Standard Form
Standard form (sometimes called scientific notation or standard index form) gives us a way of writing very big and very small numbers using powers of 10.
Why use standard form?
- Some numbers are too big or too small to write easily or for your calculator to display at all
- Imagine the number 5050 , the answer would take 84 digits to write out
- Try typing 5050 into your calculator, you will see it displayed in standard form
- Writing very big or very small numbers in standard form allows us to:
- Write them more neatly
- Compare them more easily
- Carry out calculations more easily
- Exam questions could ask for your answer to be written in standard form
How is standard form written?
- In standard form numbers are always written in the form
where
and
satisfy the following conditions:
- So there is one non – zero digit before the decimal point
- So
must be an integer
- So
for large numbers
- How many times
is multiplied by 10
- How many times
for small numbers
- How many times
is divided by 10
- How many times
How are calculations carried out with standard form?
- Your GDC will display large and small numbers in standard form when it is in normal mode
- Your GDC may display standard form as aEn
- For example,
will be displayed as
- If so, be careful to rewrite the answer given in the correct form, you will not get marks for copying directly from your GDC
- For example,
- Your GDC may display standard form as aEn
- Your GDC will be able to carry out calculations in standard form
- If you put your GDC into scientific mode it will automatically convert numbers into standard form
- Beware that your GDC may have more than one mode when in scientific mode
- This relates to the number of significant figures the answer will be displayed in
- Your GDC may add extra zeros to fill spaces if working with a high number of significant figures, you do not need to write these in your answer
- If you put your GDC into scientific mode it will automatically convert numbers into standard form
- To add or subtract numbers written in the form
without your GDC you will need to write them in full form first
- To multiply or divide numbers written in the form
without your GDC you can either write them in full form first or use the laws of indices
Exam Tip
- Your GDC will give very big or very small answers in standard form and will have a setting which will allow you to carry out calculations in scientific notation
- Make sure you are familiar with the form that your GDC gives answers in as it may be different to the form you are required to use in the exam
Worked Example
Calculate the following, giving your answer in the form , where
and
.
i)
![ai-sl-1-1-1-we-1-standard-form-part-i](../../../../../../../../cdn-cgi/image/index-5.htm?f=auto,width=1920/uploads/2022/03/ai-sl-1-1-1-we-1-standard-form-part-i.png)
ii) ![left parenthesis 7 space cross times space 10 to the power of 5 right parenthesis space minus space left parenthesis 5 space cross times space 10 to the power of 4 right parenthesis](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2223%22%20width%3D%22168%22%20wrs%3Abaseline%3D%2217%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E(%3C%2Fmo%3E%3Cmn%3E7%3C%2Fmn%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E%26%23xD7%3B%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmsup%3E%3Cmn%3E10%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmo%3E)%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E(%3C%2Fmo%3E%3Cmn%3E5%3C%2Fmn%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E%26%23xD7%3B%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmsup%3E%3Cmn%3E10%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmo%3E)%3C%2Fmo%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'math139fd9ea237c83aaa39b87594e5'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAAA4WhlYWQQC2qxAAACeAAAADZoaGVhCGsXSAAAArAAAAAkaG10eE2rRkcAAALUAAAADGxvY2EAHTwYAAAC4AAAABBtYXhwBT0FPgAAAvAAAAAgbmFtZaBxlY4AAAMQAAABn3Bvc3QB9wD6AAAEsAAAACBwcmVwa1uragAABNAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACANciEv%2F%2FAAAA1yIS%2F%2F%2F%2FKt3wAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAIAgABVAtUCgAADAAcARhiwARQAsQAAExCxAAnksQABExCwBDyxBgj0sAI8MAGxCAETELEAA%2FawBzyxAQX1sAY8sgUHABD0sAI8sQkD5rEEBfWwAzwTMwEjETMBI4BVAgBVVf4AVQKA%2FdUCK%2F3VAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'round_brackets18549f92a457f2409'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjwHLFQAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5ZmAOz2cAAAFsAAABJGhlYWQOKih8AAACkAAAADZoaGVhCvgVwgAAAsgAAAAkaG10eCA6AAIAAALsAAAADGxvY2EAAARLAAAC%2BAAAABBtYXhwBIgEWQAAAwgAAAAgbmFtZXHR30MAAAMoAAACOXBvc3QDogHPAAAFZAAAACBwcmVwupWEAAAABYQAAAAHAAAGcgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2B0AIgBtAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTAREAEzABEQASMAAZCQ%2FnABkJD%2BcALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAA%2FtACIAbQAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LADELACPLAHELAIPDEwARABIwAREAEzAAIg%2FnCQAZD%2BcJABkALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAAAAEAAPW2NYFfDzz1AAMIAP%2F%2F%2F%2F%2FVre7u%2F%2F%2F%2F%2F9Wt7u4AAP7QA7cG0AAAAAoAAgABAAAAAAABAAAHPv5OAAAXcAAA%2F%2F4DtwABAAAAAAAAAAAAAAAAAAAAAwDVAAACIAAAAiAAAAAAAAAAAAAkAAAAowAAASQAAQAAAAMACgACAAAAAAACAIAEAAAAAAAEAABNAAAAAAAAABUBAgAAAAAAAAABAD4AAAAAAAAAAAACAA4APgAAAAAAAAADAFwATAAAAAAAAAAEAD4AqAAAAAAAAAAFABYA5gAAAAAAAAAGAB8A%2FAAAAAAAAAAIABwBGwABAAAAAAABAD4AAAABAAAAAAACAA4APgABAAAAAAADAFwATAABAAAAAAAEAD4AqAABAAAAAAAFABYA5gABAAAAAAAGAB8A%2FAABAAAAAAAIABwBGwADAAEECQABAD4AAAADAAEECQACAA4APgADAAEECQADAFwATAADAAEECQAEAD4AqAADAAEECQAFABYA5gADAAEECQAGAB8A%2FAADAAEECQAIABwBGwBSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAAUgBvAHUAbgBkACAAYgByAGEAYwBrAGUAdABzACAAdwBpAHQAaAAgAGEAcwBjAGUAbgB0ACAAMQA4ADUANABSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFYAZQByAHMAaQBvAG4AIAAyAC4AMFJvdW5kX2JyYWNrZXRzX3dpdGhfYXNjZW50XzE4NTQATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlAAAAAAMAAAAAAAADnwHPAAAAAAAAAAAAAAAAAAAAAAAAAAC5B%2F8AAY2FAA%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style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iii)
![ai-sl-1-1-1-we-1-standard-form-part-iii](../../../../../../../../cdn-cgi/image/index-7.htm?f=auto,width=1920/uploads/2022/03/ai-sl-1-1-1-we-1-standard-form-part-iii.png)