**High School Number Sense Lesson 47: Adding in a Base other than 10**

Working in and between bases other than 10 is a bit of a learning curve, but these types of problems are some of my favorite ones on number sense tests. A couple of months ago we covered moving back and forth between base 10 and another base (in

**BaseNtoBase10**and

**Base10toBaseN**). Today we will add numbers that are in a base other than 10. This concept appeared

**8 times**last year, with a median placement at

**question # 47**.

**Number Dojo Level: 214**

One thing that needs to be understood clearly is that no matter what base we are in, we cannot have a single digit that is equal to or greater than the base. In other words, in base

**10**(which we're all used to), the highest single digit is

**9**. The same is true regardless of the base. So the highest digit in base 3 is 2; the highest digit in base 7 is 6, etc. Keep this in mind as we go forward. Everything else we discuss will work the same way as adding whole numbers in base 10.

**How to Solve**

- Work from right to left. Add the units digit from each addend. If their sum is
**less than the base**we're in, write that sum down. - If the sum is
**equal to or greater than the base**, divide the sum by the base and**write the remainder**down. Regroup (carry) the quotient into step 3. - Move to the next digits to the left. Add these digits from each addend. If their sum is
**less than the base**we're in, write that sum down. Otherwise, go back to step 2. (Repeat this process until you run out of digits.)

**Example 1: 23 (base 5) + 14 (base 5) = ___ (base 5)**

- Add the units digits: 3 + 4 =
**7**. 7 ≥ 5... - So divide 7 by 5 to get
**1 remainder 2**. Write the remainder (**2**) and regroup the quotient (**1**). - Move to the left and add these digits: 2 + 1 (plus the regrouped
**1**) =**4**. Write this down in front. The answer is**42**.*(Don't think of this as forty-two base five. Think of it as***four-two base five**).

**Example 2: 112 base 3 + 22 base 3 = ___ base 3**

- Add the units digits: 2 + 2 =
**4**. 4 ≥ 3... - So divide 4 by 3 to get
**1 remainder 1**. Write the remainder (**1**) and regroup the quotient (**1**). - Move to the left and add these digits: 1 + 2 (plus the regrouped
**1**) =**4**. 4 ≥ 3... - So divide 4 by 3 to get
**1 remainder 1**. Write the remainder (**1**) and regroup the quotient (**1**). - Move to the left and add these digits: 1 (plus the regrouped
**1**) =**2.**Write this down in front. The answer is**211**.

**Example 3:**

**234**

_{7}+ 56_{7}= ____{7}1. Add the units digits: 4 + 6 =

**10**. 10 ≥ 7...

2. So divide 10 by 7 to get

**1 remainder 3**. Write the remainder (

**3**) and regroup the quotient (

**1**).

3. Move to the left and add these digits. 3 + 5 (plus the regrouped

**1**) =

**9**. 9 ≥ 7...

4. So divide 9 by 7 to get

**1 remainder 2**. Write the remainder (

**2**) and regroup the quotient (

**1**).

5. Move to the left and add these digits. 2 (plus the regrouped

**1**) =

**3**. Write this down in front. The answer is

**323**.

**Example 4:**

**133**

_{4}+ 23_{4}= ____{4}1. Add the units digits: 3 + 3 =

**6**. 6 ≥ 4...

2. So divide 6 by 4 to get

**1 remainder 2**. Write the remainder (

**2**) and regroup the quotient (

**1**).

3. Move to the left and add these digits. 3 + 2 (plus the regrouped

**1**) =

**6**. 6 ≥ 4...

4. So divide 6 by 4 to get

**1 remainder 2**. Write the remainder (

**2**) and regroup the quotient (

**1**).

5. Move to the left and add these digits. 1 (plus the regrouped

**1**) =

**2**. Write this down in front. The answer is

**222**.

**Example 5:**

**524**

_{6}+ 423_{6}+ 125_{6}= ____{6}1. Add the units digits: 4 + 3 + 5 =

**12**. 12 ≥ 6...

2. So divide 12 by 6 to get

**2 remainder 0**. Write the remainder (

**0**) and regroup the quotient (

**2**).

3. Move to the left and add these digits. 2 + 2 + 2 (plus the regrouped

**2**) =

**8**. 8 ≥ 6...

4. So divide 8 by 6 to get

**1 remainder 2**. Write the remainder (

**2**) and regroup the quotient (

**1**).

5. Move to the left and add these digits. 5 + 4 + 1 (plus the regrouped

**1**) =

**11**. 11 ≥ 6...

6. So divide 11 by 6 to get

**1 remainder 5**. Write this down in front. The answer is

**1520**.

**Example 6: 16 + 8 + 2 + 1 = ___ base 2**

- Notice that each of the addends is a power of 2. Except for 4, we have one of each power of two, up to 16.
- The answer is
**11011**. *For this problem, it is important to understand expanded notation. In base 10, 11011 is 1 ten thousand + 1 thousand (+ 0 hundreds) + 1 ten + 1 one. In base 2, 11011 is 1 sixteen + 1 eight (+ 0 fours) + 1 two + 1 one.*

**Here's a free worksheet to help you practice BaseAdd:**

baseadd.pdf |

**Up Next for High School: Midpoint2Pts**