When numbers are stored, a corresponding binary number can represent every number or fractional number. For example, the fraction 1/10 can be represented in a decimal number system as 0.1. However, the same number in binary format becomes the following repeating binary decimal:

0001100110011100110011 (and so on)

This can be infinitely repeated. This number cannot be represented in a finite (limited) amount of space. Therefore, this number is rounded down by approximately -2.8E-17 when it is stored.However, there are some limitations of the IEEE 754 specification that fall into three general categories:

- aximum/minimum limitations
- recision
- epeating binary numbers

- Underflow: Underflow occurs when a number is generated that is too small to be represented. In IEEE and Excel, the result is 0 (with the exception that IEEE has a concept of -0, and Excel does not).
- Overflow: Overflow occurs when a number is too large to be represented. Excel uses its own special representation for this case (#NUM!).

- Denormalized numbers: A denormalized number is indicated by an exponent of 0. In that case, the entire number is stored in the mantissa and the mantissa has no implicit leading 1. As a result, you lose precision, and the smaller the number, the more precision is lost. Numbers at the small end of this range have only one digit of precision.Example: A normalized number has an implicit leading 1. For instance, if the mantissa represents 0011001, the normalized number becomes 10011001 because of the implied leading 1. A denormalized number does not have an implicit leading one, so in our example of 0011001, the denormalized number remains the same. In this case, the normalized number has eight significant digits (10011001) while the denormalized number has five significant digits (11001) with leading zeroes being insignificant.

Denormalized numbers are basically a workaround to allow numbers smaller than the normal lower limit to be stored. Microsoft does not implement this optional portion of the specification because denormalized numbers by their very nature have a variable number of significant digits. This can allow significant error to enter into calculations. - Positive/Negative Infinities: Infinities occur when you divide by 0. Excel does not support infinities, rather, it gives a #DIV/0! error in these cases.
- Not-a-Number (NaN): NaN is used to represent invalid operations (such as infinity/infinity, infinity-infinity, or the square root of -1). NaNs allow a program to continue past an invalid operation. Excel instead immediately generates an error such as #NUM! or #DIV/0!.

1 Sign Bit | 11 Bit Exponent | 1 Implied Bit | 52 Bit Mantissa |

The mantissa and the exponent are both stored as separate components. As a result, the amount of precision possible may vary depending on the size of the number (the mantissa) being manipulated. In the case of Excel, although Excel can store numbers from 1.79769313486232E308 to 2.2250738585072E-308, it can only do so within 15 digits of precision. This limitation is a direct result of strictly following the IEEE 754 specification and is not a limitation of Excel. This level of precision is found in other spreadsheet programs as well.

Floating-point numbers are represented in the following form, where exponent is the binary exponent:

X = Fraction * 2^(exponent - bias)

Fraction is the normalized fractional part of the number, normalized because the exponent is adjusted so that the leading bit is always a 1. This way, it does not have to be stored, and you get one more bit of precision. This is why there is an implied bit. This is similar to scientific notation, where you manipulate the exponent to have one digit to the left of the decimal point; except in binary, you can always manipulate the exponent so that the first bit is a 1, because there are only 1s and 0s.Bias is the bias value used to avoid having to store negative exponents. The bias for single-precision numbers is 127 and 1,023 (decimal) for double-precision numbers. Excel stores numbers using double-precision.

A1: 1.2E+200 B1: 1E+100 C1: =A1+B1The resulting value in cell C1 would be 1.2E+200, the same value as cell A1. In fact if you compare cells A1 and C1 using the IF function, for example IF(A1=C1), the result will be TRUE. This is caused by the IEEE specification of storing only 15 significant digits of precision. To be able to store the calculation above, Excel would require at least 100 digits of precision.

A1: 0.000123456789012345 B1: 1 C1: =A1+B1The resulting value in cell C1 would be 1.00012345678901 instead of 1.000123456789012345. This is caused by the IEEE specification of storing only 15 significant digits of precision. To be able to store the calculation above, Excel would require at least 19 digits of precision.

A1: 1.2E+200 B1: 1E+100 C1: =ROUND(A1+B1,5)

This results in 1.2E+200.

D1: =IF(C1=1.2E+200, TRUE, FALSE)

This results in the value TRUE.

- On the
**Tools**menu, click**Options**. - On the
**Calculation**tab, select the**Precision as displayed**check box.

- Click the
**Microsoft Office**button, click**Excel Options**, and then click the**Advanced**category. - In the
**When calculating this workbook**section, select the workbook that you want, and then select the**Set precision as displayed**check box.

- On the
**File**menu, click**Options**, and then click the**Advanced**category. - In the
**When calculating this workbook**section, select the workbook that you want, and then select the**Set precision as displayed**check box.

For example, if you choose a number format that shows two decimal places, and then you turn on the

000110011001100110011 (and so on)

The IEEE 754 specification makes no special allowance for any number. It stores what it can in the mantissa and truncates the rest. This results in an error of about -2.8E-17, or 0.000000000000000028 when it is stored.Even common decimal fractions, such as decimal 0.0001, cannot be represented exactly in binary. (0.0001 is a repeating binary fraction that has a period of 104 bits). This is similar to why the fraction 1/3 cannot be exactly represented in decimal (a repeating 0.33333333333333333333).

For example, consider the following simple example in Microsoft Visual Basic for Applications:

` Sub Main() MySum = 0 For I% = 1 To 10000 MySum = MySum + 0.0001 Next I% Debug.Print MySum End Sub`

- Enter the following into a new workbook:
A1: =(43.1-43.2)+1

- Right-click cell A1, and then click
**Format Cells**. On the Number tab, click Scientific under Category. Set the**Decimal places**to 15.

- In Excel 95 or earlier, enter the following into a new workbook:
A1: =1.333+1.225-1.333-1.225

- Right-click cell A1, and then click
**Format Cells**. On the Number tab, click Scientific under Category. Set the**Decimal places**to 15.

Excel 97, however, introduced an optimization that attempts to correct for this problem. Should an addition or subtraction operation result in a value at or very close to zero, Excel 97 and later will compensate for any error introduced as a result of converting an operand to and from binary. The example above when performed in Excel 97 and later correctly displays 0 or 0.000000000000000E+00 in scientific notation. For more information, click the following article numbers to view the articles in the Microsoft Knowledge Base:

172911 Incorrect result raising 10 to very large/very small power

214373 Incorrect result raising 10 to very large/very small power

For more information about floating-point numbers and the IEEE 754 specification, please see the following World Wide Web sites: 214118 How to correct rounding errors in floating-point arithmetic

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Note This article also applies to Microsoft Excel for Mac for Office 365.

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Article ID: 78113 - Last Review: 07/29/2015 15:40:00 - Revision: 14.1

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