\n\n\n\n

          b. [katex]0.0000567[\/katex]<\/p>\n\n\n\n

          We apply a similar process in writing the second example. This time, however, we move five decimal places to the right, so we expect the power of 10 to be negative. <\/p>\n\n\n\n

          [katex]\\begin{aligned}0.0000567 &= 5.67 \\times 10^{-5}\\end{aligned}[\/katex]<\/p>\n\n\n\n

          Moving [katex]0.0000567[\/katex] five decimal places to the right will leave us with 5.67, which is between 1 and 10. To retain its original value, we multiply the new base by [katex]10^{5}[\/katex]. Hence, we have [katex] 5.67 \\times 10^{-5}[\/katex].<\/p>\n\n\n\n

          How To Write Linear Equations in Standard Form?<\/h2>\n\n\n\n

          We can write the line equation\u2019s standard form by isolating the linear equation\u2019s constant on the right-hand side of the equation. Here\u2019s the standard form or also known as the general form of the linear equation:<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          The process is straightforward \u2013 isolate all the terms on the left-hand side of the equation. Apply appropriate algebraic techniques when needed. Of course, the best way to master the process of writing linear equations in standard form is through practice. When you\u2019re ready, try out the sample problem below!<\/p>\n\n\n\n

          Problem 3<\/h3>\n\n\n\n

          Rewrite the following linear equations in standard form.<\/p>\n\n\n\n

          a. [katex]y = -4x + 5[\/katex]<\/p>\n\n\n\n

          For us to isolate the constant, 5, on the right-hand side, we simply transpose [katex]-4x[\/katex] on the left-hand side or add [katex]4x[\/katex] on both sides of the equation. <\/p>\n\n\n\n

          [katex] \\begin{aligned}y &= -4x + 5\\\\y {\\,\\color{Purple} + 4x}&= -4x + 5{\\,\\color{Purple} + 4x}\\\\y + 4x &= 5\\end{aligned} [\/katex]<\/p>\n\n\n\n

          To write the equation in standard form, we rearrange the terms on the left-hand side of the equation so that [katex]4x[\/katex] is the leading term. Hence, we have [katex]\\boldsymbol{4x + y = 5}[\/katex].<\/p>\n\n\n\n

          b. [katex]y = -\\dfrac{3}{4}x + \\dfrac{1}{6}[\/katex]<\/p>\n\n\n\n

          Although all constants and coefficients are real numbers, let\u2019s multiply both sides of the equations by the least common denominator shared by the fractions to rewrite them into integers.<\/p>\n\n\n\n

          [katex] \\begin{aligned}y &= -\\dfrac{3}{4}x + \\dfrac{1}{6}\\\\y {\\,{\\color{Pink}\\cdot  12}} &= -\\dfrac{3}{4}x{\\,{\\color{Pink}\\cdot  12}} + \\dfrac{1}{6}{\\,{\\color{Pink}\\cdot  12}}\\\\ 12y &= -9x + 2\\end{aligned} [\/katex]<\/p>\n\n\n\n

          Now, isolate the constant, 2, on the right-hand side of the equation.  We can do this by adding [katex]9x[\/katex] on both sides of the equation.<\/p>\n\n\n\n

          [katex] \\begin{aligned} 12y {\\,{\\color{Pink}+   9x}}&= -9x + 2 {\\,{\\color{Pink}+   9x}}\\\\12y + 9x &=2\\\\9x + 12y&= 2\\end{aligned} [\/katex]<\/p>\n\n\n\n

          After rearranging the terms so that [katex]9x[\/katex] is the leading term. This means that the equation\u2019s standard form is [katex]9x + 12y = 2[\/katex].<\/p>\n\n\n\n

          How To Write Higher-degree Equations in Standard Form<\/h2>\n\n\n\n

          When writing quadratic equations or equations with higher degrees in standard form, we simply isolate all the terms on the left-hand side of the equation. This means that higher-degree equations\u2019 standard forms will always have 0 on their right-hand side. <\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          The main idea is that when writing equations other than linear equations in standard form, we simply move all the terms on the left-hand side so we have 0 on the right-hand side of the equation. Of course, for some cases, we\u2019ll need to apply appropriate algebraic techniques to simplify the given equation.<\/p>\n\n\n\n

          Problem 4<\/h3>\n\n\n\n

          Rewrite the quadratic equation, [katex]x(x -5) = 4x + 1[\/katex], in its standard form.<\/p>\n\n\n\n

          For us to rewrite this equation in standard form, we need to expand [katex]x(x -5)[\/katex] by distributing [katex]x[\/katex] on the left-hand side. Move all the terms on the left-hand side of the equation to write the resulting quadratic equation in standard form.<\/p>\n\n\n\n

          \\begin{aligned} x(x -5) &= 4x + 1\\\\x^2 – 5x &= 4x + 1\\\\x^2 -5x – 4x – 1 &= 0\\\\x^2 -9x – 1&= 0 \\end{aligned}<\/p>\n\n\n\n

          This means that the quadratic equation shown has a standard form of [katex]\\boldsymbol{x^2 \u2013 9x \u2013 1 =0}[\/katex].<\/p>\n\n\n\n

          The process will still be the same when rewriting cubic or even quartic equations in standard form. The important thing to remember is that the right-hand of the standard form only contains 0. <\/p>\n","protected":false},"excerpt":{"rendered":"

          In this article, we\u2019ll show you how to write common mathematical expressions in their standard forms and break down the steps for you. We\u2019ll show you an overview of how different standard forms are written and also give you enough examples to work on.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-274","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/274","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/comments?post=274"}],"version-history":[{"count":26,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/274\/revisions"}],"predecessor-version":[{"id":305,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/274\/revisions\/305"}],"wp:attachment":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/media?parent=274"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/categories?post=274"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/tags?post=274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}