# Birkhoff's variety theorem with and without free algebras by Adamek J., Trnkova V.

From "Theory and purposes of Categories", Vol. 14, No. 18, 2005, pp. 424-450.

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Additional info for Birkhoff's variety theorem with and without free algebras

Example text

602,550,000 e. 602,550,000 7 8 Chapter 1 Whole Numbers ᭤ Your Turn 8 Example 8 Round each number so that there is only one nonzero digit. a. 27,502,341 b. 6,128,200 c. 453,219 a. 36,568 Round each number so that there is only one nonzero digit. Solution: 36,568 Farthest left The digit to the right of 3 is 6, which is greater than 5; so we round up. Answer: 40,000 b. 621,905 Solution: 621,905 Farthest left The digit to the right of 6 is 2, which is less than 5; so we round down. Answer: 600,000 ᭡ Note Remember, when rounding down, the digit in the rounded place remains the same.

40. Jose is raising funds to run for a seat in his state’s senate. Currently, he has raised \$280,540, and he intends to add \$28,580 of his own money. His competitor has raised a total of \$485,300. How much more does Jose need to raise to match his competitor’s funds? 41. The photosphere, which is the outer layer of the sun, has an average temperature of about 5500 K. A typical atomic blast at ground zero has a temperature of about 3400 K. How much warmer is the sun’s temperature? 42. The sun is 92,958,349 miles from the earth.

Deﬁnition Constant: A symbol that does not vary in value. All numerals are constants. For example, the symbol 3 always means three; so it is a constant. ᭤ Do Your Turn 5 17 ᭤ Your Turn 5 Mr. Marcus needs to fence a space in his yard for a garden. It is a rectangular space 45 feet wide by 60 feet long. How much fence will he need? Objective 4 Subtract whole numbers. Subtraction is the inverse of addition. Addition and subtraction are inverse operations because they undo each other. If 5 + 4 = 9, we can undo the addition with subtraction, as in 9 - 4 = 5.