Ask Dr. Math

Middle School Archive

Browse Middle School Number Sense/About Numbers
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    About zero.
    Multiplying by zero.
    Casting out nines.
    Classifying numbers.
    Googol, googolplex.
    Number facts from the Web.
    Significant figures/digits.

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Associative Property of Multiplication and Grouping Symbols [10/22/2006]

Does the Associative Property of Multiplication make grouping symbols unnecessary, since (a x b) x c = a x (b x c) = a x b x c?

Averaging Ages [11/17/1999]

What is the average age of these 4 children...

Babylonian Numbers [11/17/1999]

Can you give me information on the Babylonian number system?

Base 3 [10/24/2001]

What is 2 + 2 in base three?

Base 5 [07/28/1997]

What is .23 in base 5? I know that 6 in base 5 is 11, 7 is 12 and so on. Would .23 be the same in base 5 as it is in base 10?

Base of Roman Numerals [04/07/2001]

What is the base of the Roman numeral system, base 10 or base 5?

Bell Numbers [10/15/1998]

How are Bell numbers generated? What are the first 12 Bell numbers?

Binary Notation [8/28/1996]

What is the smallest five digit number in decimal notation?

Binary Notation Card Trick [12/10/2002]

My 7th grade teacher had a card game that had about 7 cards. She said she could guess your number by asking you if your number was on each card. One card had even numbers and the other had odd.

CADAEIBFEC and Other NCTM Questions [10/27/1998]

CADAEIBFEC is a mnemonic for an important piece of mathematical information. What is it?

Calculators and e+ [09/27/1998]

What does the "e+" mean on a calculator?

Can Even and Odd Numbers Be Negative? [04/11/2006]

Can even and odd numbers be non-positive? For example, is 0 even? Is -4 even? Is -5 odd? I've heard both sides of this question. Who's right?

Can the Sum of 5 Odd Numbers Equal an Even Number? [01/05/2003]

What are 5 odd numbers which, when added, equal 32?

Cardinal Numbers [03/20/2001]

Is there a largest cardinal number? If there is, how do you express it?

Casting out Nines: How Is 9 the Same As 0? Why Bother? [09/25/2009]

Doctor Peterson makes a case for this old technique in the calculator age -- and makes a connection between digital roots and remainders in base ten division.

Casting Out Nines to Check Arithmetic [08/26/1999]

Can you explain how to use the 'casting out 9's' technique in a way that a 6th grader can understand?

Checkerboard Pattern Using an L Shape [10/14/2002]

Place the numbers 1-64 in L-shaped patterns on an 8x8 checkboard. Can you always get an L shape with the sum of the three digits as odd using up all the available squares on the board?

Checking When Rounding [12/10/2002]

Why is it important to check by estimating?

Classifying the Number One [11/27/1998]

My students have learned about prime and composite numbers. What classification fits the number 1?

Commas in Numbers [09/13/2001]

Do commas change the value of a number?

Comparing Individual Discounts to a Collective Discount [07/19/2005]

I am purchasing two books that each have a 10% discount. If the discount is taken on each book first or taken on the total price of the two books, will the result be same or different?

Comparing Numbers in Different Bases [07/28/1999]

If I am given a number in base 4 and asked if it is bigger or smaller than another number in another base, should I always convert both to base 10?

Comparing Sums of Odd and Even Integers [03/31/2004]

Which is greater, the sum of all of the even numbers that are less than 100, or the sum of all of the odd numbers that are less than 100? It takes too long to add them all up!

Compatible Number Estimating [01/31/2002]

My homework is on compatible numbers to estimate quotients - for example, 1,625 divided by 38.

Compatible Number Method for Estimating [09/10/2001]

We are supposed to estimate using compatible numbers. How does one estimate something like 574/8?

Compensatory Number [01/29/2002]

Is there a math definition for the term "compensatory number"?

Consecutive Integers [02/10/1999]

Can the product of four consecutive integers be a perfect square?

Conversion of Time Range into Minutes [11/13/2002]

What equation/conversion can I use to find the number of minutes between the start and end times for both 'available' and 'actual' time on a 24-hour clock?

Converting a Number from One Base to Another [02/28/2008]

Is there any method to convert a number in one base to another base without having to go through base 10 on the way?

Converting a Repeating Decimal to a Fraction [9/10/1996]

Why does this way of converting a recurring decimal to a fraction work?

Converting Base Numbers Directly Without Using Base 10 [01/26/2004]

How do you convert from one base to another without going through base 10? How do I go from base 3 to base 7, for instance? Would the same algorithm work to go from base 5 to base 12?

Converting from Hindu-Arabic Numerals to Roman Numerals [05/18/1999]

A conversion table, and recommendations for using bars over Roman numerals, with examples.

Converting Julian Date to Gregorian [03/20/2001]

Assume that the base date is 01/01/0001 and that a Gregorian date is expressed as DD/MM/YYYY...

Counting a Span Versus Finding a Difference between Two Numbers [06/03/2004]

Why is it that the difference between year 2012 and year 2003 equals 10 years but the difference between the numbers 2012 and 2003 equals 9?

Counting Days of the Week [02/02/1999]

If today is Thursday, what day would it be 120 days from now?

Counting in Base 5 [03/02/2004]

How do I count from 1 to 7 in base 5?

Counting in the Teens [09/10/1998]

Why do we count ten eleven twelve thirteen, instead of ten eleventeen twelveteen thirteen?

Counting School Supplies for the First Days of School [07/23/1997]

The rule is n(n+1)/2 for each day and n(n+1)(n+2)/6 for the entire sequence. Where did the divide by 2 and 6 come from?

Counting Techniques [12/29/1997]

Find the number of positive divisors of 17640, inclusive of 1 and 17640 itself. Find the number of odd integers between 2000 and 7000 that do not have repeated digits.

Counting the Number of Factors [01/04/1999]

Is there a systematic way to find the number less than 100 that has the greatest number of factors?

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