Adding and subtracting real numbers

You have added whole num­bers, frac­tions and dec­i­mals for years. Real num­bers include all of those, and also roots such as $\sqrt{2}$ and $\sqrt{3}$. Adding and tak­ing away real num­bers fol­lows the same rules you already know. The only new skill is decid­ing which parts can be put together and which parts must stay apart. It works just like col­lect­ing like terms in alge­bra.

Like roots and unlike roots#

$3\sqrt{2}$ means $3 \times \sqrt{2}$: three lots of $\sqrt{2}$. A root on its own, such as $\sqrt{2}$, is one lot of it.

Roots with the same num­ber under the root sign are called like roots. $\sqrt{2}$ and $3\sqrt{2}$ are like roots. $\sqrt{2}$ and $\sqrt{3}$ are unlike roots.

Like roots can be added by adding the num­bers in front, just as in alge­bra.

$$a + 3a = 4a \qquad \text{so} \qquad \sqrt{2} + 3\sqrt{2} = 4\sqrt{2}$$

Think of each root as a tile. Tiles of the same kind can be counted together. Tiles of dif­fer­ent kinds can­not.

Adding real num­bers#

Exam­ple 1#

Add $\sqrt{2}$ and $3\sqrt{2}$.

$$\sqrt{2} + 3\sqrt{2} = 4\sqrt{2}$$

Exam­ple 2#

Add $2 + 3\sqrt{3}$ and $-5 + 7\sqrt{3}$.

Write the two num­bers side by side. Then join the plain num­bers with each other and the $\sqrt{3}$ parts with each other.

$$\begin{aligned} (2 + 3\sqrt{3}) + (-5 + 7\sqrt{3}) &= 2 + 3\sqrt{3} - 5 + 7\sqrt{3} \\ &= (2 - 5) + (3 + 7)\sqrt{3} \\ &= -3 + 10\sqrt{3} \end{aligned}$$

Com­pare it with alge­bra: $2a + 5b - 7a + 4b = -5a + 9b$. The $a$ terms join together and the $b$ terms join together.

Exam­ple 3#

Add $2\sqrt{2} + 5\sqrt{3}$ and $\sqrt{2} - 7\sqrt{3}$.

$$\begin{aligned} 2\sqrt{2} + 5\sqrt{3} + \sqrt{2} - 7\sqrt{3} &= (2 + 1)\sqrt{2} + (5 - 7)\sqrt{3} \\ &= 3\sqrt{2} - 2\sqrt{3} \end{aligned}$$

A com­mon slip is to write $-12\sqrt{3}$. But $5 - 7 = -2$, not $-12$.

Exam­ple 4#

Add $-5\sqrt{7}$ and $-12\sqrt{7}$.

$$-5\sqrt{7} - 12\sqrt{7} = -17\sqrt{7}$$

Just as $-5a - 12a = -17a$. Two neg­a­tives added make a big­ger neg­a­tive.

Parts that must stay apart#

Now look at $4 + \sqrt{5}$. The $4$ is a plain num­ber and $\sqrt{5}$ is a root. They are not the same kind, so they can­not be joined into one num­ber. The answer is sim­ply $4 + \sqrt{5}$.

$4 + \sqrt{5}$ is one exact point on the num­ber line, even though it is writ­ten in two parts. In the same way, $\sqrt{2} + \sqrt{3}$ stays as it is, because the roots are dif­fer­ent. It is not $\sqrt{5}$.

Remem­ber. Join plain num­bers with plain num­bers, and like roots with like roots. Unlike parts stay apart.

Sub­tract­ing real num­bers#

First, a reminder about the words. "Sub­tract $a$ from $b$" means start with $b$ and take $a$ away: $b - a$.

So "sub­tract $-2$ from $7$" means $7 - (-2)$. Tak­ing away a neg­a­tive is the same as adding.

$$7 - (-2) = 7 + 2 = 9$$

Exam­ple 5#

Sub­tract $-2\sqrt{3}$ from $7\sqrt{3}$.

$$\begin{aligned} 7\sqrt{3} - (-2\sqrt{3}) &= 7\sqrt{3} + 2\sqrt{3} \\ &= 9\sqrt{3} \end{aligned}$$

Exam­ple 6#

Sub­tract $(2 + \sqrt{3})$ from $(2 - \sqrt{3})$.

The minus sign in front of the bracket changes the sign of every part inside it.

$$\begin{aligned} (2 - \sqrt{3}) - (2 + \sqrt{3}) &= 2 - \sqrt{3} - 2 - \sqrt{3} \\ &= (2 - 2) - (1 + 1)\sqrt{3} \\ &= -2\sqrt{3} \end{aligned}$$

A com­mon slip is to write $2\sqrt{3}$. Both roots have a minus sign in front, so the answer is neg­a­tive.

Longer sums#

When there are many like roots, gather the ones being added and the ones being taken away, as you did with dec­i­mals.

Exam­ple 7#

Find $5\sqrt{7} - 11\sqrt{7}$ and $8\sqrt{3} + 7\sqrt{3} - 21\sqrt{3}$.

$$5\sqrt{7} - 11\sqrt{7} = -6\sqrt{7}$$

$$\begin{aligned} 8\sqrt{3} + 7\sqrt{3} - 21\sqrt{3} &= 15\sqrt{3} - 21\sqrt{3} \\ &= -6\sqrt{3} \end{aligned}$$

Exam­ple 8#

Find $1052\sqrt{10} - 20108\sqrt{10} + 7128\sqrt{10} - 238\sqrt{10}$.

Num­bers added: $1052 + 7128 = 8180$. Num­bers taken away: $20108 + 238 = 20346$.

$$1052\sqrt{10} - 20108\sqrt{10} + 7128\sqrt{10} - 238\sqrt{10} = 8180\sqrt{10} - 20346\sqrt{10}$$

$20346$ is big­ger than $8180$, so the answer is neg­a­tive. Find the dif­fer­ence the other way round:

$$\begin{array}{r} 20346 \\ (-)\ 08180 \\ \hline 12166 \end{array}$$

So the answer is $-12166\sqrt{10}$.

Exam­ple 9#

Find $70128\sqrt{7} - 1038\sqrt{7} - 21238\sqrt{7} - 213\sqrt{7}$.

All three of the last terms are being taken away. Add them first: $1038 + 21238 + 213 = 22489$.

$$\begin{aligned} 70128\sqrt{7} - 22489\sqrt{7} &= (70128 - 22489)\sqrt{7} \\ &= 47639\sqrt{7} \end{aligned}$$

Here the first num­ber is big­ger, so the answer is pos­i­tive.

Com­mon mis­takes#

• Adding the num­bers under the roots. $\sqrt{2} + \sqrt{3}$ is not $\sqrt{5}$.
• Join­ing a plain num­ber with a root. $4 + \sqrt{5}$ is not $5\sqrt{5}$.
• Chang­ing the sign of only the first part when a minus stands before a bracket.
• Mix­ing up "sub­tract $a$ from $b$". It means $b - a$, not $a - b$.

Prac­tice#

1. Add $5\sqrt{3}$ and $-7\sqrt{3}$.
2. Add $2 + \sqrt{5}$ and $2 - \sqrt{5}$.
3. Add $2 + \sqrt{7}$ and $2 + 3\sqrt{7}$.
4. Add $\sqrt{5} + \sqrt{6}$ and $-3\sqrt{5} - 7\sqrt{6}$.
5. Find $8\sqrt{2} - 9\sqrt{2}$.
6. Sub­tract $7\sqrt{3}$ from $-5\sqrt{3}$.
7. Sub­tract $-8\sqrt{5}$ from $-7\sqrt{5}$.
8. Sub­tract $3\sqrt{5}$ from $-7\sqrt{5}$.
9. Find $31012\sqrt{3} - 510023\sqrt{3} - 1203\sqrt{3}$.
10. Find $-1201\sqrt{5} - 12012\sqrt{5} + 2312\sqrt{5} + 127013\sqrt{5}$.

Answers#