CERTIFICATION PURSUANT TO
18 U.S.C. SECTION 1350,
AS ADOPTED PURSUANT TO
SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002 EXHIBIT 32(b)

I, P. O. Loux, Senior Vice President, Finance and Chief Financial Officer of W.W. Grainger, Inc. (“Grainger”), certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

1. The Annual Report on Form 10-K of Grainger for the annual period ended December 31, 2004, (the “Report”) fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

2. The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of Grainger.

/s/ P. O. Loux                               
P. O. Loux
Senior Vice President, Finance
and Chief Financial Officer

February 25, 2005